Mathematics of Complexity and Dynamical Systems

Mathematics of Complexity and Dynamical Systems

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

Robert A. Meyers (Ed.)

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

123

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

Library of Congress Control Number: 2011939016

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

Preface

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

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

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

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Preface









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

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

Sections

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

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Sections

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

Sections

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

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About the Editor-in-Chief

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

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

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About the Editor-in-Chief

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

Editorial Board Members

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

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

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

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

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

Section Editors

Ergodic Theory

Fractals and Multifractals

BRYNA KRA Professor Department of Mathematics Northwestern University

SHLOMO HAVLIN Professor Department of Physics Bar Ilan University and

DANIEL BEN-AVRAHAM Professor Department of Physics Clarkson University

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Section Editors

Non-linear Ordinary Differential Equations and Dynamical Systems

FERDINAND VERHULST Professor Mathematisch Institut University of Utrecht

Solitons

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

Non-linear Partial Differential Equations

Systems and Control Theory

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

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

Perturbation Theory

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

Table of Contents

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

1

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

13

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

26

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

32

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

48

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

63

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

88

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

102

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

126

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

143

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

160

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

175

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

183

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

205

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

225

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

241

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

264

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

288

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Table of Contents

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

302

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

313

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

327

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

329

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

357

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

369

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

383

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

395

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

409

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

429

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

446

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

463

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

488

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

512

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

532

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

537

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

557

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

559

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

571

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

591

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

606

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

637

Table of Contents

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

657

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

683

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

704

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

729

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

740

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

755

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

771

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

783

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

796

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

810

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

837

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

884

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

890

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

908

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

924

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

937

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

953

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

964

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

981

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

1009

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

1043

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

1070

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

1082

XIX

XX

Table of Contents

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

1102

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

1111

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

1113

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

1115

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

1126

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

1137

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

1152

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

1172

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

1195

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

1205

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

1212

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

1236

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

1251

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

1265

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

1276

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

1290

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

1301

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

1314

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

1317

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

1325

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

1337

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

1351

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

1376

Table of Contents

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

1389

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

1400

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

1407

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

1422

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

1438

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

1457

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

1475

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

1489

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

1505

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

1520

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

1533

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

1548

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

1553

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

1561

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

1576

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

1601

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

1603

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

1618

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

1639

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

1653

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

1672

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

1689

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

1711

XXI

XXII

Table of Contents

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

1724

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

1726

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

1748

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

1771

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

1811

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

1819

Contributors

ABOU-DINA , MOUSTAFA S. Cairo University Giza Egypt

AUDIT, BENJAMIN ENS-Lyon CNRS Lyon Cedex France

AKIN, ETHAN The City College New York City USA

BAMBUSI , DARIO Università degli Studi di Milano Milano Italia

AKTOSUN, TUNCAY University of Texas at Arlington Arlington USA

BEN -AVRAHAM, D ANIEL Clarkson University Potsdam USA

ALABAU-BOUSSOUIRA , FATIHA Université de Metz Metz France

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

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

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

XXIV

Contributors

BULDYREV, SERGEY V. Yeshiva University New York USA

CORTÉS, JORGE University of California San Diego USA

BULLO, FRANCESCO University of California Santa Barbara USA

CRILLY, TONY Middlesex University London UK

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

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

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

CHIERCHIA , LUIGI Università “Roma Tre” Roma Italy

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

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

DERCOLE, FABIO Politecnico di Milano Milano Italy

Contributors

DERKS, GIANNE University of Surrey Guildford UK

GLASS, OLIVIER Université Pierre et Marie Curie Paris France

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

GOEBEL, RAFAL Loyola University Chicago USA

DOLCETTA , ITALO CAPUZZO Sapienza Universita’ di Roma Rome Italy

GRAMCHEV, TODOR Università di Cagliari Cagliari Italy

DOMOTOR, Z OLTAN University of Pennsylvania Philadelphia USA

GRASMAN, JOHAN Wageningen University and Research Centre Wageningen The Netherlands

FRANTZIKINAKIS, N IKOS University of Memphis Memphis USA

HANSSMANN, HEINZ Universiteit Utrecht Utrecht The Netherlands

GAETA , GIUSEPPE Università di Milano Milan Italy

HAVLIN, SHLOMO Bar–Ilan University Ramat–Gan Israel

GALDI , GIOVANNI P. University of Pittsburgh Pittsburgh USA

HE, JI -HUAN Donghua University Shanghai China

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

HELAL, MOHAMED A. Cairo University Giza Egypt

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

HEREMAN, W ILLY Colorado School of Mines Golden USA

GAUTHIER, JEAN-PAUL ANDRÉ University of Toulon Toulon France

HOFFMAN, CHRISTOPHER University of Washington Seattle USA

GENTILE, GUIDO Università di Roma Tre Roma Italy

HUVET, MAXIME CNRS Gif-sur-Yvette France

XXV

XXVI

Contributors

INC, MUSTAFA Fırat University Elazı˘g Turkey

KRA , BRYNA Northwestern University Evanston USA

ISIDORI , ALBERTO University of Rome La Sapienza Italy

LAKSHMANAN, M. Bharathidasan University Tiruchirapalli India

JUNCO, ANDRÉS DEL University of Toronto Toronto Canada

LEE, TAEYOUNG University of Michigan Ann Arbor USA

JURDJEVIC, VELIMIR University of Toronto Toronto Canada

LEFEBVRE, ALINE Université Paris-Sud Orsay cedex France

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

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

KAWSKI , MATTHIAS Arizona State University Tempe USA

LEOK, MELVIN Purdue University West Lafayette USA

˘ KAYA , DO GAN Firat University Elazig Turkey

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

KELLER, GERHARD Universität Erlangen-Nürnberg Erlangen Germany

MARCONI , LORENZO University of Bologna Bologna Italy

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

MARCUS, BRIAN University of British Columbia Vancouver Canada

KLEINBOCK, DMITRY Brandeis University Waltham USA

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

KOIKE, SHIGEAKI Saitama University Saitama Japan

MARSDEN, JERROLD E. California Institute of Technology Pasadena USA

Contributors

MARTÍNEZ, SONIA University of California San Diego USA

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

MAWHIN, JEAN Université Catholique de Louvain Maryland USA

OLDEMAN, BART Concordia University Montreal Canada

MCCLAMROCH, HARRIS University of Michigan Ann Arbor USA

ORBACH, RAYMOND L. University of California Riverside USA

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

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

XXVII

XXVIII

Contributors

QUAS, ANTHONY University of Victoria Victoria Canada

SANFELICE, RICARDO G. University of Arizona Tucson USA

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

SANNS, W ERNER University of Applied Sciences Darmstadt Germany

REDNER, SIDNEY Boston University Boston USA

SBANO, LUCA University of Warwick Warwick UK

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

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

Contributors

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

VANDERBAUWHEDE, ANDRÉ Ghent University Gent Belgium

TAKAYASU, HIDEKI Sony Computer Science Laboratories Inc Tokyo Japan

VERHULST, FERDINAND University of Utrecht Utrecht The Netherlands

TAKAYASU, MISAKO Tokyo Institute of Technology Tokyo Japan

VIANA , MARCELO IMPA Rio de Janeiro Brazil

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

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

XXIX

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

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

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

1

2

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

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

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

1 X

u n (x; t) ;

(1)

nD0

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

(2)

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

(3)

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

1 X

u n (x; t)

(4)

nD0

into both sides of (3) yields 1 X

un D g  L

1

nD0

R

X 1

! un

:

(5)

nD0

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

k  0;

(6)

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

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

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

1

(7)

(R(u2 )) ;

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

F(u) D

1 X

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

(8)

nD0

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

0

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

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

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

(11)

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

3

4

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

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

k 0:

(12)

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

(13)

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

1

(Ru k ) ;

(14) k 1:

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

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

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

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

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

Solitons of the KdV Equation

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

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

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

(15)

@ ; @t

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

1 X

un ;

(22)

nD0

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

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

(16)

(17)

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

L1 t

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

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

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

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

where the differential operator Lt is Lt D

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

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

A0 D F(u0 ) D u0 u0x ;

(19)

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

u0 (x; t) D

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

D

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

5

6

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

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

D

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

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

2c 2 ec(xc 1C

2 t)

2 ec(xc 2 t)

:

(28)

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

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

t > 0;

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

 ¤0;

(30)

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

(31)

where the differential operator Lt is Lt D

Kinks of the Burgers Equation

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

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

@ ; @t

(32)

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

(33)

0

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

(34)

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

2c cx

(1 C e  )

C L1 t (u x x  uu x ) :

(35)

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

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

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

2c

u n (x; t) D

cx

(1 C e  )

nD0

C

L1 t



nD0 1 X

 L1 t

!

1 X

u n (x; t) xx

! :

An

(36)

nD0

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

u0 (x; t) D

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

(37) k 0:

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

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

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

cx

2c 3 e 

D

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

 1)

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

cx

D

c 7 e  (e

2cx 

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

c 9 e (e

3cx 

 11e

12 4 (1

2cx 

Ce

t3 ;

Peakons of the Camassa–Holm Equation

C 11e cx 

cx 

 1)

)5

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

t4 ;

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

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

u(x; t) D

2ce 

cx

(1 C e  ) cx

c2

cx

 e  C

cx

(1 C e  ) C

c 6 (e

2cx 

tC

c 4 (e   1) cx

2 2 (1 C e  )2

t2

cx

 4e  C 1) cx

6 3 (1 C e  )3

t3 C   

! ;

(39)

and in a closed form by 2c

u(x; t) D 

c

1 C e  (xc t)

;

(41)

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

cx

 4e  C 1) cx 

cx 

(38)

cx 

 2 (1 C

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

 ¤0;

(40)

(42)

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

7

8

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

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

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

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

or equivalently

k 0: (51)

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

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

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

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

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

(46)

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

(47)

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

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

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

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

and

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

C0 D u0 u0 x x x ; (48)

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

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

C

nD0

L1 t

3

1 X nD0

An C 2

1 X nD0

Bn C

xxt 1 X nD0

!

Cn

:

(50)

(54)

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

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

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

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

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

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

(57)

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

(58)

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

n>1:

(59)

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

@ : @t

(62)

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

9

10

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

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

(64)

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

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

(65)

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

(69)

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

u(x; t) D

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

nD0

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

and

1 X

and so on. The series solution is thus given by

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

j x  ct j

:0

otherwise :

3 2

;

(71)

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

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

This gives the recursive relation p

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

(67) k 0:

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

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

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

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

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

j x  ct j 

 2

;

otherwise ; (72)

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

j x  ct j

 

;

otherwise : (73)

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

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

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

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

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

Anomalous Diffusion on Fractal Networks

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

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

13

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Anomalous Diffusion on Fractal Networks

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

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

D

n X iD1

!2 + si

D

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

iD1 jDiC1

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

Anomalous Diffusion on Fractal Networks

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

(1)

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

(2)

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

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

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

15

16

Anomalous Diffusion on Fractal Networks

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

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

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

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

Anomalous Diffusion on Fractal Networks

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

(5)

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

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

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

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

17

18

Anomalous Diffusion on Fractal Networks

according to  pE (r; l) D



  d dr2 exp  : 2 a2 jlj 2a2 jlj

(6)

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

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

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

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

Anomalous Diffusion on Fractal Networks

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

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

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

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

(9)

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

19

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Anomalous Diffusion on Fractal Networks

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

(10)

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

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

(11)

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

2df : ds

(12)

Anomalous Diffusion on Fractal Networks

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

for t ! 1 :

(13)

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

(14)

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

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

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

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

21

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Anomalous Diffusion on Fractal Networks

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

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

R (t) /

t 2/d w ;

t 1/d w < N 1/ f f

N 2/d f

otherwise:

;

Anomalous Diffusion on Fractal Networks

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

N cross X(t)

N 2/d f N 1 C

ND1

1 X

t 2/d w N 1 :

N cross (t)

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

r d f d f t d s /2



r t 1/d w

 :

(15)

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

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

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

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

23

24

Anomalous Diffusion on Fractal Networks

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

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

(18)

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

r ! 1/d w r ;

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

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

Anomalous Diffusion on Fractal Networks

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

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

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

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

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

VL V L D  

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

Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

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

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

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

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

NC1 X iD1 t

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

D A ˚  (˙(1) ; : : : ; ˙(N) ; c˙) t

(3)

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

27

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

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

N X

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

(4)

iD1

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

N TX

0 ¤ c D 0

Vi ˙(i) dt

(5)

iD1

i. e., that the differential form

PN

iD1 Vi

d (i) is not exact.

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

t 2R;

(6)

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

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

t 2 R;

(7)

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

T

c˙(t)dt D 0 :

c D

(8)

0

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

(9)

Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

T

c D

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

˙ V ((t))(t)dt

Z

0

Z

T

@˝

d  ((t))dt dt

D 0

Z

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

G i j () D

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

D

1 T

RTR

R1R n  v n  v D 0 @˝ 6L c¯2 6L(c)2

0

@˝

(10)

@˝

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

NC1 X

V i (p; )u(i)

(12)

W i (p; )˙(i) :

(13)

iD1

D

N X iD1

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

@˝

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

(15)

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

Z

1Z 0

@˝

1

n  v D

˙ ) ˙ (G();

(16)

0

subject to the constraint Z

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

(14)

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

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

1

˙ ) ˙  n  v D (G();

c D

1

V()  ˙

(17)

0

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

@G ˙ ˙ ;  @ (1) :: :

1

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

29

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

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

(19)

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

(21)

where

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

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

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

(22) (23)

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

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

Biological Fluid Dynamics, Non-linear Partial Differential Equations

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

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

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

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Catastrophe Theory

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

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

2 "(n C

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

Catastrophe Theory

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

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

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

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

Catastrophe Theory, Figure 1 Ring on the wire

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

Catastrophe Theory

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

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

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

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

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

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

(1)

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

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

Catastrophe Theory

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

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

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

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

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

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

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

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

Catastrophe Theory

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

3

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

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

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

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

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

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

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

Catastrophe Theory, Figure 10 Perturbations of x 3

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

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

Catastrophe Theory

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

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

codim( f ) D

dimR (n) : h@x f i

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

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Catastrophe Theory

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

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

it follows

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

F(x; u) D x C ux 3

F(x; u) D x C ux

is versal;

is universal; 2

is not versal :

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

k X

u i b i (x)

iD1

is a universal unfolding of f .

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

codim f 1 2 3 3

x3  xy2 3 x6 4 x2 y C y4 4

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

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

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

Catastrophe Theory

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

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

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

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Catastrophe Theory

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

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

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

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

Catastrophe Theory

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

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

Further Applications

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

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

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

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Catastrophe Theory

Catastrophe Theory, Figure 19 The Zeeman wheel

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

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

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

Catastrophe Theory

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

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

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

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Center Manifolds

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

dx D F(x) dt

(1)

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

(2)

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

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

Rn

is a Ck -smooth vector field,

(3)

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

Center Manifolds

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

(4)

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

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

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

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

(5)

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

(6)

(8)

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

(9)

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

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

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

(11)

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

(12)

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

(13)

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

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

(14)

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

Center Manifolds

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

(15)

Center Manifolds, Figure 2 The classical five invariant manifolds

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

(16)

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

   

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

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

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

(17)

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

Dh(0) D 0 :

(18)

51

52

Center Manifolds

Center Manifolds, Figure 3 Representation of a center manifold

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

(19)

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

(21)

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

Center Manifolds, Figure 4 Nonuniqueness of the center manifold

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

(24)

y˙ D y ;

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

(25)

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

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

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

(22)

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

Dh1 (0) D 0

h2 (0) D 0 ;

Dh2 (0) D 0 :

(26)

(23)

  1 x

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

Center Manifolds

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

(28)

˙ D0: 

Center Manifolds, Figure 5 Weak uniqueness of the center manifold

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

(27)

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

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

(29)

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

(30)

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

53

54

Center Manifolds

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

(31)

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

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

Reduction Principle

x˙ D Ax C f (x; y) ;

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

y˙ D By C g(x; y) ;

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

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

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

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

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

(33)

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

(35)

(34)

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

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

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

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

(37)

y˙ D y C x 2 ;

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

(38)

Center Manifolds

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

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

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

: :

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

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

y˙ D y ;

(42)

(43)

(44)

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

(46)

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

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

v ! Lv C G(v) ;

(41)

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

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

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



(40)

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

as x < 0 and y D 0 as x  0

(45)

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

x ! Lx :

(47)

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

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

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

55

56

Center Manifolds

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

(48)

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

(49)

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

(50)

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

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

(51)

tangent bundle TRn

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

0 n

1 k

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

for n  0 ; for n  0 : (52)

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

Center Manifolds

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

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

(55)

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

(56)

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

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

Applications

Center Manifold in Infinite-Dimensional Space

Stability

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

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

y 2 E s (x) ;

z 2 E u (x) :

(53)

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

(54)

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

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

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

57

58

Center Manifolds

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

1

;

jT2 j < :

(57)

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

(58)

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

(59)

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

S

A2

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

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

(60)

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

Center Manifolds

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

(61)

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

(62)

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

u 2 DA :

(63)

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

Au D lim

(64)

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

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

U ts f (u(s))ds 0

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

(65)

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

59

60

Center Manifolds

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

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

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

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

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

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

Chaos and Ergodic Theory

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

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

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

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

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

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

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

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

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

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

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

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

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

Chaos and Ergodic Theory

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

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

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

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

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

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

n!1

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

(1)

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

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

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

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

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

Chaos and Ergodic Theory

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

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

Z  d :

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

x 2 M : 8 :

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

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

n!1

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

D1:

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

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

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

n1 1X n

X

ıf kx

kD0 x2E(n; )

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

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

Chaos and Ergodic Theory

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

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

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

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

Chaos and Ergodic Theory

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

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

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

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

x

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

(3)

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

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

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

and

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

n

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

Chaos and Ergodic Theory

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

lim sup n!1

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

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

(4)

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

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

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

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

n1 1X  ı T k (x) n kD0

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

Chaos and Ergodic Theory

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

ı T n d 

X

Z

ˇ ˇ dˇˇ  kkk kw a n

Z  d X

X

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

 lim 

n!1

x 2 X:



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

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

(6)

(Tx) 

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

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

Z Cn

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

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

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

lim

compare to (6).

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

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

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

Chaos and Ergodic Theory

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

n!1

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

1 log s(ı; n; X) n

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

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

 ˚ 1 log s ı; n; y 2 X : n

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



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

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

x2X

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



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



!

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

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



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

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

Chaos and Ergodic Theory

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

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

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

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

Chaos and Ergodic Theory

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

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

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

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

Chronological Calculus in Systems and Control Theory

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

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

89

90

Chronological Calculus in Systems and Control Theory

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

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

) D ( pˆ ')  ( pˆ ) ;

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

(2)

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

C '  ( fˆ ) :

(3)

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

Chronological Calculus in Systems and Control Theory

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

(4)

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

(5)

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

(6)

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

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

d dt

   Z t Z ! ! exp f d D exp 0

0

d dt

 Z  exp

a

 f t ' dt :



t

f d

0

(9)

 Z  D f t ı exp



t 0

f d

: (10)

t

q() ı f d

Z t (7)

ı ft :

f d

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

0





t

Analogously the left chronological exponential satisfies

q t D q0 C b

(8)

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

Z

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

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

D q0 C

Z

0





q0 C 0

(11)

q( ) ı f d

ı f d

(12)

91

92

Chronological Calculus in Systems and Control Theory

to eventually, formally, obtain the expansion !

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

Variation of Parameters and the Chronological Logarithm

t2

0

 fk ı    ı f2 ı f1 d k : : : d2 d1

(13)

and analogously for the left chronological exponential 

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

t2

0

 f1 ı f2 ı    ı fk d k : : : d2 d1 :

(14)

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

(15)

0

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

t2 t1

  Z ! f d ı exp



t3

f d

t2

 Z ! D exp



t3 t1

f d

:

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

t1 t0

f d

!

Z

t0



Z

t1 t1

D exp D exp t0

f d (17) ( f ) d :

q˙ D f t (q) C g t (q) :

(18)

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

Z

t

˚ t D exp 0

!

f d and  t D exp

Z

t 0

( f Cg ) d; (19)

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

(20)

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

which has the unique solution !

 t D exp

Z t Z ! exp 0



 0

ad f d

g d ;

(22)

and consequently one has the variations formula !

Z

t

exp 0

(16)

 Z ! exp

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

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

 Z  !  g d ı exp



 0

ad f d

t 0

f d

 :

(23)

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

Chronological Calculus in Systems and Control Theory

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

jD1

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

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

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

(29)

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

(30)

93

94

Chronological Calculus in Systems and Control Theory

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

1

1

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

(32)

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

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

( j)

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

(33)

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

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

˙ (0) D lim ( (t)  (0)) : t!0 t

(34)

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

(35)

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

Chronological Calculus in Systems and Control Theory

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

f e

t f t g t f

C pe e e

t f tg

 pe e ge

t f t g

e

t g

fe

t g t f t g

tf

g  pe f e e

t f t g t f

g C pe e e

t g

fe

g C pe

t g t f t g 2

e e

e

e

g

g

tf

(36)

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

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

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

XZ 0

I

Z  0

Z u i s (ts )

T

Z t3    u i 2 (t2 )

t s1 0

0

t2

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

(38)

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

Z u i s (ts )

XZ

T

0

I

0

Z t3    u i 2 (t2 )

t s1

0

t2

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

(39)

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

T

I : U 7! 0

U i0s (ts )

Z

t s1

Z

 0

0

Z

t3

U i02 (t2 )

t2

 0

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

(40)

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

95

96

Chronological Calculus in Systems and Control Theory

Zinbiel Algebra and Combinatorics

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

(42)

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

(43)

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

h(w); u ˝ vi D hw; u

vi :

(44)

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

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

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

9

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

0

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

Chronological Calculus in Systems and Control Theory

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

X

au a (t) :

(46)

a2Z

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

t

S()F 0 () d with F D

0

X

U a ; (47)

a2Z

is most compactly written as S D1CS F :

(48)

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

1 X

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

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

h2H

D F

n

2

D 1C F C F C F

3

4

CF CF

5

6

C F C  

nD0

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

Y

exp ( h ˝ [h])

(51)

h2H

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

97

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Chronological Calculus in Systems and Control Theory

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

and

 hk D  hk   h  k :

(52)

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

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

Chronological Calculus in Systems and Control Theory

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

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

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

101

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

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

Control of Non-linear Partial Differential Equations

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

: in Q T D (0; T)  ˝

(1)

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

(2)

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

(3)

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

(4)

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

in

QT

(5)

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

t 2 (0; T) ;

(7)

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

t 2 (0; T) ;

where  UD

u v



 ;

BD

0 I

 ;

F D L2 (˝) :

103

104

Control of Non-linear Partial Differential Equations

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

in Q T ;

(9)

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

on ˙T ;

(10)

or Neumann type @u D f @

on ˙T ;

(11)

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

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

Control of Non-linear Partial Differential Equations

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

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

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

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

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

Z

t

t f D

e(ts)A B f (s) ds

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

(14)

0

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

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

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

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

T

˝

8s 2 [0; T] :

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

B e tA u D 0

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

(17)

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

Z

T

k f (t)k2 dt C

0

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

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

Z

T 0

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

1 C hu" (T); T gi "

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

(18)

Therefore, passing to the adjoint of T , Z

T 0

D

f" (t) C

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

105

106

Control of Non-linear Partial Differential Equations

Taking

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

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

H D L2 (˝) D F ; (19)

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

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

T

0

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

0

hence Z 0

T

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

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

T

(22)

on ˙T x2˝

8i D 1; : : : ; n :

(23)

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

(21)

“

2

Z

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

in Q T

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



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

1 ku" (T)k2 C "

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

@x i u 2 L2 (Q T )

So, T

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

(20)

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

8 f 2 L2 (˝)

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

It turns out that 1 2

B f D 1! f

QT

and note that

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

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

˝

ju0 j2 dx

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

T 0

Z !

jv(t; x)j2 dxdt  C

Z ˝

jv(T; x)j2 dx

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

on ˙T :

(24)

(25)

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

Control of Non-linear Partial Differential Equations

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

(28)

and : (t) D

1 t(T  t)

0< t< T:

Note that > 0; r

> 0;

(t) ! 1 r (t; x)

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

“

2 2s

j f (t; x)j e

C QT

Z 



Z r

T

dxdt C Cs

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

(t)dt 0

r

dH n1 (x)

(29)

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

on ˙T :

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

t ! 0;T

!1

t # 0;t " T :

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

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

! 00

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

1

on Rn n ! 00 ;

0

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

on @˝ 0 D @˝ [ @! 0 ;

on ! 0 ;

(31)

107

108

Control of Non-linear Partial Differential Equations

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

“

3 jwj2 e2s

Q 0T

C Z

2 2s

Q 0T

jhj e

Z

T

C Cs

dt 0

Z



Z

T

C Cs

dt

r

dxdt

r

dxdt

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

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

3 jwj2 e2s

Q 0T

 s3

Z

Z

T1

dt

˝n!

T0

r

dxdt

3 jwj2 e2s r dxdt Z

Z

T1



dt

˝n!

T0

jvj2 dx

Therefore, recalling the definition of h, Z

T1

dt Z

˝n!

Z

T

C

dt Z

0

dt 0

! 00 n! 0

Z

T

C

jvj2 dx  C

!

“

jhj2 e2s

Q 0T

Z

r

dxdt

dt

Z dt

0

! 00 n! 0

Z

jrvj2 e2s

2 2s

jrvj e

! 00 n! 0

r

r

Z

Z

T

dt 0

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

dt

!

jvj2 dx

0

!

dt

Z

Z

2T/3

dt T/3

Z

˝ T

v 2 (t; x) dx Z dt

0

!

v 2 (t; x) dx ;

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

jvj2 e2s

r

dx ;

u0 2L2 (˝) ;

(32)

u1 2 H 1 (˝)

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

Z

T

3 v (T; x) dx  T ˝ 2

which is exactly (24).

dx :

dx

C

T/3

Z

T

Observe that problem (32) is well-posed taking

0

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

Z 0

3  (1 C C) T

T



Z

Z

 2 2 2 jr j v C jrj2jrvj2 e2s r dx

jvj2 dx C C

˝

jvj2 dx  (1 C C)

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

Q 0T

T0

Z

2T/3 T/3

@ @

@! 0

0

Z

Z

dt “

s3

or

!

jvj2 dx

R D sup jx  x0 j : x2˝

Control of Non-linear Partial Differential Equations

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

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

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

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

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

109

110

Control of Non-linear Partial Differential Equations

a system is given by E(t) D

1 2

Z ˝

Geometrical Aspects

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

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

Z ˝

a(x)ju t j2 dx  0 ;

8t  0 :

(34)

On the other hand, if a  0, then the system is conservative, i. e., E(t) D E(0) for all t  0. Another well-investigated stabilization problem for the wave equation is when the feedback is localized on a part  0 of the boundary  , that is, 8 ˆ @ t t u  u D 0 ˆ ˆ ˆ < @u C u D 0

in ˝  R

on ˙0 D (0; 1)  0 t @ ˆ u D 0 on ˙1 D (0; 1)  ( n 0 ) ˆ ˆ ˆ :(u; @ u)(0) D (u0 ; u1 ) t

(35) In this case, the dissipation relation (34) takes the form 0

E (t) D 

Z 0

ju t j2 dH n1  0 ;

8t  0 :

In many a situation—such as to improve the quality of an acoustic hall—one seeks to reduce vibrations to a minimum: this is why stabilization is an important issue in CT. We note that the above system has a unique stationary solution—or, equilibrium—given by u  0. Stabilization theory studies all questions related to the convergence of solutions to such an equilibrium: existence of the limit, rate of convergence, different effects of nonlinearities in both displacement and velocity, effects of geometry, coupled systems, damping effects due to memory in viscoelastic materials, and so on. System (33) is said to be:  strongly stable if E(t) ! 0 as t ! 1;  (uniformly) exponentially stable if E(t)  Ce˛t E(0) for all t  0 and some constants ˛ > 0 and C  0, independent of u0 ; u1 . This note will focus on some of the above issues, such as geometrical aspects, nonlinear damping, indirect damping for coupled systems and memory damping.

A well-known property of the wave equation is the socalled finite speed of propagation, which means that, if the initial conditions u0 ; u1 have compact support, then the support of u(t; ) evolves in time at a finite speed. This explains why, for the wave equation, the geometry of ˝ plays an essential role in all the issues related to control and stabilization. The size and localization of the region in which the feedback is active is of great importance. In this paper such a region, denoted by !, is taken as a subset of ˝ of positive Lebesgue measure. More precisely, a is assumed to be continuous on ˝ and such that a0

on

˝

and

a  a0

on

!;

(36)

for some constant a0 > 0. In this case, the feedback is said to be distributed. Moreover, it is said to be globally distributed if ! D ˝ and locally distributed if ˝ n ! has positive Lebesgue measure. Two main methods have been used or developed to study stabilization, namely the multiplier method and microlocal analysis. The one that gives the sharpest results is based on microlocal analysis. It goes back to the work of Bardos, Lebeau and Rauch [17], giving geodesics sufficient conditions on the region of active control for exact controllability to hold. These conditions say that each ray of geometric optics should meet the control region. Burq and Gérard [25] showed that these results hold under weaker regularity assumptions on the domain and coefficients of the operators (see also [26,27]). These geodesics conditions are not explicit, in general, but they allow to get decay estimates of the energy under very general hypotheses. The multiplier method is an explicit method, based on energy estimates, to derive decay rates (as well as observability and exact controllability results). For boundary control and stabilization problems it was developed in the works of several authors, such as Ho [38,73], J.-L. Lions [86], Lasiecka–Triggiani, Komornik–Zuazua [76], and many others. Zuazua [123] gave an explicit geometric condition on ! for a semilinear wave equation subject to a locally distributed damping. Such a condition was then relaxed K. Liu [87] (see also [93]) who introduced the so-called piecewise multiplier method. Lasiecka and Triggiani [80,81] introduced a sharp trace regularity method which allows to estimate boundary terms in energy estimates. There also exist intermediate results between the geodesics conditions of Bardos–Lebeau–Rauch and the multiplier method, obtained by Miller [95] using differentiable escape functions.

Control of Non-linear Partial Differential Equations

Zuazua’s multiplier geometric condition can be described as follows. If a subset O of ˝ is given, one can define an "-neighborhood of O in ˝ as the subset of points of ˝ which are at distance at most " of O. Zuazua proved that if the set ! is such that there exists a point x0 2 Rn —an observation point—for which ! contains an "neighborhood of  (x 0 ) D fx 2 @˝ ; (x  x 0 )  (x)  0g, then the energy decays exponentially. In this note, we refer to this condition as (MGC). If a vanishes for instance in a neighborhood of the two poles of a ball ˝ in Rn , one cannot find an observation point x0 such that (MGC) holds. K. Liu [87] (see also [93]) introduced a piecewise multiplier method which allows to choose several observation points, and therefore to handle the above case. Introduce disjoint lipschitzian domains ˝ j of ˝, j D 1; : : : ; J, and observation points x j 2 R N , j D 1; : : : ; J and define

j (x j ) D fx 2 @˝ j ; (x  x j )   j (x)  0g Here  j stands for the unit outward normal vector to the boundary of ˝ j . Then the piecewise multiplier geometrical condition for ! is:    (PWMGC) !  N" [ JjD1 j (x j ) [ ˝n [ JjD1 ˝ j It will be denoted by (PWMGC) condition in the sequel. Assume now that a vanishes in a neighborhood of the two poles of a ball in Rn . Then, one can choose two subsets ˝ 1 and ˝ 2 containing, respectively, the two regions where a vanishes and apply the piecewise multiplier method with J D 2 and with the appropriate choices of two observation points and ". The multiplier method consists of integrating by parts expressions of the form Z TZ  t

˝

@2t u

otherwise. One can then prove that the energy satisfies an estimate of the form Z

T

E(s) ds t

Z

Z

T

 cE(t) C

2

˝

t

 u C a(x)u t Mu dx dt D 0 80  t  T ;

where u stands for a (strong) solution of (33), with an appropriate choice of Mu. Multipliers have generally the form Mu D (m(x)  ru C c u) (x) ; where m depends on the observation points and is a cut-off function. Other multipliers of the form Mu D 1 (ˇu), where ˇ is a cut-off function and 1 is the inverse of the Laplacian operator with homogeneous Dirichlet boundary conditions, have also used. The geometric conditions (MGC) or (PWMGC) serve to bound above by zero terms which cannot be controlled

!

ju t j

2

ds

8t  0 :

(37)

Once this estimate is proved, one can use the dissipation relation to prove that the energy satisfies integral inequalities of Gronwall type. This is the subject of the next section. Decay Rates, Integral Inequalities and Lyapunov Techniques The Linear Feedback Case Using the dissipation relation (34), one has Z TZ ˝

t

aju t j2 dx ds 

Z

T

E 0 (s) ds  E(t)

t

8 0  t  T: On the other hand, thanks to assumption (36) on a Z TZ !

t

u2t dx ds 

1 a0

Z TZ ˝

t



aju t j2 dx ds 1 E(t) a0

80  t  T :

By the above inequalities and (37), E satisfies Z

T

E(s) ds  cE(t) ;



a(x)ju t j C

!

Z

80  t  T :

(38)

t

Since E is a nonincreasing function and thanks to this integral inequality, Haraux [71] (see also Komornik [74]) proved that E decays exponentially at infinity, that is  E(t)  E(0) exp 1  t/c) ;

8t  c :

This proof is as follows. Define Z

1

(t) D exp(t/c)

E(s) ds

8t  0 :

t

Thanks to (38)  is nonincreasing on [0; 1), so that Z

1

(t)  (0) D

E(s) ds : 0

(39)

111

112

Control of Non-linear Partial Differential Equations

Using once again (38) with t D 0 in this last inequality and the definition of , one has Z 1 E(s) ds  cE(0) exp(t/c) 8t  0 :

one can prove that the energy E of solutions satisfies the following inequality for all 0  t  T Z

t

E (pC1)/2 (s) dt

t

Since E is a nonnegative and nonincreasing function Z t Z 1 cE(t)  E(s) ds  E(s) ds tc

T

 cE (pC1)/2 (t) C c

Z

T t

tc

 cE(0) exp((t  c)/c) ; so that (39) is proved. An alternative method is to introduce a modified (or perturbed) energy E" which is equivalent to the natural one for small values of the parameter " as in Komornik and Zuazua [76]. Then one shows that this modified energy satisfies a differential Gronwall inequality so that it decays exponentially at infinity. The exponential decay of the natural energy follows then at once. In this case, the modified energy is indeed a Lyapunov function for the PDE. The natural energy cannot be in general such a Lyapunov function due to the finite speed of propagation (consider initial data which have compact support compactly embedded in ˝n!). There are also very interesting approaches using the frequency domain approach, or spectral analysis such as developed by K. Liu [87] Z. Liu and S. Zheng [88]. In the sequel, we concentrate on the integral inequality method. This method has been generalized in several directions and we present in this note some results concerning extensions to  nonlinear feedback  indirect or single feedback for coupled system  memory type feedbacks Generalizations to Nonlinear Feedbacks Assume now that the feedback term a(x)u t in (33) is replaced by a nonlinear feedback a(x) (u t ) where is a smooth, increasing function satisfying v (v)  0 for v 2 R, linear at 1 and with polynomial growth close to zero, that is: (v) D jvj p for jvj  1 where p 2 (1; 1). Assume moreover that ! satisfies Zuazua’s multiplier geometric condition (MGC) or Liu’s piecewise multiplier method (PWMGC). Then using multipliers of the space and time variables defined as E(s)(p1)/2 Mu(x) where Mu(x) are multipliers of the form described in section 5.1 and integrating by parts expressions of the form Z T E(s)(p1)/2 t Z   2  @ t u  u C a(x) (u t ) Mu(x) dx ds D 0 ; ˝

E (p1)/2 (s) Z  Z 2 2

(u t ) C ju t j :  ˝

!

One can remark than an additional multiplicative weight in time depending on the energy has to be taken. This weight is E (p1)/2 . Then as in the linear case, but in a more involved way, thanks to the dissipation relation E 0 (t) D 

Z ˝

a(x)u t (u t ) ;

(40)

one can prove that E satisfies the following nonlinear integral inequality Z

T

E (pC1)/2 (s) ds  cE(t) ;

80  t  T :

t

Thanks to the fact that E is nonincreasing, a wellknown result by Komornik [74] shows that E is polynomially decaying, as t 2/(p1) at infinity. The above type results have been obtained by many authors under weaker form (see also [40,41,71,98,122]). Extensions to nonlinear feedbacks without growth conditions close to zero have been studied by Lasiecka and Tataru [78], Martinez [93], W. Liu and Zuazua [89], Eller Lagnese and Nicaise [56] and Alabau–Boussouira [5]. We present the results obtained in this last reference since they provide optimal decay rates. The method is as follows. Define respectively the linear and nonlinear kinetic energies (R

R!

˝

ju t j2 dx ; ja(x) (u t )j2 dx ;

and use a weight function in time f (E(s)) which is to be determined later on in an optimal way. Integrating by parts expressions of the form Z

Z

T

f (E(s)) t

 ˝

 @2t uuCa(x) (u t ) Mu(x) dx ds D 0 ;

one can prove that the energy E of solutions satisfies the

Control of Non-linear Partial Differential Equations

following inequality for all 0  t  T Z

T t

Z T E(s) f (E(s)) ds  c f (E(t)) C c f (E(s)) t Z  Z 2 2  ja(x) (u t )j C ju t j : ˝

!

types of the feedback, one can assume convexity of H only in a neighborhood of 0. One can prove from (41) that there exists an (explicit) T0 > 0 such that for all initial data, E satisfies the following nonlinear integral inequality (41)

The difficulty is to determine the optimal weight under general growth conditions on the feedback close to 0, in particular for cases for which the feedback decays to 0 faster than polynomials. Assume now that the feedback satisfies g(jvj)  j (v)j  Cg 1 (jvj) ;

8jvj  1 ;

(42)

Z

E(s) f (E(s)) ds  T0 E(t)

then, define a function F as follows: 8  ˆ < Hˆ (y) if y 2 (0; C1) ; F(y) D y ˆ :0 if y D 0 ; ˆ that is where Hˆ  stands for the convex conjugate of H, ˆ ˆ  (y) D sup fx y  H(x)g : H

Z ˝t

where ˇ is of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants. One can prove that the above formulas make sense, and in particular that F is invertible and smooth. More precisely, F is twice continuously differentiable strictly increasing, one-to-one function from [0; C1) onto [0; r02 ). Note that since the feedback is supposed to be linear at infinity, if one wants to obtain results for general growth

ja(x) (u t )j2 dx  1 (t)Hˆ 1   Z 1  a(x)u t (u t ) dx

1 (t) ˝

In a similar way, one proves that Z

ju t j2 dx  2 (t)Hˆ 1



1

2 (t)



Z ˝

a(x)u t (u t ) dx

where ˝ t and ! t are time-dependent sets of respective Lebesgue measures 1 (t) and 2 (t) on which the velocity u t (t; x) is sufficiently small. Using the above two estimates, together with the linear growth of at infinity, one proves Z

Z

T

f (E(s)) t

˝

Z

T

 t

ja(x) (u t )j2 C

Z !

ju t j2



  Z 1 f (E(s))Hˆ 1 a(x)u t (u t ) dx c ˝

Using then Young’s inequality, together with the dissipation relation (40) in the above inequality, one obtains Z

Z

T

f (E(s)) t

x2R

Then the optimal weight function f is determined in the following way   f (s) D F 1 (s/2ˇ) s 2 0; 2ˇr02 ;

(43)

This inequality is proved thanks to convexity arguˆ one can ments as follows. Thanks to the convexity of H, use Jensen’s inequality and (42), so that

!t

Moreover, is assumed to have a linear growth at infinity. We define the optimal weight function f as follows. We first extend H to a function Hˆ define on all R (  H(x) if x 2 0; r02 ; ˆ H(x) D C1 otherwise ;

80  t  T :

t

where g is continuously differentiable on R strictly increasing with g(0) D 0 and 8 2 ˆ < g 2 C ([0; r0 ]) ; r0 sufficiently small ;  p p H(:) D :g( :) is strictly convex on 0; r02 ; ˆ : g is odd :

T

˝ T

Z

ja(x) (u t )j2 C

Hˆ

 C1

 ?

Z !

ju t j2



 f (E(s) ds C C2 E(t) ;

(44)

t

where C i > 0 i D 1; 2 is a constant independent of the initial data. Using the dissipation relation (40) in the above inequality, this gives for all 0  t  T Combining this last inequality with (41) gives Z

Z

T

T

E(s) f (E(s)) ds  ˇ t

  ˆ ? f (E(s) ds C C2 E(t) (H)

t

where ˇ is chosen of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants to guarantee that the argument E of f stays in the domain of definition of f .

113

114

Control of Non-linear Partial Differential Equations

Thus (43) is proved, thanks to the fact that the weight function has been chosen so that ˆ ? ( f (E(s)) D ˇH

1 E(s) f (E(s)) 2

80  s:

Therefore E satisfies a nonlinear integral inequality with a weight function f (E) which is defined in a semi-explicit way in general cases of feedback growth. The last step is to prove that a nonincreasing and nonnegative absolutely continuous function E satisfying a nonlinear integral inequality of the form (43) is decaying at infinity, and to establish at which rate this holds. For this, one proceeds as in [5]. Let  > 0 and T0 > 0 be fixed given real numbers and F be a strictly increasing function from [0; C1) on [0; ), with F(0) D 0 and lim y!C1 F(y) D . For any r 2 (0; ), we define a function K r from (0; r] on [0; C1) by Z r dy ; (45) K r () D 1  yF (y) and a function r which is a strictly increasing function from [ F 11 (r) ; C1) onto [ F 11 (r) ; C1) by r (z)

1 D z C K r (F( ))  z; z

8z 

1 ; F 1 (r)

(46)

One can prove that if E is a nonincreasing, absolutely continuous function from [0; C1) on [0; C1), satisfying 0 < E(0) <  and the inequality Z T E(s)F 1 (E(s)) ds  T0 E(S) ; 8 0  t  T ; (47) t

then E satisfies the following estimate: ! T0 1 ; 8t  1 ; E(t)  F 1 ( t ) F (r) r T

(48)

0

where r is any real such that Z C1 1 E()F 1 (E()) d  r   : T0 0 Thus, one can apply the above result to E with  D r02 and show that lim t ! C1E(t) D 0, the decay rate being given by estimate (48). If g is polynomial close to zero, one gets back that the 2 energy E(t) decays as t p1 at infinity. If g(v) behaves as exp(1/jvj) close to zero, then E(t) decays as 1/(ln(t))2 at infinity. The usefulness of convexity arguments has been first pointed out by Lasiecka and Tataru [78] using Jensen’s

inequality and then in different ways by Martinez [93] (the weight function does not depend on the energy) and W. Liu and Zuazua [89] and Eller Lagnese and Nicaise [56]. Optimal decay rates have been obtained by Alabau–Boussouira [5,6] using a weight function determined through the theory of convex conjugate functions and Young’s (named also as Fenchel–Moreau’s) inequality. This argument was also used by W. Liu and Zuazua [89] in a slightly different way and combined to a Lyapunov technique. Optimality of estimates in [5] is proved in one-dimensional situation and for boundary dampings applying optimality results of Vancostenoble [119] (see also Martinez and Vancostenoble [118]). Indirect Damping for Coupled Systems Many complex phenomena are modeled through coupled systems. In stabilizing (or controlling) energies of the vector state, one has very often access only to some components of this vector either due to physical constraints or to cost considerations. In this case, the situation is to stabilize a full system of coupled equation through a reduced number of feedbacks. This is called indirect damping. This notion has been introduced by Russell [109] in 1993. As an example, we consider the following system: (

@2t u  u C @ t u C ˛v D 0 @2t v  v C ˛u D 0 in ˝  R ;

uD0Dv

on @˝  R :

(49)

Here, the first equation is damped through a linear distributed feedback, while no feedback is applied to the second equation. The question is to determine if this coupled system inherits any kind of stability for nonzero values of the coupling parameter ˛ from the stabilization of the first equation only. In the finite dimensional case, stabilization (or control) of coupled ODE’s can be analyzed thanks to a powerful rank type condition named Kalman’s condition. The situation is much more involved in the case of coupled PDE’s. One can show first show that the above system fails to be exponentially stable (see also [66] for related results). More generally, one can study the stability of the system ( u00 C A1 u C Bu0 C ˛v D 0 (50) v 00 C A2 v C ˛u D 0 in a separable Hilbert space H with norm jj, where A1 ; A2 and B are self-adjoint positive linear operators in H. Moreover, B is assumed to be a bounded operator. So, our analysis applies to systems with internal damping supported

Control of Non-linear Partial Differential Equations

in the whole domain ˝ such as (49); the reader is referred to [1,2] for related results concerning boundary stabilization problems (see also Beyrath [23,24] for localized indirect dampings). In light of the above observations, system (50) fails to be exponentially stable, at least when H is infinite dimensional and A1 has a compact resolvent as in (49). Indeed it is shown in Alabau, Cannarsa and Komornik [8] that the total energy of sufficiently smooth solutions of (50) decays polynomially at infinity whenever j˛j is small enough but nonzero. From this result we can also deduce that any solution of (50) is strongly stable regardless of its smoothness: this fact follows by a standard density argument since the semigroup associated with (50) is a contraction semigroup. A brief description of the key ideas of the approach developed in [2,8] is as follows. Essentially, one uses a finite iteration scheme and suitable multipliers to obtain an estimate of the form Z

j

T

E(u(t); v(t))dt  c 0

X kD0

(51)

where j is a positive integer and E denotes the total energy of the system  1  1/2 2 jA1 uj C ju0 j2 2  1  1/2 2 jA2 vj C jv 0 j2 C ˛hu; vi : C 2 Once (51) is proved, an abstract lemma due to Alabau [1,2] shows that E(u(t); v(t)) decays polynomially at 1. This abstract lemma can be stated as follows. Let A be the infinitesimal generator of a continuous semi-group exp(tA) on an Hilbert space H , and D(A) its domain. For U 0 in H we set in all the sequel U(t) D exp(tA)U 0 and assume that there exists a functional E defined on C([0; C1); H ) such that for every U 0 in H , E(exp(:A)) is a non-increasing, locally absolutely continuous function from [0; C1) on [0; C1). Assume moreover that there exist an integer k 2 N ? and nonnegative constants cp for p D 0; : : : k such that E(u; v) D

Z

T

E(U(t))dt  S

k X

c p E(U (p) (S))

pD0

80  S  T ; 8U 0 2 D(Ak ) :

T

E(U()) S

(52) U0

in Then the following inequalities hold for every D(Akn ) and all 0  S  T where n is any positive integer:

kn X   (  S)n1 E U (p) (S) ; (53) d  c (n  1)! pD0

and

E(U(t))  c

kn X   E U (p) (0) t n pD0

8t > 0 ;

  8U 0 2 D Akn ;

where c is a constant which depends on n. First (53) is proved by induction on n. For n D 1, it reduces to the hypothesis (52). Assume now that (53) holds for n and let U 0 be given in D(Ak(nC1) ). Then we have Z TZ

T

E(U()) S

c

t kn Z X pD0 S

E(u(k) (0); v (k) (0)) 8T  0;

Z

T

(  t)n1 ddt (n  1)!

  E U (p) (t) dt   8 0  S  T ; 8 U 0 2 D Akn :

Since U 0 is in D(Ak(nC1) ) we deduce that U (p) (0) D A p U 0 is in D(Ak ) for p 2 f0; : : : kng. Hence we can apply the assumption (52) to the initial data U (p) (0). This together with Fubini’s Theorem applied on the left hand side of the above inequality give (53) for n C 1. Using the property that E(U(t)) is non increasing in (53) we easily obtain the last desired inequality. Applications to wave-wave, wave-Petrowsky equations and various concrete examples hold. The above results have been studied later on by Batkai, Engel, Prüss and Schnaubelt [18] using very interesting resolvent and spectral criteria for polynomial stability of abstract semigroups. The above abstract lemma in [2] has also been generalized using interpolation theory. One should note that this integral inequality involving higher order energies of solutions is not of differential nature, unlike Haraux’s and Komornik’s integral inequalities. Another approach based on decoupling techniques and for slightly different abstract systems have been introduced by Ammar Khodja Bader and Ben Abdallah [12]. Spectral conditions have also been studied by Z. Liu [88] and later on by Z. Liu and Rao [90], Loreti and Rao [92] for peculiar abstract systems and in general for coupled equations only of the same nature (wave-wave for instance), so that a dispersion relation for the eigenvalues of the coupled system can be derived. Also these last results are given for internal stabilization only. Because of the above limitations, Z. Liu–Rao and Loreti–Rao’s results

115

116

Control of Non-linear Partial Differential Equations

are less powerful in generality than the ones given by Alabau, Cannarsa and Komornik [8] and Alabau [2]. Moreover results through energy type estimates and integral inequalities can be generalized to include nonlinear indirect dampings as shown in [7]. On the other side spectral methods are very useful to obtain optimal decay rates provided that one can determine at which speed the eigenvalues approach the imaginary axis for high frequencies.

in a Hilbert space X, where A: D(A) X ! X is an accretive self-adjoint linear operator with dense domain, and rF denotes the gradient of a Gâteaux differentiable functional F : D(A1/2 ) ! R. In particular, equation (54) fits into this framework as well as several other classical equations of mathematical physics such as the linear elasticity system. We consider the following assumptions.

Memory Dampings

Assumptions (H1)

We consider the following model problem

1. A is a self-adjoint linear operator on X with dense domain D(A), satisfying

8 Rt ˆ ˆ ˆu t t (t; x)  u(t; x) C 0 ˇ(t  s)u(s; x) ds D ˆ < ju(t; x)j u(t; x) ˆ u(t; )j@˝ D 0 ˆ ˆ ˆ :(u(0; ); u (0; )) D (u ; u ) t 0 1 (54) 2 where 0 <  N2 holds. The second member is a source term. The damping

Z

hAx; xi  Mkxk2

8x 2 D(A)

(56)

for some M > 0. 2. ˇ : [0; 1) ! [0; 1) is a locally absolutely continuous function such that Z

1

ˇ(t)dt < 1 ˇ(0) > 0

ˇ 0 (t)  0

0

for a.e. t  0 :

t

ˇ(t  s)u(s; x) ds

3. F : D(A1/2 ) ! R is a functional such that 1. F is Gâteaux differentiable at any point x 2 D(A1/2 ); 2. for any x 2 D(A1/2 ) there exists a constant c(x) > 0 such that

0

is of memory type. The energy is defined by Eu (t) D

1 ku t (t)k2L 2 (˝) dx 2   Z t 1 ˇ(s) ds kru(t)k2L 2 (˝) 1 C 2 0 1  C2  ku(t)kL  C2 (˝)

C2 Z 1 t C ˇ(t  s)kru(t)  ru(s)k2L 2 (˝) ds 2 0

The damping term produces dissipation of the energy, that is (for strong solutions) 1 Eu0 (t) D  ˇ(t)kru(t)k2 2 Z 1 t 0 C ˇ (t)kru(s)  ru(t)k2 ds  0 2 0 One can consider more general abstract equations of the form u00 (t) C Au(t) 

Z

t

jDF(x)(y)j  c(x)kyk;

where DF(x) denotes the Gâteaux derivative of F in x; consequently, DF(x) can be extended to the whole space X (and we will denote by rF(x) the unique vector representing DF(x) in the Riesz isomorphism, that is, hrF(x); yi D DF(x)(y), for any y 2 X); 3. for any R > 0 there exists a constant C R > 0 such that krF(x)  rF(y)k  C R kA1/2 x  A1/2 yk for all x; y 2 D(A1/2 ) satisfying kA1/2 xk ; kA1/2 yk  R. Assumptions (H2) 1. There exist p 2 (2; 1] and k > 0 such that ˇ 0 (t)  kˇ

ˇ(t  s)Au(s) ds D rF(u(t))

for any y 2 D(A1/2 );

1C 1p

(t) for a.e. t  0

0

t 2 (0; 1)

(55)

(here we have set

1 p

D 0 for p D 1).

Control of Non-linear Partial Differential Equations

2. F(0) D 0, rF(0) D 0, and there is a strictly increasing continuous function : [0; 1) ! [0; 1) such that (0) D 0 and jhrF(x); xij 

1/2

(kA

1/2

xk)kA

2

xk

T

m p

Z

t

Eu (t) S

8x 2 D(A ) :

ˇ(t  s)kA1/2 u(s)  A1/2 u(t)k2 dsdt

0 p pCm

 CEu

Z

T

(S)

1C mp Eu (t)'m (t)dt

!

m pCm

(58)

S

for some constant C > 0. Suppose that, for some m  1, the function ' m defined in (57) is bounded. Then, for any S0 > 0 there is a positive constant C such that Z

1

1C mp

Eu

 m m  p p (t)dt  C Eu (0) C k'm k1 Eu (S)

S

8 S  S0 :

(59)

One uses this last result first with m D 2 noticing that ' 2 is bounded and  D E 2/p . This gives a first energy decay rate as (t C1)p/2 . This estimate shows that ' 1 is bounded. Then one applies once again the last result with m D 1 and  D E 1/p . One deduces then that E decays as (t C 1)p which is the optimal decay rate expected.

0

where the multipliers Mu are of the form (s)(c1 (ˇ u)(s) C c2 (s)u) with  which is a differentiable, nonincreasing and nonnegative function, and c1 being a suitable constant, whereas c2 may be chosen dependent on ˇ. Integrating by parts the resulting relations and performing some involved estimates, one can prove that for all t0 > 0 and all T  t  t0 Z T Z T (s)E(s) ds  C(0)E(t) C (s) t t Z s  2  ˇ(s  ) A1/2 u(s)  A1/2 u() d ds ; 0

If p D 1, that is if the kernel ˇ decays exponentially, one can easily bound the last term of the above estimate by cE(t) thanks to the dissipation relation. If p 2 (2; 1), one has to proceed differently since the term Z T Z s  2 (s) ˇ(s  )  A1/2 u(s)  A1/2 u() d ds t

Z

1/2

Under these assumptions, global existence for sufficiently small (resp. all) initial data in the energy space can be proved for nonvanishing (resp. vanishing) source terms. It turns out that the above energy methods based on multiplier techniques combined with linear and nonlinear integral inequalities can be extended to handle memory dampings and applied to various concrete examples such as wave, linear elastodynamic and Petrowsky equations for instance. This allows to show in [10] that exponential as well as polynomial decay of the energy holds if the kernel decays respectively exponentially or polynomially at infinity. The method is as follows. One evaluates expressions of the form Z T Z t hu00 (s) C Au(s)  ˇ Au(s)  rF(u(s); Mui ds t

Then, we have for any T  S  0

0

cannot be directly estimated thanks to the dissipation relation. To bound this last term, one can generalize an argument of Cavalcanti and Oquendo [37] as follows. Define, for any m  1, Z t 1 'm (t) :D ˇ 1 m (t  s)kA1/2 u(s)  A1/2 u(t)k2 ds ; 0

t  0:

(57)

Bibliographical Comments For an introduction to the multiplier method, we refer the interested reader to the books of J.-L. Lions [86], Komornik [74] and the references therein. The celebrated result of Bardos Lebeau and Rauch is presented in [86]. A general abstract presentation of control problems for hyperbolic and parabolic equations can be found in the book of Lasiecka and Triggiani [80,81]. Results on spectral methods and the frequency domain approach can be found in the book of Z. Liu [88]. There also exists an interesting approach developed for bounded feedback operators by Haraux and extended to the case of unbounded feedbacks by Ammari and Tucsnak [11]. In this approach, the polynomial (or exponential) stability of the damped system is proved thanks to the corresponding observability for the undamped (conservative) system. Such observability results for weakly coupled undamped systems have been obtained for instance in [3]. Many other very interesting issues have been studied in connection to semilinear wave equations, see [34,123] and the references therein, and damped wave equations with nonlinear source terms [39]. Well-posedness and asymptotic properties for PDE’s with memory terms have first been studied by Dafermos [53,54] for convolution kernels with past history (convolution up to t D 1), by Prüss [103] and Prüss and Propst [102] in which the efficiency of different models of dampings are compared to experiments (see also

117

118

Control of Non-linear Partial Differential Equations

Londen Petzeltova and Prüss [91]). Decay estimates for the energy of solutions using multiplier methods combined with Lyapunov type estimates for an equivalent energy are proved in Munoz Rivera [97], Munoz Rivera and Salvatierra [96], Cavalcanti and Oquendo [37] and Giorgi Naso and Pata [67] and many other papers. Optimal Control As for positional control, also for optimal control problems it is convenient to adopt the abstract formulation introduced in Sect. “Abstract Evolution Equations”. Let the state space be represented by the Hilbert space H, and the state equation be given in the form (12), that is (

u0 (t) D Au(t) C B f (t)

t 2 [0; T]

u(0) D u0 :

(60)

Recall that A is the infinitesimal of a strongly continuous semigroup, etA , in H, B is a (bounded) linear operator from F (the control space) to H, and uf stands for the unique (mild) solution of (60) for a given control function f 2 L2 (0; T; H). A typical optimal control problem of interest for PDE’s is the Bolza problem which consists in 8 the cost functional ˆ < minimizing   : RT J( f ) D 0 L(t; u f (t); f (t))dt C ` u f (T) ˆ : over all controls f 2 L2 (0; T; F) :

(61)

Here, T is a positive number, called the horizon, whereas L and ` are given functions, called the running cost and final cost, respectively. Such functions are usually assumed to be bounded below. A control function f 2 L2 (0; T; F) at which the above minimum is attained is called an optimal control for problem (61) and the corresponding solution u f of (60) is said to be an optimal trajectory. Alltogether, fu f ; f g is called an optimal (trajectory/control) pair. For problem (61) the following issues will be addressed in the sections below:  the existence of controls minimizing functional J;  necessary conditions that a candidate solution must satisfy;  sufficient conditions for optimality provided by the dynamic programming method. Other problems of particular interest to CT for PDE’s are problems with an infinite horizon (T D 1), problems with a free horizon T and a final target, and problems with constraints on both control variables and state variables.

Moreover, the study of nonlinear variants of (60), including semilinear problems of the form ( u0 (t) D Au(t) C h(t; u(t); f (t)) t 2 [0; T] (62) u(0) D u0 ; is strongly motivated by applications. The discussion of all these variants, however, will not be here pursued in detail. Traditionally, in optimal control theory, state variables are denoted by the letters x; y; : : : , whereas u; v; : : : are reserved for control variables. For notational consistency, in this section u() will still denote the state of a given system and f () a control function, while  will stand for a fixed element of control space F. Existence of Optimal Controls From the study of finite dimensional optimization it is a familiar fact that the two essential ingredients to guarantee the existence of minima are compactness and lower semicontinuity. Therefore, it is clear that, in order to obtain a solution of the optimal control problem (60)–(61), one has to make assumptions that allow to recover such properties. The typical hypotheses that are made for this purpose are the following:  coercivity: there exist constants c0 > 0 and c1 2 R such that `()  c1

and L(t; u; )  c0 kk2 C c1 8(t; u; ) 2 [0; T]  H  F

(63)

 convexity: for every (t; u) 2 [0; T]  H  7! L(t; u; )

is convex on

F:

(64)

Under the above hypotheses, assuming lower semicontinuity of ` and of the map L(t; ; ), it is not hard to show that problem (60)–(61) has at least one solution. Indeed, assumption (63) allows to show that any minimizing sequence of controls f f k g is bounded in L2 (0; T; H). So, it admits a subsequence, still denoted by f f k g which converges weakly in L2 (0; T; H) to some function f . Then, by linearity, u f k (t) converges to u f (t) for every t 2 [0; T]. So, using assumption (64), it follows that f is a solution of (60)–(61). The problem becomes more delicate when the Tonelli type coercivity condition (63) is relaxed, or the state equation is nonlinear as in (62). Indeed, the convergence of u f k (t) is no longer ensured, in general. So, in order to recover compactness, one has to make further assumptions, such as the compactness of etA , or structural properties of L

Control of Non-linear Partial Differential Equations

and h. For further reading, one may consult the monographs [22,85], and [79], for problems where the running and final costs are given by quadratic forms (the so-called Linear Quadratic problem), or [84] and [59] for more general optimal control problems. Necessary Conditions Once the existence of a solution to problem (60)–(61) has been established, the next important step is to provide conditions to detect a candidate solution, possibly showing that it is, in fact, optimal. By and large the optimality conditions of most common use are the ones known as Pontryagin’s Maximum Principle named after the Russian mathematician L.S. Pontryagin who greatly contributed to the development of control theory, see [100,101]. So, suppose fu ; f g, where u D u f is a candidate optimal pair and consider the so-called adjoint system 8 0  ˆ <p (t) D A p(t) C @u L(t; u (t); f (t)) D 0 ˆ :

t 2 [0; T] a.e.

p(T) D @`(u (T)) ;

where @u L(t; u; ) and @`(u) denote the Fréchet gradients of the maps L(t; ; ) and ` at u, respectively. Observe that the above is a backward linear Cauchy problem with terminal condition, which can obviously be reduced to a forward one by the change of variable t ! T  t. So, it admits a unique mild solution, labeled p , which is called the adjoint state associated with fu ; f g. Pontryagin’s Maximum Principle states that, if fu ; f g is optimal, then hp (t); B f (t)i C L(t; u (t); f (t)) D  min hp (t); Bi C L(t; u (t); ) 2F

t 2 [0; T] a.e.

(65)

The name Maximum Principle rather than Minimum Principle, as it would be more appropriate, is due to the fact that, traditionally, attention was focussed on the maximization—instead of minimization—of the functional in (61). Even today, in most models from economics, one is interested in maximizing payoffs, such as revenues, utility, capital and so on. In that case, (65) would still be true, with a “max” instead of a “min”. At first glance, it might be hard to understand the revelance of (65) to problem (61). To explain this, introduce the function, called the Hamiltonian,  H (t; u; p) D min hp; Bi C L(t; u; ) 2F

(t; u; p) 2 [0; T]  H  H :

(66)

Then, Fermat’s rule yields B p C @ L(t; u; ) D 0 at every  2 F at which the minimum in (66) is attained. Therefore, from (65) it follows that B p (t)C@ L(t; u (t); f (t)) D 0

t 2 [0; T] a.e. (67)

which provides a much-easier-to-use optimality condition. There is a vast literature on necessary conditions for optimality for distributed parameter systems. The set-up that was considered above can be generalized in several ways: one can consider nonlinear state equations as in (62), nonsmooth running and finals costs, constraints on both state and control, problems with infinite horizon or exit times. Further reading and useful references on most of these extensions can be found in the aforementioned monographs [22,79,84,85], and in [59] which is mainly concerned with time optimal control problems. Dynamic Programming Though useful as it may be, Pontryagin’s Maximum Principle remains a necessary condition. So, without further information, it does not suffice to prove the optimality of a give trajectory/control pair. Moreover, even when the map  7! @ L(t; u; ) turns out to be invertible, the best result identity (67) can provide, is a representation of f (t) in terms of u (t) and p (t): not enough to determine f (t), in general. This is why other methods to construct optimal controls have been proposed over the years. One of the most interesting ones is the so-called dynamic programming method (abbreviated, DP), initiated by the work of R. Bellman [20]. Such a method will be briefly described below in the set-up of distributed parameter systems. Fix T > 0, s such that 0  s  T, and consider the optimal control problem to minimize Z J s;v ( f ) D

T s

    s;v L t; u s;v (t); f (t) dt C ` u (T) f f (68)

over all control functions f 2 L2 (s; T; F), where u s;v f (t) is the solution of the controlled system ( u0 (t) D Au(t) C B f (t) t 2 [s; T] (69) u(s) D v : The value function U associated to (68)-(69) is the realvalued function defined by U(s; v) D

inf

f 2L 2 (s;T;F)

J s;v ( f ) 8(s; v) 2 [0; T]  H : (70)

119

120

Control of Non-linear Partial Differential Equations

A fundamental step of DP is the following result, known as Bellman’s optimality principle. Theorem 5 For any (s; v) 2 [0; T]  H and any f 2 L2 (s; T; F) Z

r

U(s; v)  s



   s;v L t; u s;v (t); f (t) dt C U r; u (r) f f 8r 2 [s; T] :

Moreover, f  () is optimal if and only if

Now, suppose there is a control function f 2 L2 (s; T; F) such that, for all t 2 [s; T], h@v W(t; u (t)); B f (t)i C L(t; u (t); f (t)) D H (t; u (t); @v W(t; u (t))) ;

(73)

where u () D u s;v f  (). Then, from (71) and (73) it follows that d W(t; u (t)) D L(t; u(t); f (t)) ; dt whence

Z

r

U(s; v) D s



   s;v L t; u s;v f (t); f (t) dt C U r; u f (r) 8r 2 [s; T] :

The connection between DP and optimal control is based on the properties of the value function. Indeed, applying Bellman’s optimality principle, one can show that, if U is Fréchet differentiable, then 8 ˆ <@s U(s; v) C hAv; @v U(s; v)i C H (s; v; @v U(s; v)) D 0 (s; v) 2 (0; T)  D(A) ˆ : U(T; v) D `(v) v 2 H where H is the Hamiltonian defined in (66). The above equation is the celebrated Hamilton–Jacobi equation of DP. To illustrate its connections with the original optimal control problem, a useful formal argument—that can, however, be made rigorous—is the following. Consider a sufficiently smooth solution W of the above problem and let (s; v) 2 (0; T)  D(A). Then, for any trajectory/control pair fu; f g, d W(t; u(t)) D @s W(t; u(t)) C h@v W(t; u(t)); Au(t) dt C B f (t)i D h@v W(t; u(t)); B f (t)i  H (t; u(t); @v W(t; u(t)))  L(t; u(t); f (t))

(71)

W(s; v) D J s;v ( f )  U(s; v) : From the above inequality and (72) it follows that W(s; v) D U(s; v) for all (s; v) 2 (0; T)  D(A), hence for all (s; v) 2 (0; T)  H since D(A) is dense in H. So, f is an optimal control. Note 2 The above considerations lead to the following procedure to obtain optimal an optimal trajectory:  find a smooth solution of the Hamilton–Jacobi equation;  for every (t; v) 2 (0; T)  D(A) provide a feedback f (t; v) such that h@v W(t; v); B f (t; v)i C L(t; v; f (t; v)) D H (t; v; @v W(t; v))  solve the so-called closed loop equation ( u0 (t) D Au(t) C B f (t; u(t)) t 2 [s; T] u(s) D v Notice that not only is trajectory u optimal, but the corresponding control f is given in feedback form as well. Linear Quadratic Optimal Control One of the most successful applications of DP is the socalled Linear Quadratic optimal control problem. Consider problem (68)–(69) with costs L and ` given by L(t; u; ) D hM(t)u; ui C hN(t); i

by the definition of H . Therefore, integrating from s to T, Z

T

`(u(T))  W(s; v)  

and

L(t; u(t); f (t))dt; s

`(u) D hDu; ui

whence J s;v ( f )  W(s; v). Thus, taking the infimum over all f 2 L2 (s; T; F), W(s; v)  U(s; v)

8(s; v) 2 (0; T)  D(A) :

8(T; u; ) 2 [0; T]  H  F

(72)

8u 2 H ;

where  M : [0; T] ! L(H) is continuous, M(t) is symmetric and hM(t)u; ui  0 for every (t; u) 2 [0; T]  H;

Control of Non-linear Partial Differential Equations

 N : [0; T] ! F is continuous, N(t) is symmetric and hN(t); i  c0 jj2 for every (t; ) 2 [0; T]  F and some constant c0 > 0;  D 2 L(H) is symmetric and hDu; ui  0 for every u 2 H. Then, assumptions (63) and (64) are satisfied. So, a solution to (68)–(69) does exist. Moreover, it is unique because of the strict convexity of functional J s;v . In order to apply DP, one computes the Hamiltonian h i H (t; u; p) D min hp; Bi C hM(t)u; ui C hN(t); i

So, solving the closed loop equation ( u0 (t) D [A  BN 1 (t)B P(t)]u(t)

t 2 (s; T)

u(s) D v one obtains the unique optimal trajectory of problem (68)–(69). In sum, by DP one reduces the original Linear Quadratic optimal control problem to the problem of finding the solution of the Riccati equation, which is easier to solve than the Hamilton–Jacobi equation.

2F

1 D hM(t)u; ui  hBN 1 (t)B p; pi ; 4

Bibliographical Comments

where the above minimum is attained at 1  (t; p) D  N 1 (t)B p : 2

(74)

Therefore, the Hamilton–Jacobi equation associated to the problem is 8 ˆ ˆ ˆ@s W(s; v) C hAv; @v W(s; v)i C hM(s)v; vi ˆ <  1 hBN 1 (s)B @ W(s; v); @ W(s; v)i D 0 v v 4 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :w(T; v) D hDv; vi 8v 2 H It is quite natural to search a solution of the above problem in the form W(s; v) D hP(s)v; vi

8(s; v) 2 [0; T]  H ;

with P : [0; T] ! L(H) continuous, symmetric and such that hP(t)u; ui  0. Substituting into the Hamilton–Jacobi equation yields 8 0  ˆ ˆhP (s)v; vi C h[A P(s) C P(s)A]v; vi C hM(s)v; vi ˆ ˆ < hBN 1 (s)B P(s)v; P(s)vi D 0 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :hP(T)v; vi D hDv; vi 8v 2 H Therefore, P must be a solution of the so-called Riccati equation 8 0  ˆ

P(s)BN 1 (s)B P(s) D 0

8s 2 (0; T)

P(T) D D

Once a solution P() of the Riccati equation is known, the procedure described in Note 2 can be applied. Indeed, recalling (74) and the fact that @v W(t; v) D 2P(t)v, one concludes that f (t; v) D N 1 (t)B P(t)v is a feedback law.

Different variants of the Riccati equation have been successfully studied by several authors in connection with different state equations and cost functionals, including boundary control problems and problems for other functional equations, see [22,79] and the references therein. Sometimes, the solution of the Riccati equation related to a linearized model provides feedback stabilization for nonlinear problems as in [104]. Unfortunately, the DP method is hard to implement for general optimal control problems, because of several obstructions: nonsmoothness of solutions to Hamilton– Jacobi equations, selection problems that introduce discontinuities, unboundedness of the coefficients, numerical complexity. Besides the Linear Quadratic case, the so-called Linear Convex case is the other example that can be studied by DP under fairly general conditions, see [14]. For nonlinear optimal control problems some of the above difficulties have been overcome extending the notion of viscosity solutions to infinite dimensional spaces, see [45,46,47,48,49], see also [28,29,30,31,32,33] and [112]. Nevertheless, finding additional ideas to make a generalized use of DP for distributed parameter systems possible, remains a challenging problem for the next generations. Future Directions In addition to all considerations spread all over this article on promising developments of recent—as well as established—research lines, a few additional topics deserve to be mentioned. The one subject that has received the highest attention, recently, is that of numerical approximation of control problems, from the point of view of both controllability and optimal control. Here the problem is that, due to high frequency spurious numerical solutions, stable algorithms for solving initial-boundary value problems do not necessarily yield convergent algorithms for computing

121

122

Control of Non-linear Partial Differential Equations

controls. This difficulty is closely related to the existence of concentrated numerical solutions that escape the observation mechanisms. Nevertheless, some interesting results have been obtained so far, see, e. g., [124,125]. Several interesting results for nonlinear control problems have been obtained by the return method, developed initially by Coron [42] for a stabilization problem. This and other techniques have then been applied to fluid models ([68,69]), the Korteweg–de Vries equation ([105,106,107]), and Schrödinger type equations ([19]), see also [43] and the references therein. It seems likely that these ideas, possibly combined with other techniques like Carleman estimates as in [70], will lead to new exiting results in the years to come. A final comment on null controllability for degenerate parabolic equations is in order. Indeed, many problems that are relevant for applications are described by parabolic equation equations in divergence form @ t u D r (A(x)ru)Cb(x)ruCc(t; x)uC f

in

QT ;

or in the general form @ t u D Tr [A(x)r 2 u]Cb(x)ruCc(t; x)uC f

4.

5.

6.

7.

8. 9.

10.

11.

12.

in

QT ; 13.

where A(x) is a symmetric matrix, positive definite in ˝ but possibly singular on  . For instance, degenerate parabolic equations arise in fluid dynamics as suitable transformations of the Prandtl equations, see, e. g., [94]. They can also be obtained as Kolmogorov equations of diffusions processes on domains that are invariant for stochastic flows, see, e. g., [52]. The latter interpretation explains why they have been applied to biological problems, such as gene frequency models for population genetics (see, e. g., the Wright–Fischer model studied in [111]). So far, null controllability properties of degenerate parabolic equations have been fully understood only in dimension one: for some kind of degeneracy, null controllability holds true (see [36] and [9]), but, in general, one can only expect regional null controllability (see [35]). Since very little is known on null controllability for degenerate parabolic equations in higher space dimensions, it is conceivable that such a topic will provide interesting problems for future developments. Bibliography 1. Alabau F (1999) Stabilisation frontière indirecte de systèmes faiblement couplés. C R Acad Sci Paris Sér I 328:1015–1020 2. Alabau F (2002) Indirect boundary stabilization of weakly coupled systems. SIAM J Control Optim 41(2):511–541 3. Alabau-Boussouira F (2003) A two-level energy method for indirect boundary observability and controllability of weakly

14. 15.

16.

17.

18.

19. 20. 21.

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Diagrammatic Methods in Classical Perturbation Theory GUIDO GENTILE Dipartimento di Matematica, Università di Roma Tre, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Examples Trees and Graphical Representation Small Divisors Multiscale Analysis Resummation Generalizations Conclusions and Future Directions Bibliography Glossary Dynamical system Let W  R N be an open set and f : W  R ! R N be a smooth function. The ordinary differential equation x˙ D f (x; t) on W defines a continuous dynamical system. A discrete dynamical system on W is defined by a map x ! x0 D F(x), with F depending smoothly on x. Hamiltonian system Let A  Rd be an open set and H : A  Rd  R ! R be a smooth function (A  Rd is called the phase space). Consider the system of ordinary differential equations q˙k D @H (q; p; t)/@p k , p˙ k D @H (q; p; t)/@q k , for k D 1; : : : ; d. The equations are called Hamilton equations, and H is called a Hamiltonian function. A dynamical system described by Hamilton equations is called a Hamiltonian system. Integrable system A Hamiltonian system is called integrable if there exists a system of coordinates (˛; A) 2 T d  Rd , called angle-action variables, such that in these coordinates the motion is (˛; A) ! (˛ C !(A)t; A), for some smooth function !(A). Hence in these coordinates the Hamiltonian function H depends only on the action variables, H D H0 (A). Invariant torus Given a continuous dynamical system we say that the motion occurs on an invariant d-torus if it takes place on a d-dimensional manifold and its position on the manifold is identified through a coordinate in T d . In an integrable Hamiltonian system all phase space is filled by invariant tori. In a quasi-integrable

system the KAM theorem states that most of the invariant tori persist under perturbation, in the sense that the relative Lebesgue measure of the fraction of phase space filled by invariant tori tends to 1 as the perturbation tends to disappear. The persisting invariant tori are slight deformations of the unperturbed invariant tori. Quasi-integrable system A quasi-integrable system is a Hamiltonian system described by a Hamiltonian function of the form H D H0 (A) C " f (˛; A), with (˛; A) angle-action variables, " a small real parameter and f periodic in its arguments ˛. Quasi-periodic motion Consider the motion ˛ ! ˛ C ! t on T 2 , with ! D (!1 ; !2 ). If !1 /!2 is rational, the motion is periodic, that is there exists T > 0 such that !1 T D !2 T D 0 mod 2. If !1 /!2 is irrational, the motion never returns to its initial value. On the other hand it densely fills T 2 , in the sense that it comes arbitrarily close to any point of T 2 . We say in that case that the motion is quasi-periodic. The definition extends to T d , d > 2: a linear motion ˛ ! ˛ C ! t on T d is quasi-periodic if the components of ! are rationally independent, that is if ! D !1 1 C  C!d d D 0 for  2 Zd if and only if  D 0 (a  b is the standard scalar product between the two vectors a, b). More generally we say that a motion on a manifold is quasi-periodic if, in suitable coordinates, it can be described as a linear quasi-periodic motion. The vector ! is usually called the frequency or rotation vector. Renormalization group By renormalization group one denotes the set of techniques and concepts used to study problems where there are some scale invariance properties. The basic mechanism consists in considering equations depending on some parameters and defining some transformations on the equations, including a suitable rescaling, such that after the transformation the equations can be expressed, up to irrelevant corrections, in the same form as before but with new values for the parameters. Torus The 1-torus T is defined as T D R/2Z, that is the set of real numbers defined modulo 2 (this means that x is identified with y if x  y is a multiple of 2). So it is the natural domain of an angle. One defines the d-torus T d as a product of d 1-tori, that is T d D T      T . For instance one can imagine T 2 as a square with the opposite sides glued together. Tree A graph is a collection of points, called nodes, and of lines which connect the nodes. A walk on the graph is a sequence of lines such that any two successive lines in the sequence share a node; a walk is nontrivial if it

Diagrammatic Methods in Classical Perturbation Theory

contains at least one line. A tree is a planar graph with no closed loops, that is, such that there is no nontrivial walk connecting any node to itself. An oriented tree is a tree with a special node such that all lines of the tree are oriented toward that node. If we add a further oriented line connecting the special node to another point, called the root, we obtain a rooted tree (see Fig. 1 in Sect. “Trees and Graphical Representation”). Definition of the Subject Recursive equations naturally arise whenever a dynamical system is considered in the regime of perturbation theory; for an introductory article on perturbation theory see  Perturbation Theory. A classical example is provided by Celestial Mechanics, where perturbation series, known as Lindstedt series, are widely used; see Gallavotti [21] and  Perturbation Theory in Celestial Mechanics. A typical problem in Celestial Mechanics is to study formal solutions of given ordinary differential equations in the form of expansions in a suitable small parameter, the perturbation parameter. In the case of quasi-periodic solutions, the study of the series, in particular of its convergence, is made difficult by the presence of the small divisors – which will be defined later on. Under some non-resonance condition on the frequency vector, one can show that the series are well-defined to any order. The first proof of such a property was given by Poincaré [53], even if the convergence of the series remained an open problem up to the advent of KAM theory – an account can be found in Gallavotti [17] and in Arnold et al. [2]; see also  Kolmogorov–Arnold–Moser (KAM) Theory. KAM is an acronym standing for Kolmogorov [47], Arnold [1] and Moser [51], who proved in the middle of last century the persistence of most of invariant tori for quasi-integrable systems. Kolmogorov and Arnold proofs apply to analytic Hamiltonian systems, while Moser’s approach deals also with the differentiable case; the smoothness condition on the Hamiltonian function was thereafter improved by Pöschel [54]. In the analytic case, the persisting tori turn out to be analytic in the perturbation parameter, as explicitly showed by Moser [52]. In particular, this means that the perturbation series not only are well-defined, but also converge. However, a systematic analysis with diagrammatic techniques started only recently after the pioneering, fundamental works by Eliasson [16] and Gallavotti [18], and were subsequently extended to many other problems with small divisors, including dynamical systems with infinitely many degrees of freedom, such as nonlinear partial differential equations, and non-Hamiltonian systems.

Some of these extensions will be discussed in Sect. “Generalizations”. From a technical point of view, the diagrammatic techniques used in classical perturbation theory are strongly reminiscent of the Feynman diagrams used in quantum field theory: this was first pointed out by Gallavotti [18]. Also the multiscale analysis used to control the small divisors is typical of renormalization group techniques, which have been successfully used in problems of quantum field theory, statistical mechanics and classical mechanics; see Gallavotti [20] and Gentile & Mastropietro [38] for some reviews. Note that there exist other renormalization group approaches to the study of dynamical systems, and of KAMlike problems in particular, different from that outlined in this article. By confining ourselves to the framework of problems of KAM-type, we can mention the paper by Bricmont et al. [11], which also stressed the similarity of the technique with quantum field theory, and the so called dynamical renormalization group method – see MacKay [50] – which recently produced rigorous proofs of persistence of quasi-periodic solutions; see for instance Koch [46] and Khanin et al. [45]. Introduction Consider the ordinary differential equation on Rd Du D G(u) C "F(u) ;

(1)

where D is a pseudo-differential operator and G, F are real analytic functions. Assume that (1) admits a solution u(0) (t) for " D 0, that is Du(0) D G(u(0) ). The problem we are interested in is to investigate whether there exists a solution of (1) which reduces to u(0) as " ! 0. For simplicity assume G D 0 in the following. The first attempt one can try is to look for solutions in the form of power series in ", u(t) D

1 X

" k u(k) (t) ;

(2)

kD0

which, inserted into (1), when equating the left and right hand sides order by order, gives the list of recursive equations Du(0) D 0, Du(1) D F(u(0) ), Du(2) D @u F(u(0) )u(1) , and so on. In general to order k  1 one has Du(k) D

k1 X 1 s @u F(u(0) ) s! sD0

X

u(k 1 ) : : : u(k s ) ;

(3)

k 1 CCk s Dk1 k i 1

where @su F, the sth derivative of F, is a tensor with s C 1 indices (s must be contracted with the vectors

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u(k 1 ) ; : : : ; u(k s ) ), and the term with s D 0 in the sum has to be interpreted as F(u(0) ) and appears only for k D 1. For instance for F(u) D u3 the first orders give Du(1) D u(0)3 ; Du(2) D 3u(0)2 u(1) ; Du(3) D 3u(0)2 u(2) C 3u(0) u(1)2 ;

(4)

Du(4) D 3u(0)2 u(3) C 6u(0) u(1) u(2) C u(1)3 ; as is easy to check. If the operator D can be inverted then the recursions (3) provide an algorithm to compute the functions u(k) (t). In that case we say that (2) defines a formal power series: by this we mean that the functions u(k) (t) are well-defined for all k  0. Of course, even if this can be obtained, there is still the issue of the convergence of the series that must be dealt with. Examples In this section we consider a few paradigmatic examples of dynamical systems which can be described by equations of the form (1). A Class of Quasi-integrable Hamiltonian Systems Consider the Hamiltonian system described by the Hamiltonian function H (˛; A) D 12 A2 C " f (˛) ;

Td

(5)

Rd

where (˛; A) 2  are angle-action variables, with T D R/2Z, f is a real analytic function, 2-periodic in each of its arguments, and A2 D A  A, if (here and henceforth)  denotes the standard scalar product in Rd , that is a  b D a1 b1 C    C ad bd . Assume also for simplicity that f is a trigonometric polynomial of degree N. The corresponding Hamilton equations are (we shorten @x D @/@x )

 (k) ˛¨ (k) D  " f (˛) :D 

k1 X 1 sC1 @˛ f (! t) s! sD0

X

˛ (k 1 ) : : : ˛ (k s ) : (7)

k 1 CCk s Dk1 k i 1

We look for a quasi-periodic solution of (6), that is a solution of the form ˛(t) D ! t C h(! t), with h(! t) D O("). We call h the conjugation function, as it “conjugates” (that is, maps) the perturbed solution ˛(t) to the unperturbed solution ! t. In terms of the function h (6) becomes h¨ D "@˛ f (! t C h) ;

(8)

where @˛ denotes derivative with respect to the argument. Then (8) can be more conveniently written in Fourier space, where the operator D acts as a multiplication operator. If we write h(! t) D

X

e i!t h ;

2Z d

h D

1 X

" k h(k) ;

which can be written as an equation involving only the angle variables: (6)

which is of the form (1) with u D ˛, G D 0, F D @˛ f , and D D d2 /dt 2 . For " D 0, ˛ (0) (t) D ˛0 C! t is a solution of (6) for any choice of ˛0 2 T d and ! 2 Rd . Take for simplicity ˛0 D

(9)

kD1

and insert (9) into (8) we obtain  (k) (!  )2 h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) :

˛˙ D @A H (˛; A) D A ; A˙ D @˛ H (˛; A) D "@˛ f (˛) ;

˛¨ D "@˛ f (˛) ;

0: we shall see that this choice makes sense. We say that for " D 0 the Hamiltonian function (5) describes a system of d rotators. We call ! the frequency vector, and we say that ! is irrational if its components are rationally independent, that is if !   D 0 for  2 Zd if and only if  D 0. For irrational ! the solution ˛ (0) (t) describes a quasi-periodic motion with frequency vector !, and it densely fills T d . Then (3) becomes

1 s! (10)

These equations are well-defined to all orders provided ["@˛ f (˛)](k)  D 0 for all  such that !   D 0. If ! is an irrational vector we need ["@˛ f (˛)](k) 0 D 0 for the equations to be well-defined. In that case the coefficients h0(k) are left undetermined, and we can fix them arbitrarily to vanish (which is a convenient choice). We shall see that under some condition on ! a quasiperiodic solution ˛(t) exists, and densely fills a d-dimensional manifold. The analysis carried out above for ˛0 D 0 can be repeated unchanged for all values of ˛0 2 T d : ˛0

Diagrammatic Methods in Classical Perturbation Theory

represents the initial phase of the solution, and by varying ˛0 we cover all the manifold. Such a manifold can be parametrized in terms of ˛0 , so it represents an invariant torus for the perturbed system. A Simplified Model with No Small Divisors Consider the same equation as (8) with D D d2 /dt 2 replaced by 1, that is h D "@˛ f (! t C h) :

(11)

Of course in this case we no longer have a differential equation; still, we can look again for quasi-periodic solutions h(! t) D O(") with frequency vector !. In such a case in Fourier space we have  (k) h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

1 (i0 )sC1 s!

 f0 h(k11 ) : : : h(ks s ) :

(12)

For instance if d D 1 and f (˛) D cos ˛ the equation, which is known as the Kepler equation, can be explicitly solved by the Lagrange inversion theorem [55], and gives

h(k)

D

8 ˆ < ˆ :

jj  k;

i(1) kC(k)/2 ; 2 ((k  )/2)!((k C )/2)!

 C k even ;

0;

otherwise :

k

(13) We shall show in Sect. “Small Divisors” that a different derivation can be provided by using the forthcoming diagrammatic techniques. The Standard Map Consider the finite difference equation D˛ D " sin ˛ ;

(14)

on T , where now D is defined by D˛( ) :D 2˛( )  ˛( By writing ˛ D Dh D " sin(

C !)  ˛(

k1

h(k) D

X 1 2 4 sin (!/2) sD0

X

X

k 1 CCk s Dk1 0 C1 CC s D  i 2Z k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) ;

1 s!

(18)

where 0 D ˙1 and f˙1 D 1/2. Note that (17) is a discrete dynamical system. However, when passing to Fourier space, (18) acquires the same form as for the continuous dynamical systems previously considered, simply with a different kernel for D. In particular if we replace D with 1 we recover the Kepler equation. The number ! is called the rotation number. We say that ! is irrational if the vector (2; !) is irrational according to the previous definition. Trees and Graphical Representation Take ! to be irrational. We study the recursive equations (  (k) h(k) D g(!  ) "@˛ f (˛)  ;  ¤ 0 ; (19) (k)   D0; "@˛ f (˛) 0 D 0 ; where the form of g depends on the particular model we are investigating. Hence one has either g(!  ) D (!  )2 or g(!  ) D 1 or g(!  ) D (2 sin(!/2))2 according to models described in Sect. “Examples”. For  ¤ 0 we have equations which express the co0 efficients h(k) ,  2 Zd , in terms of the coefficients h(k ) ,  2 Zd , with k 0 < k, provided the equations for  D 0 are satisfied for all k  1. Recursive equations, such as (19), naturally lead to a graphical representation in terms of trees. Trees

 !) :

(15)

C h( ), (14) becomes C h) ;

In other words, by writing x D C h( ) and y D ! C h( ) h( !), with ( ; !) solving (17) for " D 0, that is ( 0 ; ! 0 ) D ( C !; !), we obtain a closed-form equation for h, which is exactly (16). In Fourier space the operator D acts as D : e i ! 4 sin2 (!/2)e i , so that, by expanding h according to (9), we can write (16) as

(16)

which is the functional equation that must be solved by the conjugation function of the standard map ( x 0 D x C y C " sin x ; (17) y 0 D y C " sin x :

A connected graph G is a collection of points (nodes) and lines connecting all of them. Denote with N(G ) and L(G ) the set of nodes and the set of lines, respectively. A path between two nodes is the minimal subset of L(G ) connecting the two nodes. A graph is planar if it can be drawn in a plane without graph lines crossing. A tree is a planar graph G containing no closed loops. Consider a tree G with a single special node v0 : this introduces a natural partial ordering on the set of lines and nodes, and one can imagine that each line carries an arrow

129

130

Diagrammatic Methods in Classical Perturbation Theory

pointing toward the node v0 . We add an extra oriented line `0 exiting the special node v0 ; the added line will be called the root line and the point it enters (which is not a node) will be called the root of the tree. In this way we obtain a tree defined by N( ) D N(G ) and L( ) D L(G ) [ `0 . A labeled tree is a rooted tree together with a label function defined on the sets L( ) and N( ). We call equivalent two rooted trees which can be transformed into each other by continuously deforming the lines in the plane in such a way that the lines do not cross each other. We can extend the notion of equivalence also to labeled trees, by considering equivalent two labeled trees if they can be transformed into each other in such a way that the labels also match. In the following we shall deal mostly with nonequivalent labeled trees: for simplicity, where no confusion can arise, we call them just trees. Given two nodes v; w 2 N( ), we say that w  v if v is on the path connecting w to the root line. We can identify a line ` through the node v it exits by writing ` D `v . We call internal nodes the nodes such that there is at least one line entering them, and end-points the nodes which have no entering line. We denote with L( ), V ( ) and E( ) the set of lines, internal nodes and end-points, respectively. Of course N( ) D V( ) [ E( ). The number of unlabeled trees with k nodes (and hence with k lines) is bounded by 22k , which is a bound on the number of random walks with 2k steps [38]. For each node v denote by S(v) the set of the lines entering v and set sv D jS(v)j. Hence sv D 0 if v is an endnode, and sv  1 if v is an internal node. One has X X sv D sv D k  1 ; (20) v2N( )

v2V ( )

this can be easily checked by induction on the order of the tree. An example of unlabeled tree is represented in Fig. 1. For further details on graphs and trees we refer to the literature; cf. for instance Harary [43]. Labels and Diagrammatic Rules We associate with each node v 2 N( ) a mode label v 2 Zd , and with each line ` 2 L( ) a momentum label ` 2 Zd , with the constraint X X w D v C ` ; (21) `v D w2N( ) wv

`2S(v)

which represents a conservation rule for each node. Call T k; the set of all trees with k nodes and momentum  associated with the root line. We call k and  the order and the momentum of , respectively.

Diagrammatic Methods in Classical Perturbation Theory, Figure 1 An unlabeled tree with 17 nodes

Diagrammatic Methods in Classical Perturbation Theory, Figure 2 Graph element

Diagrammatic Methods in Classical Perturbation Theory, Figure 3 Graphical representation of the recursive equations

We want to show that trees naturally arise when studying (19). Let h(k) be represented with the graph element in Fig. 2 as a line with label  exiting from a ball with label (k). Then we can represent (19) graphically as depicted in Fig. 3. Simply represent each factor h(ki i ) on the right hand side as a graph element according to Fig. 2. The lines of all such graph elements enter the same node v0 . This is a graphical expedient to recall the conservation rule: the momentum  of the root line is the sum of the mode label 0 of the node v0 plus the sum of the momenta of the lines entering v0 . The first few orders k  4 are as depicted in Fig. 4. For each node the conservation rule (21) holds: for instance for k D 2 one has  D 1 C 2 , for k D 3 one has  D 1 C `1 and `1 D 2 C 3 in the first tree and  D 1 C 2 C 3 in

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 4 Trees of lower orders

the second tree, and so on. Moreover one has to sum over all possible choices of the labels v , v 2 N( ), which sum up to . Given any tree 2 T k; we associate with each node v 2 N( ) a node factor F v and with each line ` 2 L( ) a propagator g` , by setting Fv :D

1 (iv )s v C1 fv ; sv !

g` :D g(!  ` ) ;

and define the value of the tree as  Y   Y Fv g` : Val( ) :D v2N( )

(22)

(23)

`2L( )

The propagators g` are scalars, whereas each F v is a tensor with sv C 1 indices, which can be associated with the sv C 1 lines entering or exiting v. In (23) the indices of the tensors F v must be contracted: this means that if a node v is connected to a node v0 by a line ` then the indices of F v and Fv 0 associated with ` are equal to each other, and eventually one has to sum over all the indices except that associated with the root line. For instance the value of the tree in Fig. 4 contributing to h(2) is given by Val( ) D (i1 )2 f1 (i2 ) f2 g(!  )g(!  2 ) ;

with 1 C 2 D , while the value of the last tree in Fig. 4 contributing to h(4) is given by Val( ) D

(i1 )4 f1 (i2 ) f2 (i3 ) f3 (i4 ) f4 3!  g(!  )g(!  2 )g(!  3 )g(!  4 ) ;

with 1 C 2 C 3 C 4 D . It is straightforward to prove that one can write X h(k) D Val( ) ;  ¤0; k  1:

(24)

2T k;

This follows from the fact that the recursive equations (19) can be graphically represented through Fig. 3: one iterates the graphical representation of Fig. 3 until only graph elements of order k D 1 appear, and if is of order 1 (cf. Fig. 4) then Val( ) D (i) f g(!  ). Each line ` 2 L( ) can be seen as the root line of the tree consisting of all nodes and lines preceding `. The choice h0(k) D 0 for all k  1 implies that no line can have zero momentum: in other words we have ` ¤ 0 for all ` 2 L( ). Therefore in order to prove that (9) with h(k) given by (24) solves formally, that is order by order, the Eqs. (19),

131

132

Diagrammatic Methods in Classical Perturbation Theory

we have only to check that ["@˛ f (! t C h(! t))](k) 0 D 0 for all k  1. If we define g` D 1 for ` D 0, then also the second relation in (19) can be graphically represented as in Fig. 3 by setting  D 0 and requiring h0(k) D 0, which yields that the sum of the values of all trees on the right hand side must vanish. Note that this is not an equation to solve, but just an identity that has to be checked to hold at all orders. For instance for k D 2 (the case k D 1 is trivial) the identity ["@˛ f (! t C h(! t))](2) 0 D 0 reads (cf. the second line in Fig. 4) X (i1 )2 f1 (i2 ) f2 g(!  2 ) D 0 ; 1 C2 D0

which is found to be satisfied because the propagators are even in their arguments. Such a cancellation can be graphically interpreted as follows. Consider the tree with mode labels 1 and 2 , with 1 C 2 D 0: its value is (i1 )2 f1 (i2 ) f2 g(!  2 ). One can detach the root line from the node with mode label 1 and attach it to the node with mode label 2 , and reverse the arrow of the other line so that it points toward the new root line. In this way we obtain a new tree (cf. Fig. 5): the value of the new tree is (i1 ) f1 (i2 )2 f2 g(!  1 ), where g(!  1 ) D g(!  2 ) D g(!  2 ), so that the values of the two trees contain a common factor (i1 ) f1 (i2 ) f2 g(!  2 ) times an extra factor which is (i1 ) for the first tree and (i2 ) for the second tree. Hence the sum of the two values gives zero. The cancellation mechanism described above can be generalized to all orders. Given a tree one considers all trees which can be obtained by detaching the root line and attaching to the other nodes of the tree, and by reversing the arrows of the lines (when needed) to make them point toward the root line. Then one sums together the values of all the trees so obtained: such values contain a common factor times a factor iv , if v is the node which the root line exits (the only nontrivial part of the proof is to check that the combinatorial factors match each other: we refer to Gentile & Mastropietro [37] for details). Hence the sum gives zero, as the sum of all the mode labels vanishes. For instance for k D 3 the cancellation operates by considering the three trees in Fig. 5: such trees can be considered to be obtained from each other by shifting the root line and consistently reversing the arrows of the lines.

In such a case the combinatorial factors of the node factors are different, because in the second tree the node factor associated with the node with mode label 2 contains a factor 1/2: on the other hand if 1 ¤ 3 there are two nonequivalent trees with that shape (with the labels 1 and 3 exchanged between themselves), whereas if 1 D 3 there is only one such tree, but then the first and third trees are equivalent, so that only one of them must be counted. Thus, by using that 1 C 2 C 3 D 0 – which implies g(!  (2 C 3 )) D g(!  1 ) and g(!  (1 C 2 )) D g(!  3 ) – in all cases we find that the sum of the values of the trees gives a common factor (i1 ) f1 (i2 )2 f2 (i3 ) f3 g(!  3 )g(!  1 ) times a factor 1 or 1/2 times i(1 C2 C3 ), and hence vanishes: once more the property that g is even is crucial. Small Divisors We want to study the convergence properties of the series h(! t) D

X

e i!t h ;

h D

2Z d

" k h(k) ;

(25)

kD1

which has been shown to be well–defined as a formal power series for the models considered in Sect. “Examples”. Recall that the number of unlabeled trees of order k is bounded by 22k . To sum over the labels we can confine ourselves to the mode labels, as the momenta are uniquely determined by the mode labels. If f is a trigonometric polynomial of degree N, that is f D 0 for all  such that jj :D j1 j C    C jd j > N, we have that h(k) D 0 for all jj > kN (which can be easily proved by induction), and moreover we can bound the sum over the mode labels of any tree of order k by (2N C 1)d k . Finally we can bound Y Y jv js vC1  N s vC1  N 2k ; (26) v2N( )

v2N( )

because of (20). For the model (11), where g` D 1 in (22), we can bound ˇ ˇ ˇ X ˇˇ ˇ ˇ (k) ˇ ˇh(k) ˇ  22k (2N C 1)d k N 2k ˚ k ; ˇh ˇ  2Z d

(27) ˚ D max j f j ; jjN

which shows that the series (25) converges for " small enough, more precisely for j"j < "0 , with "0 :D C0 (4N 2 ˚(2N C 1)d )1 ;

Diagrammatic Methods in Classical Perturbation Theory, Figure 5 Trees to be considered together to prove that ["@f(˛)]20 D 0

1 X

(28)

where C0 D 1. Hence the function h(! t) in that case is analytic in ". For d D 1 and f (˛) D cos ˛, we can eas-

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 6 Trees to be considered together to prove that ["@f(˛)]30 D 0

ily provide an exact expression for the coefficients h(k) : all the computational difficulties reduce to a combinatorial check, which can be found in Gentile & van Erp [42], and the formula (13) is recovered. However for the models where g` ¤ 1, the situation is much more involved: the propagators can be arbitrarily close to zero for  large enough. This is the so-called small divisor problem. The series (25) is formally well–defined, assuming only an irrationality condition on !. But to prove the convergence of the series, we need a stronger condition. For instance one can require the standard Diophantine condition j!  j >

jj

8 ¤ 0 ;

(29)

for suitable positive constants and . For fixed  > d 1, the sets of vectors which satisfy (29) for some constant

> 0 has full Lebesgue measure in Rd [17]. We can also impose a weaker condition, known as the Bryuno condition, which can be expressed by requiring B(!) :D

1 X 1 1 log <1: k min0<jj2k j!  j 2

(30)

kD0

We call Diophantine vectors and Bryuno vectors the vectors satisfying (29) and (30), respectively. Note that any Diophantine vector satisfies (30). If we assume only analyticity on f – that is, we remove the assumption that f be a trigonometric polynomial, – then we need a Diophantine condition, such as (29) or (30), also to show that the series (25) are well-defined as formal power series: the condition is needed, as one can easily check, in order to sum over the mode labels. In the case of the standard map j!  j must be replaced with min p2Z j!  pj, and the function B(!) can be expressed in terms of the best approximants of the number ! [15]. For the models with small divisors considered in Sect. “Examples” convergence of the series (25) can be proved for ! satisfying (29) or (30). But this requires more work, and, in particular, a detailed discussion of the product of propagators in (24). In the following we shall consider the case of Diophantine vectors, by referring to the

bibliography for the Bryuno vectors (cf. Sect. “Generalizations”).

Multiscale Analysis Consider explicitly the case g(!  ) D (!  )2 in (19) and ! satisfying (29). We can introduce a label characterizing the size of the propagator: we say that  2 Zd is on scale (

n 1;

if 2n  j!  j < 2(n1) ;

n D0;

if  j!  j ;

(31)

where is the constant appearing in (29), and we say that a line ` has a scale label n` D n if ` is on scale n. If Nn ( ) denotes the number of lines ` 2 L( ) with scale n` D n, then we can bound in (23) 1 ˇ Y ˇ Y ˇ ˇ g` ˇ  2k 22nNn ( ) ; ˇ `2L( )

(32)

nD0

so that the problem is reduced to bounding N n ( ). The product of propagators gives problems when the small divisors “accumulate”. To make more precise the idea of accumulation we introduce the notion of cluster. Once all lines of a tree have been given their scale labels, for any n  0 we can identify the maximal connected sets of lines with scale not larger than n. If at least one among such lines has scale equal to n we say that the set is a cluster on scale n. For instance consider the tree in Fig. 1, and assign the mode labels to the nodes: this uniquely fixes the momenta, and hence the scale labels, of the lines. Suppose that we have found the scales as in Fig. 7a. Then a cluster decomposition as in Fig. 7b follows. Given a cluster T call L(T) the set of lines of contained in T, and denote by N(T) the set of nodes connected by such lines. We define k T D jN(T)j the order of the cluster T. Any cluster has either one or no exiting line, and can have an arbitrary number of entering lines. We call selfenergy clusters the clusters which have one exiting line and only one entering line and are such that both lines have the same momentum. This means that if T is a self-energy

133

134

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 7 Example of clusters with their scales

cluster and `1 and `2 are the lines entering and exiting T, respectively, then `1 D `2 , so that X

v D 0 :

(33)

v2N(T)

By definition of scale, both lines `1 and `2 have the same scale, say n, and that, by definition of cluster, one has n` < n for all ` 2 L(T). We define the value of the self-energy cluster T whose entering line has momentum  as the matrix VT (!  ) :D

 Y v2N(T)

Fv

 Y

 g` ;

(34)

`2L(T)

where all the indices of the node factors must be contracted except those associated with the line `1 entering T and with the line `2 exiting T. We can extend the notion of self-energy clusters also to a single node, by saying that v is a self-energy cluster if sv D 1 and the line entering v has the same momentum as the exiting line. In that case (34) has to be interpreted as VT (!  ) D Fv : in particular it is independent of !  . The simplest self-energy cluster one can think of consists of only one node v, but then (33) implies v D 0,

Diagrammatic Methods in Classical Perturbation Theory, Figure 8 Self-energy clusters of order k D 2

so that Fv D 0, see (22), hence the corresponding value is zero. Thus the simplest non-trivial self-energy clusters contain at least two nodes, and are represented by the clusters T 1 , T 2 and T 3 of Fig. 8. In all cases one has 1 C2 D 0. By using the definition (34) one has VT1 (x) D (i1 )2 f1 (i2 )2 f2 g(!  2 C x) and VT2 (x) D VT3 (x) D (i1 )3 f1 (i2 ) f2 g(!  2 )/2, with x D !  . Hence for x D 0 the sum of the three values gives zero. It is not difficult to see that also the first derivatives of the values of all self-energy clusters of order 2 sum up to zero, that is the sum of the values of all possible self-energy clusters of order k D 2 gives zero up to order x2 . (The only caveat is that, if we want to derive VT (x), the sharp multiscale decomposition in (31) can be a little annoying, hence it can be more convenient to replace it with a smooth decomposition through C 1 compact support functions; we refer to the literature for details). This property generalizes to all orders k, and the underlying cancellation mechanism is essentially the same that ensures the validity of the second relation in (19). The reason why it is important to introduce the selfenergy clusters is that if we could neglect them then the product of small divisors would be controlled. Indeed, let us denote by Rn ( ) the number of lines on scale n which do exit a self-energy cluster, and set Nn ( ) D N n ( ) 

Diagrammatic Methods in Classical Perturbation Theory

Rn ( ). Then an important result, known as the Siegel– Bryuno lemma, is that Nn ( )  c2n/ k ;

(35)

for some constant c, where k is the order of and  is the Diophantine exponent in (29). The bound (35) follows from the fact that if Nn ( ) ¤ 0 then Nn ( )  E(n; k) :D 2N k2(n2)/  1, which can be proved by induction on k as follows. Given a tree let `0 be its root line, let `1 ; : : : ; `m , m  0, be the lines on scales  n which are the closest to `0 , and let 1 ; : : : ; m the trees with root lines `1 ; : : : ; `m , respectively (cf. Fig. 9 – note that by construction all lines ` in the subgraph T have scales n` < n, so that if n`0  n then T is necessarily a cluster). If either `0 is not on scale n or it is on scale n but exits a self-energy cluster then Nn ( ) D Nn ( 1 ) C    C Nn ( m ) and the bound Nn ( )  E(n; k) follows by the inductive hypothesis. If `0 does not exit a self-energy cluster and n`0 D n then Nn ( ) D 1 C Nn ( 1 ) C    C Nn ( m ), and the lines `1 ; : : : ; `m enter a cluster T with k T D k  (k1 C    C k m ), where k1 ; : : : ; k m are the orders of 1 ; : : : ; m , respectively. If m  2 the bound Nn ( )  E(n; k) follows once more by the inductive hypothesis. If m D 0 then Nn ( ) D 1; on the other hand for `0 to be on scale n`0 D n one must have j!  `0 j < 2nC1 (see (31)), which, by the Diophantine condition (29), implies N k  j`0 j > 2(n1)/ , hence E(n; k) > 1. If m D 1 call 1 and 2 the momenta of the lines `0 and `1 , respectively. By construction T cannot be a self-energy cluster, hence 1 ¤ 2 , so that, by the Diophantine condition (29), 2nC2  j!  1 j C j!  2 j  j!  (1  2 )j

; > j1  2 j because n`0 D n and n`1  n. Thus, one has X N kT  jv j  j1  2 j > 2(n2)/ ;

hence T must contain “many nodes”. In particular, one finds also in this case Nn ( ) D 1 C Nn ( 1 )  1 C E(n; k1 )  1 C E(n; k)  E(n; k T )  E(n; k), where we have used that E(n; k T )  1 by (37). The argument above shows that small divisors can accumulate only by allowing self-energy clusters. That accumulation really occurs is shown by the example in Fig. 10, where a tree of order k containing a chain of p self-energy clusters is depicted. Assume for simplicity that k/3 is an integer: then if p D k/3 the subtree 1 with root line ` is of order k/3. If the line ` entering the rightmost self-energy cluster T p has momentum , also the lines exiting the p self-energy clusters have the same momentum . Suppose that jj  N k/3 and j!  j  /jj (this is certainly possible for some ). Then the value of the tree grows like a1k (k!) a 2 , for some constants a1 and a2 : a bound of this kind prevents the convergence of the perturbation series (25). If no self-energy clusters could occur (so that Rn ( ) D 0) the Siegel–Bryuno lemma would allow us to bound in (32) 1 Y

22nNn ( ) D

nD0

1 Y



22nNn ( )

nD0 1   X  exp C1 k n2n/  C2k ;

(38)

nD0

for suitable constants C1 and C2 . In that case convergence of the series for j"j < "0 would follow, with "0 defined as in (26) with C0 D 2 /C2 . However, there are selfenergy clusters and they produce factorials, as the example in Fig. 10 shows, so that we have to deal with them.

(36) Resummation (37)

v2N(T)

Let us come back to the Eq. (10). If we expand g(!  )["@˛ f (˛)](k)  in trees according to the diagrammatic rules described in Sect. “Trees and Graphical Representation”, we can distinguish between contributions in which the root line exits a self-energy cluster T, that we can write as X g(!  ) VT (!  )h(kk T ) ; (39) T : k T
Diagrammatic Methods in Classical Perturbation Theory, Figure 9 Construction for the proof of the Siegel–Bryuno lemma

and all the other contributions, that we denote by ["@˛ f (˛)](k)  . We can shift the contributions (39) to the left hand side of (10) and divide by g(!  ), so to obtain X VT (!  )h(kk T ) D(!  )h(k)  T : k T
D ["@˛ f (! t C h)](k)  :

(40)

135

136

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 10 Example of accumulation of small divisors because of the self-energy clusters

where D(!  ) D 1/g(!  ) D (!  )2 . By summing over k and setting M(!  ; ") D

1 X kD1

"k

X

VT (!  ) ;

(41)

T : k T Dk

 1 g` :D g [n] (!; ") D D(!  )  M[n] (!  ; ") ; (44)

then (40) gives D(!  )h D ["@˛ f (! t C h)] ;

with

D(!  ) :D D(!  )  M(!  ; ") :

(42)

The motivation for proceeding in this way is that, at the price of changing the operator D into D, hence of changing the propagators, lines exiting self-energy clusters no longer appear. Therefore, in the tree expansion of the right hand side of the equation, we have eliminated the selfenergy clusters, that is the source of the problem of accumulation of small divisors. Unfortunately the procedure described above has a problem: M(!  ; ") itself is a sum of self-energy clusters, which can still contain some other self-energy clusters on lower scales. So finding a good bound for M(!  ; ") could have the same problems as for the values of the trees. To deal with such a difficulty we modify the prescription by proceeding recursively, in the following sense. Let us start from the momenta  which are on scale n D 0. Since there are no self-energy clusters with exiting line on scale n D 0, for such  one has M(!  ; ") D 0. Next, we consider the momenta  which are on scale n D 1, and we can write (42), where now all self-energy clusters T whose values contribute to M(!  ; ") cannot contain any self-energy clusters, because the lines ` 2 T are on scale n` D 0. Then, we consider the momenta  which are on scale n D 2: again all the self-energy clusters contributing to M(!  ; ") do not contain any self-energy clusters, because the lines on scale n D 0; 1 cannot exit self-energy clusters by the construction of the previous step, and so on. The conclusion is that we have obtained a different expansion for h(! t), that we call a resummed series, h(! t) D

X 2Z d

e i!t h ;

where the self-energy clusters do not appear any more in the tree expansion and the propagators must be defined recursively: the propagator g` of a line ` on scale n` D n and momentum ` D  is the matrix

h D

1 X kD1

" k h[k] (") ; (43)

X

M[n] (!  ; ") :D

" k T VT (!  ) ;

(45)

T : n T
where the value VT (!  ) is written in accord with (34), with all the lines `0 2 L(T) on scales n`0 < n and the corresponding propagators g`0 expressed in terms of matrices M[n `0 ] (! `0 ; ") as in (44). By construction, the new propagators depend on ", so that the coefficients h[k] (") depend explicitly on ": hence (43) is not a power series expansion. The coefficients h[k] (") still admit a tree expansion h[k] (") D

X

Val( );

R 2T k;

Val( ) :D

 Y v2N( )

Fv

 ¤ 0; k  1 ;

 Y

 (46) g [n ` ] (!  ` ; ") ;

`2L( )

R which replaces (24). In particular T k; is defined as the set of renormalized trees of order k and momentum , where “renormalized” means that the trees do not contain any self-energy clusters. Now the propagators are matrices: the contractions of the indices in (46) yields that if ` connects v to v0 the indices of F v and Fv 0 associated with ` must be equal to the column and row indices of g` , respectively. R Since for any tree 2 T k; one has Nn ( ) D  Nn ( ), we can bound the product of propagators according to (36) provided the propagators on scale n can still be bounded proportionally to 22n . This is certainly not obvious, because of the extra term M[n] (!  ; ") appearing in (44). It is a remarkable cancellation that M[n] (x; ") vanishes up to second order, that is M[n] (x; ") D O(x 2 ), so that, by

Diagrammatic Methods in Classical Perturbation Theory

taking into account also that VT (x) ¤ 0 requires k T  2, we can write, for some constant C, [n] M[n] (x; ") D "2 x 2 M (x; ") ;

   [  M n](x; ")  C ; (47)

where k  k denotes – say – the uniform norm. The cancellation leading to (47) can be proved as discussed in Sect. “Multiscale Analysis” – where it has been explicitly proved for k D 2 in the absence of resummation. Now the propagators are matrices, but one can prove (by induction on n) that M[n] (x; ") D (M[n] (x; "))T D (M[n] (x; ")) , with T and † denoting transposition and adjointness, respectively, and this is enough to see that the same cancellation mechanism applies [23]. Thus (47) implies that  1     [n]   g (x; ") D  x 2  "2 x 2 M[n] (x; ")   2 ; (48)  x2  and the same argument as used in Sect. “Small Divisors” implies that the series in (43) for h converges for j"j < "0 , where "0 is defined as in (26) with C0 D 2 2 /C2 . Moreover (47) also implies that h(! t) is analytic in " (notwithstanding that the expansion is not a power expansion), so that we can say a posteriori that the original power series (25) also converges. Generalizations The case where the function G(u) in (1) does not vanish is notationally more involved, and it is explicitly discussed in Gentile [28]. Extensions of the results described above to Hamiltonian functions more general than (5) require only some notational complications, and can be found in Gentile & Mastropietro [37] for anisochronous systems and in Bartuccelli & Gentile [3] for isochronous systems. The case (5) with f a real analytic function was first discussed by using trees in Chierchia & Falcolini [12]. Lower-Dimensional Tori A tree formalism for hyperbolic lower-dimensional tori was introduced and studied in Gallavotti [19], Gentile [26] and Gallavotti, Gentile & Mastropietro [25], for systems consisting of a set of rotators interacting with a pendulum. In that case also the stable and unstable manifolds (whiskers) of the tori were studied with diagrammatic techniques. For Hamiltonian functions of the form (5) solutions of the form (43) describe quasi-periodic motions with

frequency vector ! on a d-dimensional manifold. If we fix t, say t D 0, and keep ˛0 as a parameter, we obtain a parametrization of the manifold in terms of ˛0 2 T d , hence the manifold is an invariant torus. We say that the torus is a maximal torus. We can study the problem of persistence of lowerdimensional tori by considering unperturbed solutions ˛(t) D ˛0 C ! t, where the components of ! are rationally dependent. For instance we can imagine that there exist s linearly independent vectors b 1 ; : : : ;b s 2 Zd such that !  b  k D 0 for k D 1; : : : ; s. In that case, we say that the unperturbed torus is a resonant torus of order s. We can imagine performing a linear change of variables which transforms the frequency vector into a new vector, that we ¯ 0), where !¯ 2 Rr still denote with !, such that ! D (!; s and 0 2 R , with r C s D d, and !¯ is irrational. This nat¯ ˇ), with ˛¯ 2 T r and urally suggests that we write ˛ D (˛; s ˇ 2 T . More generally, here and henceforth in this subsection for any vector v 2 Rd we denote by v¯ the vector in Rr whose components are the first r components of v. For instance for the initial phase ˛0 we write ˛0 D (˛¯ 0 ; ˇ0 ). In general the resonant torus is destroyed by the perturbation, and only some lower-dimensional tori persist under perturbation. We assume on !¯ one of the Diophantine conditions of Sect. “Small Divisors” in Rr , for instance j!¯  ¯ j > /j¯j for all ¯ 2 Zr , ¯ ¤ 0. To prove the existence of a maximal torus for the system with Hamiltonian function (5) we needed no condition on the perturbation f . On the contrary to prove existence of lower-dimensional tori, we need some nondegeneracy condition: by defining Z f0 (ˇ) D

Tr

d˛¯ ¯ ˇ) ; f (˛; (2)r

(49)

if @ˇ f0 (ˇ ) D 0 for some ˇ then we assume that the matrix @2ˇ f0 (ˇ ) is positive definite (more generally one could assume it to be non-singular, that is det @2ˇ f0 (ˇ ) ¤ 0). The formal analysis can be carried out as in Sect. “Trees and Graphical Representation”, with the only difference being that now the compatibility conditions ["@˛ f (˛)](k)  D 0 have to be imposed for all  such that ¯ D 0, because !   D !¯  ¯ . It turns out to be an identity only for the first r components (this is trivial for k D 1, whereas it requires some work for k > 1). For the last s components, for k D 1 it reads @ˇ f0 (ˇ0 ) D 0, hence it fixes ˇ0 to be a stationary point for f0 (ˇ), while for higher values of k it fixes the corrections of higher order of these values (to do this we need the non-degeneracy condition). Thus, we are free to choose only ˛¯ 0 as a free parameter, since the last s components of ˛0 have to be fixed.

137

138

Diagrammatic Methods in Classical Perturbation Theory

Clusters and self-energy clusters are defined as in Sect. “Multiscale Analysis”. Note that only the first r components ¯ of the momenta  intervene in the definition of the scales – again because !   D !¯  ¯ . In particular, in the definition of self-energy clusters, in (33) we must replace v with ¯ v . Thus, already to first order the value of a self-energy cluster can be non-zero: for k T D 1, that is for T containing only a node v with mode label (¯v ; ˜v ) D (0; ˜ v ), the matrix VT (x; ") is of the form  0 VT (x) D 0 (b˜ v ) i; j D e



0 ; b˜ v

i ˜ v ˇ0

(50)

(i˜v;i )(i˜v; j ) f(0;˜v ) ;

with i; j D r C 1; : : : ; d. If we sum over ˜ v 2 Zs and multiply times ", we obtain 

 0 0 ; 0 "B (51) X ˜ D e i ˇ0 (i˜ i )(i˜ j ) f(0;˜) D @ˇ i @ˇ j f0 (ˇ0 ) :

M0 :D B i; j

˜ 2Z s

The s  s block B is non-zero: in fact, the non-degeneracy condition yields that it is invertible. To higher orders one finds that the matrix M[n] (x; "), with x D !   D !¯  ¯ , is a self-adjoint matrix and M[n] (x; ") D (M[n] (x; "))T , as in the case of maximal tori. Moreover the corresponding eigenvalues [n] i (x; ") 2 x 2 ) for i D 1; : : : ; r and [n] (x; (x; ") D O(" satisfy [n] i i ") D O(") for i D r C 1; : : : ; d; this property is not trivial because of the off-diagonal blocks (which in general do not vanish at orders k  2), and to prove it one has to use the self-adjointness of the matrix M[n] (x; "). More 2 precisely one has [n] i (x; ") D "a i C O(" ) for i > r, where a rC1 ; : : : ; ad are the s eigenvalues of the matrix B in (51). From this point on the discussion proceeds in a very different way according to the sign of " (recall that we are assuming that a i > 0 for all i > r). 2 For " < 0 one has [n] i (x; ") D a i " C O(" ) < 0 for i > r, so that we can bound the last s eigenvalues of x 2  M[n] (x; ") with x2 , and the first r with x 2 /2 by the same argument as in Sect. “Resummation”. Hence we obtain easily the convergence of the series (43); of course, analyticity at the origin is prevented because of the condition " < 0. We say in that case that the lower-dimensional tori are hyperbolic. We refer to Gallavotti & Gentile [22] and Gallavotti et al. [23] for details. The case of elliptic lower-dimensional tori – that is " > 0 when a i > 0 for all i > r – is more difficult. Essentially the idea is as follows (we only sketch the strat-

egy: the details can be found in Gentile & Gallavotti [36]). One has to define the scales recursively, by using a variant, first introduced in Gentile [27], of the resummation technique described in Sect. “Resummation”. We say that  is on scale 0 if j!¯  ¯ j  and on scale [ 1] otherwise: for  on scale 0 we write (42) with M D M0 , as given in (51). This defines the propagators of the lines ` on scale n D 0 as 1  g` D g [0] (!  ` ) D (!¯  ¯ ` )2  M0 :

(52)

Denote by  i the eigenvalues of M 0 : given  on scale [ 1] wepsay that  is on scale 1 if 21  min iD1;:::;d pj(!¯  ¯ )2   i j, and on scale [ 2] if min iD1;:::;d j(!¯  ¯ )2   i j < 21 . For  on scale 1 we write (42) with M replaced by M[0] (!  ¯ ; "), which is given by M 0 plus the sum of the values of all self-energy clusters T on scale n T D 0. Then the propagators of the lines ` on scale n` D 1 is defined as  1 g` D g [1] (! ` ) D (!¯  ¯ ` )2  M[0] (!¯  ¯ ` ; ") : (53) [n] Call [n] i (x; ") the eigenvalues of M (x; "): given  on scale [ 2]qwe say that  is on scale 2 if 22 

¯  ¯ ; ")j, and on scale [ 3] j(!¯  ¯ )2  [0] i (! q ¯  ¯ ; ")j < 22 . For  if min iD1;:::;d j(!¯  ¯ )2  [0] i (! on scale 2 we write (42) with M replaced by M[1] (!¯  ¯ ; "), which is given by M[0] (!¯  ¯ ; ") plus the sum of the values of all self-energy clusters T on scale n T D 1. Thus, the propagators of the lines ` on scale n` D 2 will be defined as min iD1;:::;d

 1 ; (54) g` D g [2] (! ` ) D (!¯  ¯ ` )2  M[1] (!¯  ¯ ` ; ") and so on. The propagators are self-adjoint matrices, hence their norms can be bounded through the corresponding eigenvalues. In order to proceed as in Sects. “Multiscale Analysis” and “Resummation” we need some Diophantine conditions on these eigenvalues. We can assume for some  0 >  ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (!¯  ¯ ; ")jˇ >

i ˇ ˇ j¯j0

8¯ ¤ 0 ; (55)

for all i D 1; : : : ; d and n  0. These are known as the first Melnikov conditions. Unfortunately, things do not proceed so plainly. In order to prove a bound like (35), possibly with a different  0 replacing , we need to compare the propagators of the

Diagrammatic Methods in Classical Perturbation Theory

lines entering and exiting clusters T which are not selfenergy clusters. This requires replacing (36) with 2nC2  ˇ ˇ q q ˇ ˇ ˇ!¯  (¯1 ¯2 ) ˙ j[n] (!¯  ¯ 1 ; ")j ˙ j[n] (!¯  ¯ 2 ; ")jˇ i j ˇ ˇ

; > j¯1  ¯ 2 j0 (56) for all i; j D 1; : : : ; d and choices of the signs ˙, and hence introduces further Diophantine conditions, known as the second Melnikov conditions. The conditions in (56) turn out to be too many, because for all n  0 and all ¯ 2 Zr such that ¯ D ¯ 1  ¯ 2 there are infinitely many conditions to be considered, one per pair (¯1 ; ¯2 ). However we can impose both the condi¯  ¯ ; "), tions (55) and (56) not for the eigenvalues [n] i (! ¯ (") independent of  and then but for some quantities [n] i use the smoothness of the eigenvalues in x to control ¯  ¯ ; ") in terms of (!¯  ¯ )2  [n] (!¯  ¯ )2  [n] i (! i ("). Even¯ we have to tually, beside the Diophantine condition on !, impose the Melnikov conditions ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (")jˇ > 0 ; i ˇ ˇ j¯j ˇ ˇ (57) q q ˇ ˇ ˇj!¯  ¯ j ˙ j[n] (")j ˙ j[n] (")jˇ > 0 ; i j ˇ ˇ j¯j for all ¯ ¤ 0 and all n  0. Each condition in (57) leads us to eliminate a small interval of values of ". For the values of " which are left define h(! t) according to (43) and (46), with the new definition of the propagators. If " is small enough, say j"j < "0 , then the series (43) converges. Denote by E [0; "0 ] the set of values of " for which the conditions (57) are satisfied. One can prove that E is a Cantor set, that is a perfect, nowhere dense set. Moreover E has large relative Lebesgue measure, in the sense that lim

"!0

meas(E \ [0; "]) D1; "

(58)

provided  0 in (57) is large enough with respect to . The property (58) yields that, notwithstanding that we are eliminating infinitely many intervals, the measure of the union of all these intervals is small. If a i < 0 for all i > r we reason in the same way, simply exchanging the role of positive and negative ". On the contrary if a i D 0 for some i > r, the problem becomes much more difficult. For instance if s D 1 and arC1 D 0, then in general perturbation theory in " is not possible, not even at a formal level. However, under some conditions,

one can still construct fractional series in ", and prove that the series can be resummed [24]. Other Ordinary Differential Equations The formalism described above extends to other models, such as skew-product systems [29] and systems with strong damping in the presence of a quasi-periodic forcing term [32]. As an example of skew-product system  one can con sider the linear differential equation x˙ D A C " f (! t) x on SL(2; R), where  2 R, " is a small real parameter, ! 2 Rn is an irrational vector, and A; f 2 sl(2; R), with A is a constant matrix and f an analytic function periodic in its arguments. Trees for skew-products were considered by Iserles and Nørsett [44], but they used expansions in time, hence not suited for the study of global properties, such as quasi-periodicity. Quasi-periodically forced one-dimensional systems with strong damping are described by the ordinary differential equations x¨ C x˙ C g(x) D f (! t), where x 2 R, " D 1/ is a small real parameter, ! 2 Rn is irrational, and f ,g are analytic functions (g is the “force”), with f periodic in its arguments. We refer to the bibliography for details and results on the existence of quasi-periodic solutions. Bryuno Vectors The diagrammatic methods can be used to prove the any unperturbed maximal torus with frequency vector which is a Bryuno vector persists under perturbation for " small enough [31]. One could speculate whether the Bryuno condition (30) is optimal. In general the problem is open. However, in the case of the standard map – see Sect. “The Standard Map”, – one can prove [6,15] that, by considering the radius of convergence "0 of the perturbation series as a function of !, say "0 D 0 (!), then there exists a universal constant C such that jlog 0 (!) C 2B(!)j  C :

(59)

In particular this yields that the invariant curve with rotation number ! persists under perturbation if and only if ! is a Bryuno number. The proof of (59) requires the study of a more refined cancellation than that discussed in Sect. “Resummation”. We refer to Berretti & Gentile [6] and Gentile [28] for details. Extensions to Bryuno vectors for lower-dimensional tori can also be found in Gentile [31]. For the models considered in Sect. “Other Ordinary Differential Equations” we refer to Gentile [29] and Gentile et al. [33].

139

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Diagrammatic Methods in Classical Perturbation Theory

Partial Differential Equations Existence of quasi-periodic solutions in systems described by one-dimensional nonlinear partial differential equations (finite-dimensional tori in infinite-dimensional systems) was first studied by Kuksin [48], Craig and Wayne [14] and Bourgain [9]. In these systems, even the case of periodic solutions yields small divisors, and hence requires a multiscale analysis. The study of persistence of periodic solutions for nonlinear Schrödinger equations and nonlinear wave equations, with the techniques discussed here, can be found in Gentile & Mastropietro [39], Gentile et al. [40], and Gentile & Procesi [41]. The models are still described by (1), with G(u) D 0, but now D is given by D D @2t   C  in the case of the wave equation and by D D i@ t   C  in the case of the Schrödinger equation, where  is the Laplacian and  2 R. In dimension 1, one has  D @2x . If we look for periodic solutions with frequency ! it can be convenient to pass to Fourier space, where the operator D acts as   D : e i!ntCi mx ! ! 2 n2 C m2 C  e i!ntCi mx ; (60) for the wave equation; a similar expression holds for the Schrödinger equation. Therefore the kernel of D can be arbitrarily close to zero for n and m large enough. Then one can consider, say, (1) for x 2 [0; ] and F(u) D u3 , with Dirichlet boundary conditions u(0) D u() D 0, and study the existence of periodic solutions with frequency ! close to some of the unperturbed frequencies. We refer to the cited bibliography for results and proofs.

Conclusions and Future Directions The diagrammatic techniques described above have been applied also in cases where no small divisors appear; cf. Berretti & Gentile [4] and Gentile et al. [34]. Of course, such problems are much easier from a technical point of view, and can be considered as propaedeutic examples to become familiar with the tree formalism. Also the study of lower-dimensional tori becomes easy for r D 1 (periodic ¯ j  j!j ¯ for all ¯ ¤ 0, so solutions): in that case one has j!¯ ¯ 2k , that the product of the propagators is bounded by j!j and one can proceed as in Sect. “Small Divisors” to obtain analyticity of the solutions. In the case of hyperbolic lower-dimensional tori, if ! is a two-dimensional Diophantine vector of constant type (that is, with  D 1) the conjugation function h can be proved to be Borel summable [13]. Analogous considerations hold for the one-dimensional systems in the presence

of friction and of a quasiperiodic forcing term described in Sect. “Other Ordinary Differential Equations”; in that case one has Borel summability also for one-dimensional !, that is for periodic forcing [33]. It would be interesting to investigate whether Borel summability could be obtained for higher values of . Recently existence of finite-dimensional tori in the nonlinear Schrödinger equation in higher dimensions was proved by Bourgain [10]. It would be interesting to investigate how far the diagrammatic techniques extend to deal with such higher dimensional generalizations. The main problem is that (the analogues of) the second Melnikov conditions in (57) cannot be imposed. In certain cases the tree formalism was extended to non-analytic systems, such as some quasi-integrable systems of the form (5) with f in a class of Cp functions for some finite p [7,8]. However, up to exceptional cases, the method described here seems to be intrinsically suited in cases in which the vector fields are analytic. The reason is that in order to exploit the expansion (3), we need that F be infinitely many times differentiable and we need a bound on the derivatives. It is a remarkable property that the perturbation series can be given a meaning also in cases where the solutions are not analytic in ". An advantage of the diagrammatic method is that it allows rather detailed information about the solutions, hence it could be more convenient than other techniques to study problems where the underlying structure is not known or too poor to exploit general abstract arguments. Another advantage is the following. If one is interested not only in proving the existence of the solutions, but also in explicitly constructing them with any prefixed precision, this requires performing analytical or numerical computations with arbitrarily high accuracy. Then high perturbation orders have to be reached, and the easiest and most direct way to proceed is just through perturbation theory: so the approach illustrated here allows a unified treatment for both theoretical investigations and computational ones. The resummation technique described in Sect. “Resummation” can also be used for computational purposes. With respect to the naive power series expansion it can reduce the computation time required to approximate the solution within a prefixed precision. It can also provide accurate information on the analyticity properties of the solution. For instance, for the Kepler equation, Levi-Civita at the beginning of the last century described a resummation rule (see [49]), which gives immediately the radius of convergence of the perturbation series. Of course, in the case of small divisor problems, everything becomes much more complicated.

Diagrammatic Methods in Classical Perturbation Theory

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21. Gallavotti G (2006) Classical mechanics. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 1. Elsevier, Oxford 22. Gallavotti G, Gentile G (2002) Hyperbolic low-dimensional invariant tori and summations of divergent series. Comm Math Phys 227(3):421–460 23. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics. Springer, Berlin 24. Gallavotti G, Gentile G, Giuliani A (2006) Fractional Lindstedt series. J Math Phys 47(1):1–33 25. Gallavotti G, Gentile G, Mastropietro V (1999) A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems. J Math Phys 40(12):6430–6472 26. Gentile G (1995) Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dyn Stab Syst 10(3):269–308 27. Gentile G (2003) Quasi-periodic solutions for two-level systems. Comm Math Phys 242(1-2):221–250 28. Gentile G (2006) Brjuno numbers and dynamical systems. Frontiers in number theory, physics, and geometry. Springer, Berlin 29. Gentile G (2006) Diagrammatic techniques in perturbation theory. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 2. Elsevier, Oxford 30. Gentile G (2006) Resummation of perturbation series and reducibility for Bryuno skew-product flows. J Stat Phys 125(2): 321–361 31. Gentile G (2007) Degenerate lower-dimensional tori under the Bryuno condition. Ergod Theory Dyn Syst 27(2):427–457 32. Gentile G, Bartuccelli MV, Deane JHB (2005) Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J Math Phys 46(6):1–21 33. Gentile G, Bartuccelli MV, Deane JHB (2006) Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J Math Phys 47(7):1–10 34. Gentile G, Bartuccelli MV, Deane JHB (2007) Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies. Rev Math Phys 19(3):307–348 35. Gentile G, Cortez DA, Barata JCA (2005) Stability for quasiperiodically perturbed Hill’s equations. Comm Math Phys 260(2):403–443 36. Gentile G, Gallavotti G (2005) Degenerate elliptic tori. Comm Math Phys 257(2):319–362 37. Gentile G, Mastropietro V (1996) Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev Math Phys 8(3):393–444 38. Gentile G, Mastropietro V (2001) Renormalization group for one-dimensional fermions. A review on mathematical results. Renormalization group theory in the new millennium III. Phys Rep 352(4–6):273–437 39. Gentile G, Mastropietro V (2004) Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J Math Pures Appl 83(8):1019–1065 40. Gentile G, Mastropietro V, Procesi M (2005) Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Comm Math Phys 256(2):437– 490 41. Gentile G, Procesi M (2006) Conservation of resonant peri-

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Books and Reviews Berretti A, Gentile G (2001) Renormalization group and field theoretic techniques for the analysis of the Lindstedt series. Regul Chaotic Dyn 6(4):389–420 Gentile G (1999) Diagrammatic techniques in perturbations theory, and applications. Symmetry and perturbation theory. World Science, River Edge

Discrete Control Systems

Discrete Control Systems TAEYOUNG LEE1 , MELVIN LEOK2 , HARRIS MCCLAMROCH1 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA 2 Department of Mathematics, Purdue University, West Lafayette, USA Article Outline Glossary Definition of the Subject Introduction Discrete Lagrangian and Hamiltonian Mechanics Optimal Control of Discrete Lagrangian and Hamiltonian Systems Controlled Lagrangian Method for Discrete Lagrangian Systems Future Directions Acknowledgments Bibliography Glossary Discrete variational mechanics A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space. Lie group A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element. Symplectic A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position-momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume-preserving as well. Definition of the Subject Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader

field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control systems aims to address this deficiency by restricting the approximation to the choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure. Introduction The dynamics of Lagrangian and Hamiltonian systems have unique geometric properties; the Hamiltonian flow is symplectic, the total energy is conserved in the absence of non-conservative forces, and the momentum maps associated with symmetries of the system are preserved. Many interesting dynamics evolve on a non-Euclidean space. For example, the configuration space of a spherical pendulum is the two-sphere, and the configuration space of rigid body attitude dynamics has a Lie group structure, namely the special orthogonal group. These geometric features determine the qualitative behavior of the system, and serve as a basis for theoretical study. Geometric numerical integrators are numerical integration algorithms that preserve structures of the continuous dynamics such as invariants, symplecticity, and the configuration manifold (see [14]). The exact geometric properties of the discrete flow not only generate improved qualitative behavior, but also provide accurate and efficient numerical techniques. In this article, we view a geometric integrator as an intrinsically discrete dynamical system, instead of concentrating on the numerical approximation of a continuous trajectory.

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Numerical integration methods that preserve the simplecticity of a Hamiltonian system have been studied (see [28,36]). Coefficients of a Runge–Kutta method are carefully chosen to satisfy a simplecticity criterion and order conditions to obtain a symplectic Runge–Kutta method. However, it can be difficult to construct such integrators, and it is not guaranteed that other invariants of the system, such as a momentum map, are preserved. Alternatively, variational integrators are constructed by discretizing Hamilton’s principle, rather than discretizing the continuous Euler–Lagrange equation (see [31,34]). The resulting integrators have the desirable property that they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long times (see [2]). Lie group methods are numerical integrators that preserve the Lie group structure of the configuration space (see [18]). Recently, these two approaches have been unified to obtain Lie group variational integrators that preserve the geometric properties of the dynamics as well as the Lie group structure of the configuration space without the use of local charts, reprojection, or constraints (see [26,29,32]). Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. Discrete optimal control problems involve finding a control input for a discrete dynamic system such that an optimality objective is achieved with prescribed constraints. Optimality conditions are derived from the discrete equations of motion, described by a set of discrete equations. This approach is in contrast to traditional techniques where a discretization appears at the last stage to solve the optimality condition numerically. Discrete mechanics and optimal control approaches determine optimal control inputs and trajectories more accurately with less computational load (see [19]). Combined with an indirect optimization technique, they are substantially more efficient (see [17,23,25]). The geometric approach to mechanics can provide a theoretical basis for innovative control methodologies in geometric control theory. For example, these techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion. While the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. By applying geometric control algorithms to discrete mechanics that preserve geometric properties, we

obtain an exact numerical implementation of the geometric control theory. In particular, the method of controlled Lagrangian systems is based on the idea of adopting a feedback control to realize a modification of either the potential energy or the kinetic energy, referred to as potential shaping or kinetic shaping, respectively. These ideas are applied to construct a real-time digital feedback controller that stabilizes the inverted equilibrium of the cart-pendulum (see [10,11]). In this article, we will survey discrete Lagrangian and Hamiltonian mechanics, and their applications to optimal control and feedback control theory. Discrete Lagrangian and Hamiltonian Mechanics Mechanics studies the dynamics of physical bodies acting under forces and potential fields. In Lagrangian mechanics, the trajectory of the object is derived by finding the path that extremizes the integral of a Lagrangian over time, called the action integral. In many classical problems, the Lagrangian is chosen as the difference between kinetic energy and potential energy. The Legendre transformation provides an alternative description of mechanical systems, referred to as Hamiltonian mechanics. Discrete Lagrangian and Hamiltonian mechanics has been developed by reformulating the theorems and the procedures of Lagrangian and Hamiltonian mechanics in a discrete time setting (see, for example, [31]). Therefore, discrete mechanics has a parallel structure with the mechanics described in continuous time, as summarized in Fig. 1 for Lagrangian mechanics. In this section, we describe discrete Lagrangian mechanics in more detail, and we derive discrete Euler–Lagrange equations for several mechanical systems. Consider a mechanical system on a configuration space Q, which is the space of possible positions. The Lagrangian depends on the position and velocity, which are elements of the tangent bundle to Q, denoted by TQ. Let L : TQ ! R be the Lagrangian of the system. The discrete Lagrangian, Ld : Q  Q ! R is an approximation to the exact discrete Lagrangian, Z h (q ; q ) D L(q01 (t); q˙01 (t)) dt ; (1) Lexact 0 1 d 0

where q01 (0) D q0 , q01 (h) D q1 ; and q01 (t) satisfies the Euler–Lagrange equation in the time interval (0; h). A discrete action sum Gd : Q NC1 ! R, analogous to the action integral, is given by Gd (q0 ; q1 ; : : : ; q N ) D

N1 X kD0

Ld (q k ; q kC1 ) :

(2)

Discrete Control Systems

Discrete Control Systems, Figure 1 Procedures to derive continuous and discrete equations of motion

The discrete Hamilton’s principle states that ıGd D 0 for any ıq k , which yields the discrete Euler–Lagrange (DEL) equation, D2 Ld (q k1 ; q k ) C D1 Ld (q k ; q kC1 ) D 0 :

(3)

This yields a discrete Lagrangian flow map (q k1 ; q k ) 7! (q k ; q kC1 ). The discrete Legendre transformation, which from a pair of positions (q0 ; q1 ) gives a position-momentum pair (q0 ; p0 ) D (q0 ; D1 Ld (q0 ; q1 )) provides a discrete Hamiltonian flow map in terms of momenta. The discrete equations of motion, referred to as variational integrators, inherit the geometric properties of the continuous system. Many interesting Lagrangian and Hamiltonian systems, such as rigid bodies evolve on a Lie group. Lie group variational integrators preserve the nonlinear structure of the Lie group configurations as well as geometric properties of the continuous dynamics (see [29,32]). The basic idea for all Lie group methods is to express the update map for the group elements in terms of the group operation, g1 D g0 f0 ;

(4)

where g0 ; g1 2 G are configuration variables in a Lie group G, and f0 2 G is the discrete update represented

by a right group operation on g 0 . Since the group element is updated by a group operation, the group structure is preserved automatically without need of parametrizations, constraints, or re-projection. In the Lie group variational integrator, the expression for the flow map is obtained from the discrete variational principle on a Lie group, the same procedure presented in Fig. 1. But, the infinitesimal variation of a Lie group element must be carefully expressed to respect the structure of the Lie group. For example, it can be expressed in terms of the exponential map as ˇ d ˇˇ ıg D g exp  D g ; d ˇ D0 for a Lie algebra element  2 g. This approach has been applied to the rotation group SO(3) and to the special Euclidean group SE(3) for dynamics of rigid bodies (see [24,26,27]). Generalizations to arbitrary Lie groups gives the generalized discrete Euler–Poincaré (DEP) equation,   Te L f0  D2 Ld (g0 ; f0 )  Adf0  Te L f1  D2 Ld (g1 ; f1 ) (5) CTe L g 1  D1 Ld (g1 ; f1 ) D 0 ; for a discrete Lagrangian on a Lie group, Ld : G G ! R. Here L f : G ! G denotes the left translation map given by L f g D f g for f ; g 2 G, Tg L f : Tg G ! T f g G is the

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tangential map for the left translation, and Ad g : g ! g is the adjoint map. A dual map is denoted by a superscript  (see [30] for detailed definitions). We illustrate the properties of discrete mechanics using several mechanical systems, namely a mass-spring system, a planar pendulum, a spherical pendulum, and a rigid body. Example 1 (Mass-spring System) Consider a mass-spring system, defined by a rigid body that moves along a straight frictionless slot, and is attached to a linear spring. Continuous equation of motion: The configuration space is Q D R, and the Lagrangian L : R  R ! R is given by ˙ D L(q; q)

1 2 1 2 m q˙  q ; 2 2

(6)

where q 2 R is the displacement of the body measured from the point where the spring exerts no force. The mass of the body and the spring constant are denoted by m;  2 R, respectively. The Euler–Lagrange equation yields the continuous equation of motion. m q¨ C q D 0 :

(7)

Discrete equation of motion: Let h > 0 be a discrete time step, and a subscript k denotes the kth discrete variable at t D kh. The discrete Lagrangian Ld : RR ! R is an approximation of the integral of the continuous Lagrangian (6) along the solution of (7) over a time step. Here, we choose the following discrete Lagrangian. 

 q k C q kC1 q kC1  q k Ld (q k ; q kC1 ) D hL ; 2 h 1 h D m(q kC1  q k )2  (q k C q kC1 )2 : 2h 8 (8) Direct application of the discrete Euler–Lagrange equation to this discrete Lagrangian yields the discrete equations of motion. We develop the discrete equation of motion using the discrete Hamilton’s principle to illustrate the principles more explicitly. Let Gd : R NC1 ! R be the discrete P N1 action sum defined as Gd D kD0 Ld (q k ; q kC1 ), which approximates the action integral. The infinitesimal variation of the action sum can be written as ıGd D

 m h (q kC1  q k )  (q k C q kC1 ) h 4 kD0

 m h C ıq k  (q kC1  q k )  (q k C q kC1 ) : h 4 N1 X

ıq kC1

Since ıq0 D ıq N D 0, the summation index can be rewritten as ıGd D

N1 X kD1

n m ıq k  (q kC1  2q k C q k1 ) h 

o h (q kC1 C 2q k C q k1 ) : 4

From discrete Hamilton’s principle, ıGd D 0 for any ıq k . Thus, the discrete equation of motion is given by m (q kC1  2q k C q k1 ) h h (q kC1 C 2q k C q k1 ) D 0 : C 4

(9)

For a given (q k1 ; q k ), we solve the above equation to obtain q kC1 . This yields a discrete flow map (q k1 ; q k ) 7! (q k ; q kC1 ), and this process is repeated. The discrete Legendre transformation provides the discrete equation of motion in terms of the velocity as     h2  h2  ˙ q kC1 D h q k C 1  qk ; 1C 4m 4m h h q˙ kC1 D q˙ k  qk  q kC1 : 2m 2m

(10) (11)

For a given (q k ; q˙ k ), we compute q kC1 and q˙ kC1 by (10) and (11), respectively. This yields a discrete flow map (q k ; q˙k ) 7! (q kC1 ; q˙kC1 ). It can be shown that this variational integrator has second-order accuracy, which follows from the fact that the discrete action sum is a second-order approximation of the action integral. Numerical example: We compare computational properties of the discrete equations of motion given by (10) and (11) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg,  D 1 kg/s2 so that the natural p frequency is 1 rad/s. The initial conditions are q0 D 2 m, q˙0 D 0, and the total energy is E D 1 Nm. The simulation time is 200 sec, and the stepsize h D 0:035 of the discrete equations of motion is chosen such that the CPU times are the same for both methods. Figure 2a shows the computed total energy. The variational integrator preserves the total energy well. The mean variation is 2:73271013 Nm. But, there is a notable dissipation of the computed total energy for the Runge–Kutta method Example 2 (Planar Pendulum) A planar pendulum is a mass particle connected to a frictionless, one degree-offreedom pivot by a rigid massless link under a uniform gravitational potential. The configuration space is the one-

Discrete Control Systems

Discrete Control Systems, Figure 2 Computed total energy (RK45: blue, dotted, VI: red, solid)

sphere S1 D fq 2 R2 j kqk D 1g. While it is common to parametrize the one-sphere by an angle, we develop parameter-free equations of motion in the special orthogonal group SO(2), which is a group of 22 orthogonal matrices with determinant 1, i. e. SO(2) D fR 2 R22 j RT R D I22 ; det[R] D 1g. SO(2) is diffeomorphic to the one-sphere. It is also possible to develop global equations of motion on the one-sphere directly, as shown in the next example, but here we focus on the special orthogonal group in order to illustrate the key steps to develop a Lie group variational integrator. We first exploit the basic structures of the Lie group SO(2). Define a hat map ˆ, which maps a scalar ˝ to a 22 skew-symmetric matrix ˝ˆ as ˝ˆ D



0 ˝

 ˝ : 0

The set of 22 skew-symmetric matrices forms the Lie algebra so(2). Using the hat map, we identify so(2) with R. An inner product on so(2) can be induced from the inner product on R as h˝ˆ 1 ; ˝ˆ 2 i D 1/2tr[˝ˆ 1T ˝ˆ 2 ] D ˝1 ˝2 for any ˝1 ; ˝2 2 R. The matrix exponential is a local diffeomorphism from so(2) to SO(2) given by

cos ˝ exp ˝ˆ D sin ˝

 sin ˝ cos ˝



ˆ D 1 ml 2 ˝ 2 C mg l e T Re2 L(R; ˝) 2 2 (13) 1 2 ˆ ˆ D ml h˝; ˝i C mg l e2T Re2 ; 2 where the constant g 2 R is the gravitational acceleration. The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The second expression is used to define a discrete Lagrangian later. We choose the bases of the inertial frame and the body-fixed frame such that the unit vector along the gravity direction in the inertial frame, and the unit vector along the pendulum axis in the body-fixed frame are represented by the same vector e2 D [0; 1] 2 R2 . Thus, for example, the hanging attitude is represented by R D I22 . Here, the rotation matrix R 2 SO(2) represents the linear transformation from a representation of a vector in the body-fixed frame to the inertial frame. Since the special orthogonal group is not a linear vector space, the expression for the variation should be carefully chosen. The infinitesimal variation of a rotation matrix R 2 SO(2) can be written in terms of its Lie algebra element as ˇ ˇ ˇ d ˇˇ R exp ˆ D Rˆ exp ˆ ˇˇ D Rˆ ; (14) ıR D d ˇ D0

:

The kinematics equation for R 2 SO(2) can be written in terms of a Lie algebra element as R˙ D R˝ˆ :

as

(12)

Continuous equations of motion: The Lagrangian for a planar pendulum L : SO(2)so(2) ! R can be written

D0

where  2 R so that ˆ 2 so(2). The infinitesimal variation of the angular velocity is induced from this expression and (12) as ı ˝ˆ D ıRT R˙ C RT ı R˙ ˆ˙ ˆ T R˙ C RT (R˙ ˆ C R) D (R)

(15)

D ˆ ˝ˆ C ˝ˆ ˆ C ˆ˙ D ˆ˙ ; where we used the equality of mixed partials to compute ı R˙ as d/dt(ıR).

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RT ˆ Define the action integral to be G D 0 L(R; ˝)dt. The infinitesimal variation of the action integral is obtained by using (14) and (15). Hamilton’s principle yields the following continuous equations of motion. g ˝˙ C e2T Re1 D 0 ; (16) l R˙ D R˝ˆ :

(17)

where we use the fact that F ˆ F T D ˆ for any F 2 SO(2) and ˆ 2 so(2). Define an action sum Gd : SO(2) NC1 ! R as Gd D P N1 kD0 Ld (R k ; F k ). Using (21) and (22), the variation of the action sum is written as ıGd D

N1 X kD1

If we parametrize the rotation matrix as

 cos  sin RD ; sin cos

  1 T ml 2 F k1  F k1 2h 

these equations are equivalent to g (18) ¨ C sin D 0 : l Discrete equations of motion: We develop a Lie group variational integrator on SO(2). Similar to (4), define F k 2 SO(2) such that R kC1 D R k F k :

(19)

Thus, F k D RTk R kC1 represents the relative update between two integration steps. If we find F k 2 SO(2), the orthogonal structure is preserved through (19) since multiplication of orthogonal matrices is also orthogonal. This is a key idea of Lie group variational integrators. Define the discrete Lagrangian Ld : SO(2)SO(2) ! R to be 1 ml 2 hF k  I22 ; F k  I22 i Ld (R k ; F k ) D 2h h h C mg l e2T R k e2 C mg l e2T R kC1 e2 2 2 (20) 1 h 2 D ml tr[I22  F k ] C mg l e2T R k e2 2h 2 h C mg l e2T R kC1 e2 ; 2 which is obtained by an approximation h˝ˆ k ' RT (R kC1

2

  1  F k  F kT  hmg l e2T R k e1 ; ˆ k : 2h

From the discrete Hamilton’s principle, ıGd D 0 for any ˆ k . Thus, we obtain the Lie group variational integrator on SO(2) as 

3

   2h2 g T T  F k  F kT  F kC1  F kC1 e R kC1 e1 D 0; (23) l 2 R kC1 D R k F k :

(24)

For a given (R k ; F k ) and R kC1 D R k F k , (23) is solved to find F kC1 . This yields a discrete map (R k ; F k ) 7! (R kC1 ; F kC1 ). If we parametrize the rotation matrices R and F with and  and if we assume that   1, these equations are equivalent to 1 hg ( kC1  2 k C k ) C sin k D 0 : h l The discrete version of the Legendre transformation provides the discrete Hamiltonian map as follows. F k  F kT D 2h˝ˆ 

2

h2 g T e R k e1 ; l 2

R kC1 D R k F k ; ˝ kC1 D ˝ k 

(25) (26)

hg T hg T e2 R k e1  e R kC1 e1 : 2l 2l 2

(27)

k

 R k ) D F k  I22 , applied to the continuous Lagrangian given by (13). As for the continuous time case, expressions for the infinitesimal variations should be carefully chosen. The infinitesimal variation of a rotation matrix is the same as (14), namely ıR k D R k ˆ k ;

(21)

for  k 2 R, and the constrained variation of F k is obtained from (19) as ıF k D ıRTk R kC1 C RTk ıR kC1 D ˆ k F k C F k ˆ kC1 D F k (ˆ  kC1  F kT ˆ k F k ) (22)  kC1  ˆ k ) ; D F k (ˆ

For a given (R k ; ˝ k ), we solve (25) to obtain F k . Using this, (R kC1 ; ˝ kC1 ) is obtained from (26) and (27). This yields a discrete map (R k ; ˝ k ) 7! (R kC1 ; ˝ kC1 ). Numerical example: We compare the computational properties of the discrete equations of motion given by (25)–(27) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg, l D 9:81 m. The initial conditions are 0 D /2 rad, ˝ D 0, and the total energy is E D 0 Nm. The simulation time is 1000 sec, and the step-size h D 0:03 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 2b shows the computed total energy for both methods. The variational integrator preserves the total energy well. There is no drift in the computed total energy, and the

Discrete Control Systems

mean variation is 1:0835102 Nm. But, there is a notable dissipation of the computed total energy for the Runge– Kutta method. Note that the computed total energy would further decrease as the simulation time increases.

Since q˙ D !  q for some angular velocity ! 2 R3 satisfying !  q D 0, this can also be equivalently written as

Example 3 (Spherical Pendulum) A spherical pendulum is a mass particle connected to a frictionless, two degree-offreedom pivot by a rigid massless link. The mass particle acts under a uniform gravitational potential. The configuration space is the two-sphere S2 D fq 2 R3 j kqk D 1g. It is common to parametrize the two-sphere by two angles, but this description of the spherical pendulum has a singularity. Any trajectory near the singularity encounters numerical ill-conditioning. Furthermore, this leads to complicated expressions involving trigonometric functions. Here we develop equations of motion for a spherical pendulum using the global structure of the two-sphere without parametrization. In the previous example, we develop equations of motion for a planar pendulum using the fact that the one-sphere S1 is diffeomorphic to the special orthogonal group SO(2). But, the two-sphere is not diffeomorphic to a Lie group. Instead, there exists a natural Lie group action on the two-sphere. That is the 3-dimensional special orthogonal group SO(3), a group of 33 orthogonal matrices with determinant 1, i. e. SO(3) D fR 2 R33 j RT R D I33 ; det[R] D 1g. The special orthogonal group SO(3) acts on the two-sphere in a transitive way; for any q1 ; q2 2 S2 , there exists a R 2 SO(3) such that q2 D Rq1 . Thus, the discrete update for the two-sphere can be expressed in terms of a rotation matrix as (19). This is a key idea to develop a discrete equations of motion for a spherical pendulum. Continuous equations of motion: Let q 2 S2 be a unit vector from the pivot point to the point mass. The Lagrangian for a spherical pendulum can be written as

(32)

1 ˙ D ml 2 q˙  q˙ C mg l e3  q ; L(q; q) 2

(28)

where the gravity direction is assumed to be e3 D [0; 0; 1] 2 R3 . The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The infinitesimal variation of the unit vector q can be written in terms of the vector cross product as ıq D   q ;

(29)

where  2 R3 is constrained to be orthogonal to the unit vector, i. e.  q D 0. Using this expression for the infinitesimal variation, Hamilton’s principle yields the following continuous equations of motion. g ˙ C (q  (q  e3 )) D 0 : q¨ C (q˙  q)q l

(30)

g q  e3 D 0 ; l q˙ D !  q :

!˙ 

(31)

These are global equations of motion for a spherical pendulum; these are much simpler than the equations expressed in term of angles, and they have no singularity. Discrete equations of motion: We develop a variational integrator for the spherical pendulum defined on S2 . Since the special orthogonal group acts on the two-sphere transitively, we can define the discrete update map for the unit vector as q kC1 D F k q k

(33)

for F k 2 SO(3). The rotation matrix F k is not uniquely defined by this condition, since exp(qˆ k ), which corresponds to a rotation about the qk direction fixes the vector qk . Consequently, if F k satisfies (33), then F k exp(qˆ k ) does as well. We avoid this ambiguity by requiring that F k does not rotate about qk , which can be achieved by letting F k D exp( fˆk ), where f k  q k D 0. Define a discrete Lagrangian Ld : S2  S2 ! R to be Ld (q k ; q kC1 ) D

1 ml 2 (q kC1  q k )  (q kC1  q k ) 2h h h C mg l e3  q k C mg l e3  q kC1 : 2 2

The variation of qk is the same as (29), namely ıq k D  k  q k

(34)

for  k 2 R3 with a constraint  k  q k D 0. Using this discrete Lagrangian and the expression for the variation, discrete Hamilton’s principle yields the following discrete equations of motion for a spherical pendulum.   h2 g q k  e3  q k q kC1 D h! k C 2l 2 !1/2    h2 g  C 1   h! k C qk ; q k  e3   2l hg hg ! kC1 D ! k C q k  e3 C q kC1  e3 : 2l 2l

(35) (36)

Since an explicit solution for F k 2 SO(3) can be obtained in this case, the rotation matrix F k does not appear in the equations of motion. This variational integrator on S2 exactly preserves the unit length of qk , the constraint

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Discrete Control Systems, Figure 3 Numerical simulation of a spherical pendulum (RK45: blue, dotted, VI: red, solid)

q k  ! k D 0, and the third component of the angular velocity ! k  e3 which is conserved since gravity exerts no moment along the gravity direction e3 . Numerical example: We compare the computational properties of the discrete equations of motion given by (35) and (36) with a 4(5)th order variable step size Runge–Kutta method for (31) and (32). We choose m D 1p kg, l D 9:81 m. The p initial conditions are q0 D [ 3/2; 0; 1/2], !0 D 0:1[ 3; 0; 3] rad/sec, and the total energy is E D 0:44 Nm. The simulation time is 200 sec, and the step-size h D 0:05 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 3 shows the computed total energy and the unit length errors. The variational integrator preserves the total energy and the structure of the two-sphere well. The mean total energy deviation is 1:5460104 Nm, and the mean unit length error is 3:2476  1015 . But, there is a notable dissipation of the computed total energy for the Runge–Kutta method. The Runge–Kutta method also fails to preserve the structure of the two-sphere. The mean unit length error is 1:0164102 . Example 4 (Rigid Body in a Potential Field) Consider a rigid body under a potential field that is dependent on the position and the attitude of the body. The configuration space is the special Euclidean group, which is a semidirect product of the special orthogonal group and Eus R3 . clidean space, i. e. SE(3) D SO(3)  Continuous equations of motion: The equations of motion for a rigid body can be developed either from Hamilton’s principle (see [26]) in a similar way as Example 2, or directly from the generalized discrete Euler–Poincaré equation given at (5). Here, we summarize the results. Let m 2 R and J 2 R33 be the mass and the moment of inertia matrix of a rigid body. For (R; x) 2 SE(3), the linear

transformation from the body-fixed frame to the inertial frame is denoted by the rotation matrix R 2 SO(3), and the position of the mass center in the inertial frame is denoted by a vector x 2 R3 . The vectors ˝; v 2 R3 are the angular velocity in the body-fixed frame, and the translational velocity in the inertial frame, respectively. Suppose that the rigid body acts under a configuration-dependent potential U : SE(3) ! R. The continuous equations of motion for the rigid body can be written as R˙ D R˝ˆ ;

(37)

x˙ D v ;

(38)

J ˝˙ C ˝  J˝ D M ;

(39)

@U ; @x

(40)

m˙v D 

where the hat map ˆ is an isomorphism from R3 to 3  3 skew-symmetric matrices so(3), defined such that xˆ y D x  y for any x; y 2 R3 . The moment due to the potential M 2 R3 is obtained by the following relationship: T ˆ D @U R  RT @U : M @R @R

(41)

The matrix @U/@R 2 R33 is defined such that [@U/@R] i j D@U/@[R] i j for i; j 2 f1; 2; 3g, where the i, jth element of a matrix is denoted by [] i j . Discrete equations of motion: The corresponding discrete equations of motion are given by

b

h J ˝k C

h2 ˆ M k D F k Jd  Jd F kT ; 2

R kC1 D R k F k ;

(42) (43)

Discrete Control Systems

Discrete Control Systems, Figure 4 Numerical simulation of a dumbbell rigid body (LGVI: red, solid, RK45 with rotation matrices: blue, dash-dotted, RK45 with quaternions: black, dotted)

h2 @U k ; 2m @x k h h D F kT J˝ k C F kT M k C M kC1 ; 2 2 h Uk h @U kC1 D mv k   ; 2 @x k 2 @x kC1

x kC1 D x k C hv k 

(44)

J˝ kC1

(45)

mv kC1

(46)

where Jd 2 R33 is a non-standard moment of inertia matrix defined as Jd D1/2tr[J]I33  J. For a given (R k ; x k ; ˝ k ; v k ), we solve the implicit Eq. (42) to find F k 2 SO(3). Then, the configuration at the next step R kC1 ;x kC1 is obtained by (43) and (44), and the moment and force M kC1;(@U kC1 )/(@x kC1 ) can be computed. Velocities ˝ kC1 ;v kC1 are obtained from (45) and (46). This defines a discrete flow map, (R k ; x k ; ˝ k ; v k ) 7! (R kC1 ; x kC1 ; ˝ kC1 ; v kC1 ), and this process can be repeated. This Lie group variational integrator on SE(3) can be generalized to multiple rigid bodies acting under their mutual gravitational potential (see [26]). Numerical example: We compare the computational properties of the discrete equations of motion given by (42)–(46) with a 4(5)th order variable step size Runge– Kutta method for (37)–(40). In addition, we compute the attitude dynamics using quaternions on the unit threesphere S3 . The attitude kinematics Eq. (37) is rewritten in terms of quaternions, and the corresponding equations are integrated by the same Runge–Kutta method. We choose a dumbbell spacecraft, that is two spheres connected by a rigid massless rod, acting under a central gravitational potential. The resulting system is a restricted full two body problem. The dumbbell spacecraft model has an analytic expression for the gravitational potential, resulting in a nontrivial coupling between the attitude dynamics and the orbital dynamics. As shown in Fig. 4a, the initial conditions are chosen such that the resulting motion is a near-circular orbit

combined with a rotational motion. Figure 4b and c show the computed total energy and the orthogonality errors of the rotation matrix. The Lie group variational integrator preserves the total energy and the Lie group structure of SO(3). The mean total energy deviation is 2:5983104 , and the mean orthogonality error is 1:8553  1013 . But, there is a notable dissipation of the computed total energy and the orthogonality error for the Runge–Kutta method. The mean orthogonality errors for the Runge– Kutta method are 0.0287 and 0.0753, respectively, using kinematics equation with rotation matrices, and using the kinematics equation with quaternions. Thus, the attitude of the rigid body is not accurately computed for Runge– Kutta methods. It is interesting to see that the Runge– Kutta method with quaternions, which is generally assumed to have better computational properties than the kinematics equation with rotation matrices, has larger total energy error and orthogonality error. Since the unit length of the quaternion vector is not preserved in the numerical computations, orthogonality errors arise when converted to a rotation matrix. This suggests that it is critical to preserve the structure of SO(3) in order to study the global characteristics of the rigid body dynamics. The importance of simultaneously preserving the symplectic structure and the Lie group structure of the configuration space in rigid body dynamics can be observed numerically. Lie group variational integrators, which preserve both of these properties, are compared to methods that only preserve one, or neither, of these properties (see [27]). It is shown that the Lie group variational integrator exhibits greater numerical accuracy and efficiency. Due to these computational advantages, the Lie group variational integrator has been used to study the dynamics of the binary near-Earth asteroid 66391 (1999 KW 4 ) in joint work between the University of Michigan and the Jet Propulsion Laboratory, NASA (see [37]).

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Optimal Control of Discrete Lagrangian and Hamiltonian Systems Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. The equations of motion for a system are constrained by Lagrange multipliers, and necessary conditions for optimality is obtained by the calculus of variations. The solution for the corresponding two point boundary value problem provides the optimal control input. Alternatively, a sub-optimal control law is obtained by approximating the control input history with finite data points. Discrete optimal control problems involve finding a control input for a given system described by discrete Lagrangian and Hamiltonian mechanics. The control inputs are parametrized by their values at each discrete time step, and the discrete equations of motion are derived from the discrete Lagrange–d’Alembert principle [21], ı

N1 X kD0

Ld (q k ; q kC1 ) D

N1 X

[Fd (q k ; q kC1 )  ıq k

kD0

C FdC (q k ; q kC1 )  ıq kC1 ] ; which modifies the discrete Hamilton’s principle by taking into account the virtual work of the external forces. Discrete optimal control is in contrast to traditional techniques such as collocation, wherein the continuous equations of motion are imposed as constraints at a set of collocation points, since this approach induces constraints on the configuration at each discrete timestep. Any optimal control algorithm can be applied to the discrete Lagrangian or Hamiltonian system. For an indirect method, our approach to a discrete optimal control problem can be considered as a multiple stage variational problem. The discrete equations of motion are derived by the discrete variational principle. The corresponding variational integrators are imposed as dynamic constraints to be satisfied by using Lagrange multipliers, and necessary conditions for optimality, expressed as discrete equations on multipliers, are obtained from a variational principle. For a direct method, control inputs can be optimized by using parameter optimization tools such as a sequential quadratic programming. The discrete optimal control can be characterized by discretizing the optimal control problem from the problem formulation stage. This method has substantial computational advantages when used to find an optimal control law. As discussed in the previous section, the discrete dynamics are

more faithful to the continuous equations of motion, and consequently more accurate solutions to the optimal control problem are obtained. The external control inputs break the Lagrangian and Hamiltonian system structure. For example, the total energy is not conserved for a controlled mechanical system. But, the computational superiority of the discrete mechanics still holds for controlled systems. It has been shown that the discrete dynamics is more reliable even for controlled system as it computes the energy dissipation rate of controlled systems more accurately (see [31]). For example, this feature is extremely important in computing accurate optimal trajectories for long term spacecraft attitude maneuvers using low energy control inputs. The discrete dynamics does not only provide an accurate optimal control input, but also enables us to find it efficiently. For the indirect optimal control approach, optimal solutions are usually sensitive to a small variation of multipliers. This causes difficulties, such as numerical ill-conditioning, when solving the necessary conditions for optimality expressed as a two point boundary value problem. Sensitivity derivatives along the discrete necessary conditions do not have numerical dissipation introduced by conventional numerical integration schemes. Thus, they are numerically more robust, and the necessary conditions can be solved computationally efficiently. For the direct optimal control approach, optimal control inputs can be obtained by using a larger discrete step size, which requires less computational load. We illustrate the basic properties of the discrete optimal control using optimal control problems for the spherical pendulum and the rigid body model presented in the previous section. Example 5 (Optimal Control of a Spherical Pendulum) We study an optimal control problem for the spherical pendulum described in Example 3. We assume that an external control moment u 2 R3 acts on the pendulum. Control inputs are parametrized by their values at each time step, and the discrete equations of motion are modified to include the effects of the external control inputs by using the discrete Lagrange–d’Alembert principle. Since the component of the control moment that is parallel to the direction along the pendulum has no effect, we parametrize the control input as u k D q k w k for w k 2 R3 . The objective is to transfer the pendulum from a given initial configuration (q0 ; !0 ) to a desired configuration (qd ; ! d ) during a fixed maneuver time N, while minimizing the square of the weighted l2 norm of the control moments. N N X X h T h min J D uk uk D (q k  w k )T (q k  w k ) : wk 2 2 kD0

kD0

Discrete Control Systems

Discrete Control Systems, Figure 5 Optimal control of a spherical pendulum

We solve this optimal control problem by using a direct numerical optimization method. The terminal boundary condition is imposed as an equality constraint, and the 3(N C 1) control input parameters fw k g N kD0 are numerically optimized using sequential quadratic programming. This method is referred to as a DMOC (Discrete Mechanics and Optimal Control) approach (see [19]). Figure 5 shows a optimal solution transferring the spherical pendulum from a hanging configuration given by (q0 ; !0 ) D (e3 ; 031 ) to an inverted configuration (qd ; ! d ) D (e3 ; 031 ) during 1 second. The time step size is h D 0:033. Experiment have shown that the DMOC approach can compute optimal solutions using larger step sizes than typical Runge–Kutta methods, and consequently, it requires less computational load. In this case, using a second-order accurate Runge–Kutta method, the same optimization code fails while giving error messages of inaccurate and singular gradient computations. It is presumed that the unit length errors of the Runge– Kutta method, shown in Example 3, cause numerical instabilities for the finite-difference gradient computations required for the sequential quadratic programming algorithm. Example 6 (Optimal Control of a Rigid Body in a Potential Field) We study an optimal control problem of a rigid body using a dumbbell spacecraft model described in Example 4 (see [25] for detail). We assume that external control forces u f 2 R3 , and control moment u m 2 R3 act on the dumbbell spacecraft. Control inputs are parametrized by their values at each time step, and the Lie group variational integrators are modified to include the effects of the external control inputs by using discrete Lagrange– d’Alembert principle. The objective is to transfer the dumbbell from a given initial configuration (R0 ; x0 ; ˝0 ; v0 ) to a desired configuration (R d ; x d ; ˝ d ; v d ) during a fixed maneuver

time N, while minimizing the square of the l2 norm of the control inputs. N1 X

min J D

u kC1

kD0

T h  f T h f Wm u m u kC1 Wf u kC1C u m kC1 kC1 ; 2 2

where Wf ; Wm 2 R33 are symmetric positive definite matrices. Here we use a modified version of the discrete equations of motion with first order accuracy, as it yields a compact form for the necessary conditions. Necessary conditions for optimality: We solve this optimal control problem by using an indirect optimization method, where necessary conditions for optimality are derived using variational arguments, and a solution of the corresponding two-point boundary value problem provides the optimal control. This approach is common in the optimal control literature; here the optimal control problem is discretized at the problem formulation level using the Lie group variational integrator presented in Sect. “Discrete Lagrangian and Hamiltonian Mechanics”. Ja D

N1 X kD0

h  f T f f h  m T m m u W u kC1 C W u kC1 u 2 kC1 2 kC1

C 1;T k fx kC1 C x k C hv k g

 @U kC1 f 2;T mv kC1 C mv k  h C hu kC1 C k @x kC1  _ logm(F k  RTk R kC1 ) C 3;T k  ˚  C 4;T J˝ kC1 C F kT J˝ k C h M kC1 C u m kC1 ; k where 1k ; 2k ; 3k ; 4k 2 R3 are Lagrange multipliers. The matrix logarithm is denoted by logm : SO(3) ! so(3) and the vee map _ : so(3) ! R3 is the inverse of the hat map introduced in Example 4. The logarithm form of (43) is used, and the constraint (42) is considered implicitly using constrained variations. Using similar expressions for the

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Discrete Control Systems, Figure 6 Optimal control of a rigid body

variation of the rotation matrix and the angular velocity given in (14) and (15), the infinitesimal variation can be written as N1 n o X f ;T f hıu k W f u k C 2k1 ıJa D kD1

˚ 4 C hıu m;T Wm u m k C  k1 k ˚ C zTk  k1 C ATk  k ; where  k D [1k ; 2k ; 3k ; 4k ] 2 R12 , and z k 2 R12 represents the infinitesimal variation of (R k ; x k ; ˝ k ; v k ), given by z k D [logm(RTk ıR k )_ ; ıx k ; ı˝ k ; ıv k ]. The matrix A k 2 R1212 is defined in terms of (R k ; x k ; ˝ k ; v k ). Thus, necessary conditions for optimality are given by f

u kC1 D Wf1 2k ;

(47)

1 4 um kC1 D Wm  k ;

(48)

 k D ATkC1  kC1

(49)

together with the discrete equations of motion and the boundary conditions. Computational approach: Necessary conditions for optimality are expressed in terms of a two point boundary problem. The problem is to find the optimal discrete flow, multipliers, and control inputs to satisfy the equations of motion, optimality conditions, multiplier equations, and boundary conditions simultaneously. We use a neighboring extremal method (see [12]). A nominal solution satisfying all of the necessary conditions except the boundary conditions is chosen. The unspecified initial multiplier is updated by successive linearization so as to satisfy the specified terminal boundary conditions in the limit. This is also referred to as the shooting method. The main advantage of the neighboring extremal method is that the number of iteration variables is small.

The difficulty is that the extremal solutions are sensitive to small changes in the unspecified initial multiplier values. The nonlinearities also make it hard to construct an accurate estimate of sensitivity, thereby resulting in numerical ill-conditioning. Therefore, it is important to compute the sensitivities accurately to apply the neighboring extremal method. Here the optimality conditions (47) and (48) are substituted into the equations of motion and the multiplier equations, which are linearized to obtain sensitivity derivatives of an optimal solution with respect to boundary conditions. Using this sensitivity, an initial guess of the unspecified initial conditions is iterated to satisfy the specified terminal conditions in the limit. Any type of Newton iteration can be applied. We use a line search with backtracking algorithm, referred to as Newton–Armijo iteration (see [22]). Figure 6 shows optimized maneuvers, where a dumbbell spacecraft on a reference circular orbit is transferred to another circular orbit with a different orbital radius and inclination angle. Figure 6a shows the violation of the terminal boundary condition according to the number of iterations on a logarithmic scale. Red circles denote outer iterations in the Newton–Armijo iteration to compute the sensitivity derivatives. The error in satisfaction of the terminal boundary condition converges quickly to machine precision after the solution is close to the local minimum at around the 20th iteration. These convergence results are consistent with the quadratic convergence rates expected of Newton methods with accurately computed gradients. The neighboring extremal method, also referred to as the shooting method, is numerically efficient in the sense that the number of optimization parameters is minimized. But, in general, this approach may be prone to numerical ill-conditioning (see [3]). A small change in the initial multiplier can cause highly nonlinear behavior of the terminal attitude and angular momentum. It is difficult

Discrete Control Systems

to compute the gradient for Newton iterations accurately, and the numerical error may not converge. However, the numerical examples presented in this article show excellent numerical convergence properties. The dynamics of a rigid body arises from Hamiltonian mechanics, which have neutral stability, and its adjoint system is also neutrally stable. The proposed Lie group variational integrator and the discrete multiplier equations, obtained from variations expressed in the Lie algebra, preserve the neutral stability property numerically. Therefore the sensitivity derivatives are computed accurately. Controlled Lagrangian Method for Discrete Lagrangian Systems The method of controlled Lagrangians is a procedure for constructing feedback controllers for the stabilization of relative equilibria. It relies on choosing a parametrized family of controlled Lagrangians whose corresponding Euler–Lagrange flows are equivalent to the closed loop behavior of a Lagrangian system with external control forces. The condition that these two systems are equivalent results in matching conditions. Since the controlled system can now be viewed as a Lagrangian system with a modified Lagrangian, the global stability of the controlled system can be determined directly using Lyapunov stability analysis. This approach originated in Bloch et al. [5] and was then developed in Auckly et al. [1]; Bloch et al. [6,7,8,9]; Hamberg [15,16]. A similar approach for Hamiltonian controlled systems was introduced and further studied in the work of Blankenstein, Ortega, van der Schaft, Maschke, Spong, and their collaborators (see, for example, [33,35] and related references). The two methods were shown to be equivalent in Chang et al. [13] and a nonholonomic version was developed in Zenkov et al. [40,41], and Bloch [4]. Since the method of controlled Lagrangians relies on relating the closed-loop dynamics of a controlled system with the Euler–Lagrange flow associated with a modified Lagrangian, it is natural to discretize this approach through the use of variational integrators. In Bloch et al. [10,11], a discrete theory of controlled Lagrangians was developed for variational integrators, and applied to the feedback stabilization of the unstable inverted equilibrium of the pendulum on a cart. The pendulum on a cart is an example of an underactuated control problem, which has two degrees of freedom, given by the pendulum angle and the cart position. Only the cart position has control forces acting on it, and the stabilization of the pendulum has to be achieved indirectly through the coupling between the pendulum and the cart. The controlled Lagrangian is obtained by modifying the

kinetic energy term, a process referred to as kinetic shaping. Similarly, it is possible to modify the potential energy term using potential shaping. Since the pendulum on a cart model involves both a planar pendulum, and a cart that translates in one-dimension, the configuration space is a cylinder, S1 R. Continuous kinetic shaping: The Lagrangian has the form kinetic minus potential energy ˙ D L(q; q)

1  ˙2 ˛ C 2ˇ( ) ˙ s˙ C s˙2  V(q); 2

(50)

and the corresponding controlled Euler–Lagrange dynamics is d @L @L D0;  dt @ ˙ @ d @L Du; dt @˙s

(51) (52)

where u is the control input. Since the potential energy is translation invariant, i. e., V(q) D V( ), the relative equilibria D e , s˙ D const are unstable and given by non-degenerate critical points of V( ). To stabilize the relative equilibria D e , s˙ D const with respect to , kinetic shaping is used. The controlled Lagrangian in this case is defined by 1 ˙ D L( ; ˙ ; s˙ C ( ) ˙ ) C  (( ) ˙ )2 ; (53) L; (q; q) 2 where ( ) D ˇ( ). This velocity shift corresponds to a new choice of the horizontal space (see [8] for details). The dynamics is just the Euler–Lagrange dynamics for controlled Lagrangian (53), d @L; @L;  D0; dt @ ˙ @ d @L; D0: dt @˙s

(54) (55)

The Lagrangian (53) satisfies the simplified matching conditions of Bloch et al. [9] when the kinetic energy metric coefficient in (50) is constant.  Setting u D d ( ) ˙ /dt defines the control input, makes Eqs. (52) and (55) identical, and results in controlled momentum conservation by dynamics (51) and (52). Setting  D 1/  makes Eqs. (51) and (54) identical when restricted to a level set of the controlled momentum. Discrete kinetic shaping: Here, we adopt the following notation: q kC1/2 D

q k C q kC1 ; 2 q k D q kC1  q k ;

q k D ( k ; s k ) :

155

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Discrete Control Systems

Then, a second-order accurate discrete Lagrangian is given by, Ld (q k ; q kC1 ) D hL(q kC1/2 ; q k /h) : The discrete dynamics is governed by the equations @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D0; @ k @ k @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D u k ; @s k @s k

(56) (57)

where u k is the control input. Similarly, the discrete controlled Lagrangian is, ; (q kC1/2 ; q k /h) ; L; d (q k ; q kC1 ) D hL

with discrete dynamics given by, @L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0; @ k @ k

@L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0: @s k @s k

Numerical example: Simulating the behavior of the discrete controlled Lagrangian system involves viewing Eqs. (58)–(59) as an implict update map (q k2 ; q k1 ) 7! (q k1 ; q k ). This presupposes that the initial conditions are given in the form (q0 ; q1 ); however it is generally preferable to specify the initial conditions as (q0 ; q˙0 ). This is achieved by solving the boundary condition,

(58) (59)

@L (q0 ; q˙0 ) C D1 Ld (q0 ; q1 ) C Fd (q0 ; q1 ) D 0 ; @q˙ for q1 . Once the initial conditions are expressed in the form (q0 ; q1 ), the discrete evolution can be obtained using the implicit update map. We first consider the case of kinetic shaping on a level surface, when  is twice the critical value, and without dissipation. Here, h D 0:05 sec, m D 0:14 kg, M D 0:44 kg, and l D 0:215 m. As shown in Fig. 7, the dynamics is stabilized, but since there is no dissipation, the oscillations are sustained. The s dynamics exhibits both a drift and oscillations, as potential shaping is necessary to stabilize the translational dynamics.

Equation (59) is equivalent to the discrete controlled momentum conservation: Future Directions pk D  ; where @L; d (q k ; q kC1 ) @s k (1 C )ˇ( kC1/2 ) k C s k D : h

pk D 

Setting uk D 

 k ( kC1/2 )   k1 ( k1/2 ) h

makes Eqs. (57) and (59) identical and allows one to represent the discrete momentum equation (57) as the discrete momentum conservation law p k D p. The condition that (56)–(57) are equivalent to (58)– (59) yield the discrete matching conditions. The dynamics determined by Eqs. (56)–(57) restricted to the momentum level p k D p is equivalent to the dynamics of Eqs. (58)– (59) restricted to the momentum level p k D  if and only if the matching conditions  D hold.

1 ;



D

p ; 1 C 

Discrete Receding Horizon Optimal Control: The existing work on discrete optimal control has been primarily focused on constructing the optimal trajectory in an open loop sense. In practice, model uncertainty and actuation errors necessitate the use of feedback control, and it would be interesting to extend the existing work on optimal control of discrete systems to the feedback setting by adopting a receding horizon approach. Discrete State Estimation: In feedback control, one typically assumes complete knowledge regarding the state of the system, an assumption that is often unrealistic in practice. The general problem of state estimation in the context of discrete mechanics would rely on good numerical methods for quantifying the propagation of uncertainty by solving the Liouville equation, which describes the evolution of a phase space distribution function advected by a prescribed vector field. In the setting of Hamiltonian systems, the solution of the Liouville equation can be solved by the method of characteristics (Scheeres et al. [38]). This implies that a collocational approach (Xiu [39]) combined with Lie group variational integrators, and interpolation based on noncommutative harmonic analysis on Lie groups could yield an efficient means of propagating uncertainty, and serve as the basis of a discrete state estimation algorithm.

Discrete Control Systems

Discrete Control Systems, Figure 7 Discrete controlled dynamics with kinetic shaping and without dissipation. The discrete controlled system stabilizes the  motion about the equilibrium, but the s dynamics is not stabilized; since there is no dissipation, the oscillations are sustained

Forced Symplectic–Energy-Momentum Variational Integrators: One of the motivations for studying the control of Lagrangian systems using the method of controlled Lagrangians is that the method provides a natural candidate Lyapunov function to study the global stability properties of the controlled system. In the discrete theory, this approach is complicated by the fact that the energy of a discrete Lagrangian system is not exactly conserved, but rather oscillates in a bounded fashion. This can be addressed by considering the symplectic-energy-momentum [20] analogue to the discrete Lagrange-d’Alembert principle, ı

N1 X kD0

Ld (q k ; q kC1 ; h k ) D

N1 X

[Fd (q k ; q kC1 ; h k )  ıq k

kD0

C FdC (q k ; q kC1 ; h k )  ıq kC1 ] ; where the timestep hk is allowed to vary, and is chosen to satisfy the variational principle. The variations in hk yield an Euler–Lagrange equation that reduces to the conservation of discrete energy in the absence of external forces. By developing a theory of controlled Lagrangians around a geometric integrator based on the symplecticenergy-momentum version of the Lagrange–d’Alembert principle, one would potentially be able to use Lyapunov techniques to study the global stability of the resulting numerical control algorithms.

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cations to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications. Toronto, Canada, 28–31 August 2005, pp 962–967 Lee T, Leok M, McClamroch NH (2006) Optimal control of a rigid body using geometrically exact computations on SE(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 2710–2715 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem. Comput Method Appl Mech Eng 196:2907–2924 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem in orbital mechanics. Celest Mech Dyn Astron 98(2):121–144 Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Leok M (2004) Foundations of Computational Geometric Mechanics. Ph D thesis, California Instittute of Technology Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer, New York Marsden JE, West M (2001) Discrete mechanics and variational integrators. In: Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler–Poincarée and Lie–Poisson equations. Nonlinearity 12(6):1647–1662 Maschke B, Ortega R, van der Schaft A (2001) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr 45:1498–1502 Moser J, Veselov AP (1991) Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun Math Phys 139:217–243 Ortega R, Spong MW, Gómez-Estern F, Blankenstein G (2002) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans Autom Contr 47:1218–1233 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286 Scheeres DJ, Fahnestock EG, Ostro SJ, Margot JL, Benner LAM, Broschart SB, Bellerose J, Giorgini JD, Nolan MC, Magri C, Pravec P, Scheirich P, Rose R, Jurgens RF, De Jong EM, Suzuki S (2006) Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314:1280–1283 Scheeres DJ, Hsiao F-Y, Park RS, Villac BF, Maruskin JM (2006) Fundamental limits on spacecraft orbit uncertainty and distribution propagation. J Astronaut Sci 54:505–523 Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Comm Comput Phys 2:293–309 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2000) Matching and stabilization of low-dimensional nonholonomic systems. In: Proceedings of the IEEE Conference on Decision and Control. Sydney Convention and Exhibition Centre, Sydney, NSW Australia; 12–15 December 2000, pp 1289–1295 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2002) Flat nonholonomic matching. In: Proceedings of the American Control Conference. Hilton Anchorage, Anchorage, AK, 8–10 May 2002, pp 2812–2817

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Books and Reviews Bloch AM (2003) Nonholonomic Mechanics and Control. Interdisciplinary Appl Math, vol 24. Springer Bullo F, Lewis AD (2005) Geometric control of mechanical systems. Texts in Applied Mathematics, vol 49. Springer, New York Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365

Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer Marsden JE, West M (2001) Discrete mechanics and variational integrators. In Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286

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Dispersion Phenomena in Partial Differential Equations PIERO D’ANCONA Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy

  kukH s D (1  )s/2 uL q : Recall that this definition does not reduce to the preceding one when q D 1. We shall also use the homogeneous space H˙ qs , with norm   kukH˙ s D ()s/2 u L q :

Article Outline Glossary Definition of the Subject Introduction The Mechanism of Dispersion Strichartz Estimates The Nonlinear Wave Equation The Nonlinear Schrödinger Equation Future Directions Bibliography Glossary Notations Partial derivatives are written as ut or @ t u, @˛ D @˛x 11  @˛x nn , the Fourier transform of a function is defined as Z F f () D b f () D ei x f (x)dx and we frequently use the mute constant notation A . B to mean A  CB for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context). Evolution equations Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t. Cauchy problem A system of evolution equations, combined with a set of initial conditions at an initial time t D t0 . The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem. Blow up In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the L1 norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution. Sobolev spaces we shall use two instances of Sobolev space: the space W k;1 with norm kukW k;1 D

and the space H qs with norm

X j˛jk

k@˛ ukL 1

Dispersive estimate a pointwise decay estimate of the form ju(t; x)j  Ct ˛ , usually for the solution of a partial differential equation. Definition of the Subject In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume which increases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, defined in a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss of energy, transferred to an external system (heat dissipation being the typical example). This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical, linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in the Lp norms, and sharp estimates are available. In addition, detailed information on the profile of the solutions can be obtained, producing an accurate description of the evolution. For nonlinear equations, the theory now is able to explain and quantify several complex phenomena such as the splitting of solutions into the sum of a train of solitons, plus a radiation part which disperses and decays as time increases. Already in the 1960s it was realized that precise information on the decay of free waves could be a powerful tool to investigate the effects of linear perturbations (describing interactions with electromagnetic fields) and even nonlinear perturbations of the equations. One of the first important applications was the discovery that a nonlinear PDE may admit global small solutions, provided the rate of decay is sufficiently strong compared with the concentration effect produced by the nonlinear terms. This situation is very different from the ODE setting, where dispersion is absent. Today, dispersive and Strichartz estimates represent the backbone of the theory of nonlinear evolution equations. Their applications include local and global existence

Dispersion Phenomena in Partial Differential Equations

results for nonlinear equations, existence of low regularity solutions, scattering, qualitative study of evolution equations on manifolds, and many others. Introduction To a large extent, mathematical physics is the study of qualitative and quantitative properties of the solutions to systems of differential equations. Phenomena which evolve in time are usually described by nonlinear evolution equations, and their natural setting is the Cauchy problem: the state of the system is assigned at an initial time, and the following evolution is, or should be, completely determined by the equations. The most basic question concerns the local existence and uniqueness of the solution; we should require that, on one hand, the set of initial conditions is not too restrictive, so that a solution exists, and on the other hand it is not too lax, so that a unique solution is specified. For physically meaningful equations it is natural to require in addition that small errors in the initial state propagate slowly in time, so that two solutions corresponding to slightly different initial data remain close, at least for some time (continuous dependence on the data). This set of requirements is called local well posedness of the system. Global well posedness implies the possibility of extending these properties to arbitrary large times. However, local well posedness is just the first step in understanding a nonlinear equation; basically, it amounts to a check that the equation and its Cauchy problem are meaningful from the mathematical and physical point of view. But the truly interesting questions concern global existence, asymptotic behavior and more generally the structure and shape of solutions. Indeed, if an equation describes a microscopic system, all quantities measured in an experiment correspond to asymptotic features of the solution, and local properties are basically useless in this context. The classical tool to prove local well posedness is the energy method; it can be applied to many equations and leads to very efficient proofs. But the energy method is intrinsically unable to give answers concerning the global existence and behavior of solutions. Let us illustrate this point more precisely. Most nonlinear evolution equations are of the form Lu D N(u) with a linear part L (e. g., the d’Alembert or Schrödinger operator) and a nonlinear term N(u). The two terms are in competition: the linear flow is well behaved, usually with

several conserved quantities bounded by norms of the initial data, while the nonlinear term tends to concentrate the peaks of u and make them higher, increasing the singularity of the solution. At the same time, a small function u is made even smaller by a power-like nonlinearity N(u). The energy method tries to use only the conserved quantities, which remain constant during the evolution of the linear flow. Essentially, this means to regard the equation as an ODE of the form y 0 (t) D N(y(t)). This approach is very useful if the full (nonlinear) equation has a positive conserved quantity. But in general the best one can prove using the energy method is local well posedness, while a blow up in finite time can not be excluded, even if the initial data are assumed to be small, as the trivial example y 0 D y 2 clearly shows. To improve on this situation, the strength of the linear flow must be used to a full extent. This is where dispersive phenomena enter the picture. A finer study of the operator L shows that there are quantities, different from the standard L2 type norms used in the energy estimates, which actually decay during the evolution. Hence the term L has an additional advantage in the competition with N which can lead to global existence of small solutions. See Sect. “The Nonlinear Wave Equation” for more details. This circle of ideas was initiated at the beginning of the 1970s and rapidly developed during the 1980s in several papers, (see [26,27,31,32,33,34,38,40,41,42,50] to mention just a few of the fundamental contributions on the subject). Global existence of small solutions was proved for many nonlinear evolution equations, including the nonlinear wave, Klein–Gordon and Schrödinger equations. The proof of the stability of vacuum for the Einstein equations [15,35] can be regarded as an extreme development of this direction of research. Another early indication of the importance of Lp estimates came from the separate direction of harmonic analysis. In 1970, at a time when general agreement in the field was that the L2 framework was the only correct one for the study of the wave equation, R. Strichartz ([44,45], see also [10,11,46]) proved the estimate u t t  u D F ;

u(0; x) D u t (0; x) D 0

H)

kukL p (RnC1 ) . kFkL p0 (RnC1 )

for p D 2(n C 1)/(n  1). This was a seminal result, both for the emphasis on the Lp approach, and for the idea of regarding all variables including time on the same level, an idea which would play a central role in the later developments. At the same time, it hinted at a deep connection

161

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between dispersive properties and the techniques of harmonic analysis. The new ideas were rapidly incorporated in the mainstream theory of partial differential equations and became an important tool ([22,23,52]); the investigation of Strichartz estimates is still in progress. The freedom to work in an Lp setting was revealed as useful in a wide variety of contexts, including linear equations with variable coefficients, scattering, existence of low regularity solutions, and several others. We might say that this point of view created a new important class of dispersive equations, complementing and overlapping with the standard classification into elliptic, hyperbolic, parabolic equations. Actually this class is very wide and contains most evolution equations of mathematical physics: the wave, Schrödinger and Klein–Gordon equations, Dirac and Maxwell systems, their nonlinear counterparts including wave maps, Yang–Mills and Einstein equations, Korteweg de Vries, Benjamin–Ono, and many others. In addition, many basic physical problems are described by systems obtained by coupling these equations; among the most important ones we mention the Dirac–Klein–Gordon, Maxwell–Klein–Gordon, Maxwell– Dirac, Maxwell–Schrödinger and Zakharov systems. Finally, a recent line of research pursues the extension of the dispersive techniques to equations with variable coefficients and equations on manifolds. The main goal of this article is to give a quick overview of the basic dispersive techniques, and to show some important examples of their applications. It should be kept in mind however that the field is evolving at an impressive speed, mainly due to the contribution of ideas from Fourier and harmonic analysis, and it would be difficult to give a self-contained account of the recent developments, for which we point at the relevant bibliography. In Sect. “The Mechanism of Dispersion” we analyze the basic mechanism of dispersion, with special attention given to two basic examples (Schrödinger and wave equation). Section “Strichartz Estimates” is a coincise exposition of Strichartz estimates, while Sects. “The Nonlinear Wave Equation” and “The Nonlinear Schrödinger Equation” illustrate some typical applications of dispersive techniques to problems of global existence for nonlinear equations. In the last Sect. “Future Directions” we review some of the more promising lines of research in this field.

The same mechanism lies behind dispersive phenomena, and indeed two dual explanations are possible: a) in physical space, dispersion is characterized by a finite speed of propagation of the signals; data concentrated on some region tend to spread over a larger volume following the evolution flow. Correspondingly, the L1 and other norms of the solution tend to decrease; b) in Fourier space, dispersion is characterized by oscillations in the Fourier variable , which increase for large jj and large t. This produces cancelation and decay in suitable norms. We shall explore these dual points of view in two fundamental cases: the Schrödinger and the wave equation. These are the most important and hence most studied dispersive equations, and a fairly complete theory is available for both. The Schrödinger Equation in Physical Space The solution u(t; x) of the Schrödinger equation iu t C u D 0 ;

u(0; x) D f (x)

admits several representations which are not completely equivalent. From spectral theory we have an abstract representation as a group of L2 isometries u(t; x) D e i t f ;

(1)

then we have an explicit representation using the fundamental solution 2 1 i jxj 4t f e (4 it)n/2 Z jx yj2 1 i 4t e f (y)dy ; D (4 it)n/2 Rn

u(t; x) D

finally, we can represent the solution via Fourier transform as Z 2 n f ()d : (3) e i(tjj Cx)b u(t; x) D (2) Notice for instance that the spectral and Fourier representations are well defined for f 2 L2 , while it is not immediately apparent how to give a meaning to (2) if f 62 L1 . From (1) we deduce immediately the dispersive estimate for the Schrödinger equation, which has the form 1

The Mechanism of Dispersion

je i t f j  (4)n/2

The Fourier transform puts in duality oscillations and translations according to the elementary rule

Since the L2 norm of u is constant in time

F ( f (x C h)) D ei xh F f :

(2)

t n/2

ke i t f kL 2 D k f kL 2 ;

k f kL 1 :

(4)

(5)

Dispersion Phenomena in Partial Differential Equations

this might suggest that the constant L2 mass spreads on a region of volume t n i. e. of diameter t. Apparently, this is in contrast with the notion that the Schrödinger equation has an infinite speed of propagation. Indeed, this is a correct but incomplete statement. To explain this point with an explicit example, consider the evolution of a wave packet 1

from zero, and this already suggests a decay of the order t n/2 . This argument can be made precise and gives an alternative proof of (4). We also notice that the Fourier transform in both space and time of u(t; x) D e i t f is a temperate distribution in S0 (RnC1 ) which can be written, apart from a constant, ˜ ) D ı(  jj2) u(;

2

x 0 ;0 D e 2 jxx 0 j e i(xx 0 )0 : A wave packet is a mathematical representation of a particle localized near position x0 , with frequency localized near  0 ; notice indeed that its Fourier transform is 1

2

b  x 0 ;0 D ce 2 j0 j e ix 0 for some constant c. Exact localization both in space and frequency is of course impossible by Heisenberg’s principle; however the approximate localization of the wave packet is a very good substitute, since the gaussian tails decay to 0 extremely fast. The evolution of a wave packet is easy to compute explicitly:   1 (x  x0  2t0 )2 i i t  n2 e x 0 ;0 D (1C2it) exp  e 2 (1 C 4t 2 )

and is supported on the paraboloid  D jj2. This representation of the solution gives additional insight to its behavior in certain frequency regimes, and has proved to be of fundamental importance when studying the existence of low regularity solutions to the nonlinear Schrödinger equation [6,9]. The Wave Equation in Physical Space For the wave equation, the situation is reversed: the most efficient proof of the dispersive estimate is obtained via the Fourier representation of the solution. This is mainly due to the complexity of the fundamental solution; we recall the relevant formulas briefly. The solution to the homogeneous wave equation

where the real valued function  is given by  D

t(x  x0 )2 C (x  x0  t0 )  0 : (1 C 4t 2 )

We notice that the particle “moves” with velocity 20 and the height of the gaussian spreads precisely at a rate t n/2 . The mass is essentially concentrated in a ball of radius t, as suspected. This simple remark is at the heart of one of the most powerful techniques to have appeared in harmonic analysis in recent years, the wave packet decomposition, which has led to breakthroughs in several classical problems (see e. g. [7,8,48,51]).

w t t  w D 0 ;

2

(t; x; ) D tjj C x  

H)

r  D 2t C x

so that for each (t; x) we have a unique stationary point  D x/2t; the n curvatures at that point are different

w t (0; x) D g(x)

can be written as u(t; x) D cos(tjDj) f C

sin(tjDj) g jDj

where we used the symbolic notations cos(tjDj) f D F 1 cos(tjj)F f ; sin(tjDj) sin(tjj) Fg g D F 1 jDj jj (recall that F denotes the Fourier transform in space variables). Notice that

The Schrödinger Equation in Fourier Space In view of the ease of proof of (4), it may seem unnecessary to examine the solution in Fourier space, however a quick look at this other representation is instructive and prepares the discussion for the wave equation. If we examine (3), we see that the integral is a typical oscillatory integral which can be estimated using the stationary phase method. The phase here is

w(0; x) D f (x) ;

cos(tjDj) f D

@ sin(tjDj) f; @t jDj

hence the properties of the first operator can be deduced from the second. Moreover, by Duhamel’s formula, we can express the solution to the nonhomogeneous wave equation w t t  w D F(t; x) ;

w(0; x) D 0 ;

w t (0; x) D 0

as Z

t

u(t; x) D 0

sin((t  s)jDj) F(u(s; ))ds : jDj

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Dispersion Phenomena in Partial Differential Equations

of the solution remains constant, in view of the energy estimate

In other words, a study of the operator S(t)g D

sin(tjDj) g jDj

ku t (t; )k2L 2 Ckru(t; )k2L 2 D ku t (0; )k2L 2 Ckru(0; )k2L 2 ;

is sufficient to deduce the behavior of the complete solution to the nonhomogeneous wave equation with general data. The operator S(t) can be expressed explicitly using the fundamental solution of the wave equation. We skip the one-dimensional case since then from the expression of the general solution u(t; x) D (t C x) C

(t  x)

it is immediately apparent that no dispersion can occur. In odd space dimension n  3 we have !   n3 Z 2 1 sin(tjDj) 1 g(x  y)d y ; g D cn @t jDj t t jyjDt

on the other hand, the velocity of propagation for the wave equation is exactly 1, so that the constant L2 energy is spread on a volume which increases at a rate t n ; as a consequence, we would expect the same rate of decay t n/2 as for the Schrödinger equation. However, a more detailed study of the shape of the solution reveals that it tends to concentrate in a shell of constant thickness around the light cone. For instance, in odd dimension larger than 3 initial data with support in a ball B(0; R) produce a solution which at time t is supported in the shell B(0; t C R) n B(0; t  R). Also in even dimension the solution tends to concentrate along the light cone jtj D jxj, with a faster decreasing tail far from the cone. Thus, the energy spreads over a volume of size t n1 , in accordance with the rate of decay in the dispersive estimate.

and in even space dimension n  2 sin(tjDj) g D cn jDj



1 @t t

 n2 2

1 t

Z jyj
g(x  y) p dy t 2  jyj2

!

for some constant cn . These formulas can be expanded and the resulting integrals estimated in order to compute the explicit decay rate of the solution; this approach was followed in [50] (see also [43] for a more concise proof based on the fundamental solution). We shall prove the dispersive estimate using Fourier methods in the next section, but here it is instructive to consider the particularly simple case n D 3, when the above representation reduces to Kirchhoff’s formula Z sin(tjDj) c g(x  y)d y : gD jDj t jyjDt Using the divergence theorem it is easy to deduce that ˇ ˇ ˇ sin(tjDj) ˇ 1 ˇ ˇ . kr f kL 1 : g ˇ jDj ˇ t Recall that the rate of decay for the Schrödinger equation in dimension 3 is t 3/2 ; for the wave equation the rate of decay is slower, in addition the estimate shows a loss of one derivative with respect to the initial data. Actually this estimate is optimal, as it is evident from the explicit representation of the solution. In dimension n the correct n1 rate of decay is t  2 and the loss of derivative is of order (n  1)/2, as we shall see in the next section. This result is in apparent contrast with our physical space picture. Indeed the L2 norm of the first derivatives

The Wave Equation in Fourier Space Since 2i

e i tjDj ei tjDj sin(tjDj) gD g g; jDj jDj jDj

2 cos(tjDj) f D e i tjDj f C ei tjDj f ; we can unify the study of the various components of the solution by considering the operator Z f d : e i tjDj f D F 1 e i tjj F f D (2)n e i(xCtjj)b The phase of the last integral is (t; x; ) D tjj C x  

H)

r  D t

 Cx jj

and we notice immediately that the structure of stationary points is more complex than for Schrödinger: if jtj D jxj, i. e. if the point (t; x) is on the light cone, the phase has a half line of critical points jtj D jxj

H)

 x D jj t

which of course are degenerate since only n  1 curvatures are different from zero; on the other hand, when (t; x) is not on the light cone, the phase is nondegenerate at all points. In the nondegenerate case jtj ¤ jxj, we notice that r  D t

 Cx jj

H)

jr j  jjtj  jxjj

Dispersion Phenomena in Partial Differential Equations

hence by direct integration by parts we obtain ˇ ˇZ ˇ ˇ ˇ  C( f )jjtj  jxjj(n1) : ˇ e i(xCtjj)b f d ˇ ˇ

(6)

Notice that the singularity jj of the phase makes it impossible to perform the integration by parts more than n  1 times, but this is largely sufficient for our purposes here. In the degenerate case jtj D jxj, we can eliminate the degeneracy by passing to polar coordinates: Z 1 Z i(xCtjj)b f d D e i t n1 e 0 Z f ( !)d!d :  e i x !b j!jD1

Now the phase in the inner integral has exactly two isolated critical points for each value of x; by a standard application of the stationary phase method we can obtain an estimate of the form ˇZ ˇ ˇ ˇ ˇ e i(xCtjj)b ˇ  C( f )jxj n1 2 : f d (7) ˇ ˇ The power (n  1)/2 is directly related to the dimension n  1 of the unit sphere. From the above computations it is possible to obtain a very detailed description of the shape of the solution (see e. g. [20]). However, here we are only interested in the rate of decay of the L1 norm of the solution. Thus, using estimate (6) when jxj < jtj/2 and (7) when jxj > jtj/2 we obtain je i tjDj f j  C( f )jtj

n1 2

:

It is also possible to make precise the form of the constant C( f ) appearing in the estimate. This can be done in several ways; one of the most efficient exploits the scaling properties of the wave equation (see [10,23]). In order to state the optimal estimate we briefly recall the definition of some function spaces. The basic ingredient is the homogeneous Paley–Littlewood partition of unity which has the form X  j () D 1 ;  j () D 0 (2 j ) ; j2Z

sup  j D f : 2 j1  jj  2 jC1 g for a fixed 0 2 C01 (Rn ) (the following definitions are independent of the precise choice of 0 ). To each symbol  j () we associate by the standard calculus the operator  j (D) D F 1  j ()F . Now the homogeneous Besov norm s can be defined as follows: B˙ 1;1 X s D 2 js k j (D) f kL 1 (8) k f kB˙ 1;1 j2Z

and the corresponding Besov space can be obtained by completing C01 (see [4] for a thorough description of these definitions). Then the optimal dispersive estimate for the wave equation is je i tjDj f j  Ck f k

n1

B˙ 1;12

jtj

n1 2

:

(9)

We might add that in odd space dimension a slightly better estimate can be proved involving only the standard ˙ n1 2 instead of the Besov norm. Sobolev norm W The singularity in t at the right hand side of (9) can be annoying in applications, thus it would be natural to try to replace jtj by (1 C jtj) by some different proof. However, this is clearly impossible since both sides have the same scaling with respect to the transformation u(t; x) ! u(t; x) which takes a solution of the homogeneous wave equation into another solution. Moreover, if such an estimate were true, taking the limit as t ! 0 we would obtain a false Sobolev embedding. Thus, the only way to get a nonsingular estimate is to replace the Besov norm of the data by some stronger norm. Indeed, for jtj < 1 we can use the L2 conservation which implies, for any > 0, je i tjDj f j  ke i tjDj f kH n/2C D k f kH n/2C : Combining the two estimates, we arrive at   n1 je i tjDj f j . k f k n1 C k f k (1Cjtj) 2 : (10) n/2C H 2 B˙ 1;1

Other Dispersive Equations Dispersive estimates can be proved for many other equations; see for instance [3] for general results concerning the equations of the form iu t D P(D)u ;

P polynomial

which correspond to integrals of the form Z e i(xx iCP())b f ()d : Here we would like to add some more details on two equations of particular importance for physics. The Klein– Gordon equation u t t  u C u D 0 ;

u(0; x) D f (x) ;

u t (0; x) D g(x)

shows the same decay rate as the Schrödinger equation, although for most other properties it is very close to the wave

165

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Dispersion Phenomena in Partial Differential Equations

equation. This phenomenon can be easily explained. In physical space, we notice that the solution has finite speed of propagation less or equal to 1, so that for compactly supported initial data the solution spreads uniformly on a volume of order tn , and there is no concentration along the light cone as for the wave equation. On the other hand, the Fourier representation of the solution is sin(thDi) u(t; x) D cos(thDi) f C g hDi

cos(thDi) f D F 1 cos(tjhij)F f ;

with hi D (1 C i. e. hDi D (1  Thus, we see that, similarly to the wave equation, the study of the solutions can be reduced to the basic operator Z f ()d : e i thDi f D F 1 e i thi F f D (2)n e i(thiCx)b It is easy to check that the phase has an isolated stationary point if and only if jtj < jxj, and no stationary point if jtj  jj:  Cx; hi

in accord with the claimed dispersion rate of order t n/2 . However, the proof is not completely elementary (for a very detailed study, see Section 7.2 of [25]). The optimal estimate is now je

i thDi

f j  Ck f k

n

2 B 1;1

jtj

 n2

sup  j D f : 2

and the 4  4 Dirac matrices can be written     I 0 k 0 ˛k D ; ˇD 2 ; k 0 0 I2 in terms of the Pauli matrices    1 0 0 I2 D ; 1 D 0 1 1    0 i 1 2 D ; 3 D i 0 0

j  0;

 jj  2 jC1 g ;

sup 1 D fjxj  1g for some 1 ; 0 2 C01 (Rn ). A second important example is the Dirac system. From a mathematical point of view (and setting the values of the

1 0

k D 1; 2; 3

 ;

 0 : 1

Then the solution u(t; x) D e i t D f of the massless Dirac system with initial value u(0; x) D f (x) satisfies the dispersive estimate ju(t; x)j 

C k f kB˙ 2 1;1 t

while in the massive case we have ju(t; x)j 

j2N

j1

3 1X ˛k @k i

(11)

(see the Appendix to [18]). Here we used the nonhomoges which is defined exactly as the honeous Besov norm B1;1 mogeneous one (8) by using the inhomogeneous version of the Palye–Littlewood partition of unity: X 1 () C  j () D 1 ;  j () D 0 (2 j ) for

iu t C Du C ˇu D 0

kD1

)1/2 .

r  D t

in the massless case, and

DD

sin(thDi) sin(tjhij) Fg g D F 1 hDi jhij

(t; x; ) D thi C x   ;

iu t C Du D 0

in the massive case. Here u : R t  R3x ! C 4 , the operator D is defined as

where we used the symbolic notations

jj2)1/2

speed of light and Planck’s constant equal to 1 for simplicity of notation) this is a 4  4 constant coefficient system of the form

C 3

t2

kfk

5

2 B 1;1

(see Sect. 7 of [16]). Strichartz Estimates Since the Schrödinger flow e i t f is an L2 isometry, it may be tempting to regard it as some sort of “rotation” of the Hilbert space L2 . This picture is far from true: there exist small subspaces of L2 which contain the flow for almost all times. This surprising phenomenon, which holds in greater generality for any unbounded selfadjoint operator on an abstract Hilbert space, is known as Kato smoothing and was discovered by T. Kato in 1966 [29], see also [39]. This fact is just a corollary of a quantitative estimate, the Kato smoothing estimate, which in the case of the Laplace

Dispersion Phenomena in Partial Differential Equations

operator takes the following form: there exist closed unbounded operators A on L2 such that kAe

i t

f kL 2 L 2 . k f kL 2 :

As a consequence, the flow belongs for a. e. t to the domain of the operator A, which can be a very small subspace of L2 . Two simple examples of smoothing estimates are the following: V(x) 2 L2 \ L n \ L nC H)

kV(x)e i t f kL 2 L 2 . k f kL 2

(12)

and kjxj1 e i t f kL 2 L 2 . k f kL 2 :

(13)

On the other hand, Strichartz proved in 1977 [46] that pD

2(n C 2) n H)

2n

L 2 L n2

ke

i t

f kL p (RnC1 ) . k f kL 2

(14)

. k f kL 2

(15)

while (13) would follow from the slightly stronger ke i t f k

2n

L 2 L n2 ;2

The homogeneous Strichartz estimates have the general form ke i t f kL p L q . k f kL 2 for suitable values of the couple (p; q). Notice that we can restrict the L p L q norm at the right hand side to an arbitrary time interval I (i. e. to L p (I; L q (Rn ))). Notice also that by the invariance of the flow under the scaling u(t:x) 7! u(2 t; x) the possible couples (p; q) are bound to satisfy n n 2 C D : p q 2

. k f kL 2

(16)

where L p;2 denotes a Lorentz space. Both (15) and (16) are true, although their proof is far from elementary. Such estimates are now generally called Strichartz estimates. They can be deduced from the dispersive estimates, hence they are in a sense weaker and contain less information. However, for this same reason they can be obtained for a large number of equations, even with variable coefficients, and their flexibility and wide range of applications makes them extremely useful. The full set of Strichartz estimates, with the exception of the endpoint (see below) was proved in [22], where it was also applied to the problem of global well posedness for the nonlinear Schrödinger equation. The corresponding estimates for the wave equation were obtained in [23], however they turned out to be less useful because of the loss of derivatives. A fundamental progress was made in [30], where not only the most difficult endpoint case was solved, but it also outlined a general procedure to deduce the Strichartz estimates from the dispersive estimates, with applications to any evolution equation. In the following we shall focus on the two main examples, the Schrödinger and the wave equation, and we refer to the original papers for more details and the proofs.

(17)

The range of possible values of q is expressed by the inequalities 2q

thus showing that the smoothing effect could be measured also in the Lp norms. If we try to reformulate (12) in terms of an Lp norm, we see that it would follow from ke i t f k

The Schrödinger Equation

2n n2

2  q  1 if

if

n  2;

n D1;

q ¤1;

(18) (19)

the index p varies accordingly between 1 and 2 (between 4 and 2 in one space dimension). Such couples (p; q) are called (Schrödinger) admissible. It is easy to visualize the admissible couples by plotting the usual diagram in the (1/p; 1/q) space (see Fig 1–3). Note that the choice (p; q) D (1; 2), the point E in the diagrams, corresponds to the conservation of the L2 norm, while at the other extreme we have for n  2 point P corresponding to the couple   2n (p; q) D 2; : (20) n2 This is usually called the endpoint and is excluded in dimension 2, allowed in dimension  3. The original Strichartz estimate corresponds to the case p D q and is denoted by S. After the proof in [22] of the general Strichartz estimates, it was not clear if the endpoint case P could be reached or not; the final answer was given in [30] where the endpoint estimates were proved to be true for all dimension larger than three. For the two-dimensional case and additional estimates, see [47]. Notice that all admissible couples can be obtained by interpolation between the endpoint and the L2 conservation, thus in a sense the endpoint contains the maximum of information concerning the Lp smoothing properties of the flow. There exist an inhomogeneous form of the estimates, which is actually equivalent to the homogeneous one, and can be stated as follows:  Z t   i(ts)   e F(s)ds . kFkL p˜ 0 L q˜0 (21)   0

L p Lq

167

168

Dispersion Phenomena in Partial Differential Equations

Dispersion Phenomena in Partial Differential Equations, Figure 3 Schrödinger admissible couples, n  3 Dispersion Phenomena in Partial Differential Equations, Figure 1 Schrödinger admissible couples, n D 1

neous global estimates are optimal since other combination of indices are excluded by scale invariance; however, the local Strichartz estimates can be slightly improved allowing for indices outside the admissible region (see [21]). We also mention that a much larger set of estimates can be deduced from the above ones by combining them with Sobolev embeddings; in the above diagrams, all couples in the region bounded by the line of admissibility, the axes and the line 1/p D 2 can be reached, while the external region is excluded by suitable counterexamples (see [30]). Strichartz Estimates for the Wave Equation

Dispersion Phenomena in Partial Differential Equations, Figure 2 Schrödinger admissible couples, n D 2

Most of the preceding section has a precise analogue for the wave equation. Since the decay rate for the dispersive estimates is t (n1)/2 for the wave equation instead of t n/2 as for the Schrödinger equation, this causes a shift n ! n 1 in the numerology of indices. A more serious difference is the loss of derivatives: this was already apparent in the original estimate by Strichartz [44], which can be written ke i tjDj f k

˜ which can be for all admissible couples (p; q) and ( p˜ ; q), chosen independently. Here p0 denotes the conjugate exponent to p. These estimates continue to be true in their local version, with the norms at both sides restricted to a fixed time interval I R. Notice that the inhomoge-

L

2(nC1) n1

. k f kH˙ 1/2 ;

n  2:

The general homogeneous Strichartz estimates for the wave equation take the form ke i tjDj f kL p L q . k f k

1 nC1 n1

˙p H

;

Dispersion Phenomena in Partial Differential Equations

now the (wave) admissible couples of indices must satisfy the scaling condition 2 n1 n1 C D p q 2

(22)

and the constraints 2  p  1, 2q

2(n  1) n3

2q1

if

n  3;

if

q¤1

(23)

n D2;

(24)

The Nonlinear Wave Equation In this section we shall illustrate, using the semilinear wave equation as a basic example, what kind of improvements can be obtained over the classical energy method via the additional information given by dispersive techniques. From the linear theory it is well known that the correct setting of the Cauchy problem for the wave equation u t t  u D F(t; x) ;

requires two initial data at t D 0: u(0; x) D u0 (x) ;

while p varies accordingly between 2 and 1 (between 4 and 1 in dimension 2). Recall that in dimension 1 the solutions of the wave equation do not decay since they can be decomposed into traveling waves. We omit the graphic representation of these sets of conditions since it is completely analogous to the Schrödinger case (apart from the shift n ! n  1). The inhomogeneous estimates may be written  Z t   i(ts)   e F(s)ds   0

nC1  1p n1

˙q LpH

. kFk

nC1  1 n1

0 ˙ p˜ L p˜ H q˜0

(25)

u(t; x) : R  Rn ! C

u t (0; x) D u1 (x) :

For this equation the energy method is particularly simple. If we multiply the equation by u t we can recast it in the form   @ t ju t j2 C jruj2 D 2
. ku1 kL 2 C kru0 kL 2 C kFkL 1 L 2 T

in terms of the homogeneous Sobolev norms kgkH˙ sq D kjDjs gkL q ;

jDjs D (1  )s/2 :

Other Equations The general procedure of Keel and Tao can be applied to a large variety of equations. For the convenience of the reader we list here the precise form of the homogeneous Strichartz estimates in some important cases. For the Klein–Gordon equation we have ke i thDi f kL p L q . k f k

1 nC2 n

where we are using the shorthand notation p

L T L q D L p (0; T; L q (Rn )) : Estimate (26) is the basic energy estimate, which is enough to prove the existence of a global solution u 2 C(R; H 1 (Rn )) \ C 1 (R; L2 (Rn )) of the linear wave equation for data in (u0 ; u1 ) 2 H 1  L2 and a forcing term in F 2 L1T L2 for all T. Uniqueness and continuous dependence on the data are simple consequences of the energy estimate. Moreover, if we differentiate the equation k times, the same computations lead to the higher order energy estimates

˙p H

with (p; q) Schrödinger admissible (recall that the rate of dispersion is t n/2 for Klein–Gordon). For the massless Dirac system we have ke i t D f kL p L q . k f k

(26)

2

˙p H

;

n D3;

T

5

H 3p

;

nD3

with (p; q) Schrödinger admissible in dimension 3.

T

. ku1 kH k Cku0 kH kC1 CkFkL 1 H k ; T

k D 0; 1; 2; : : : (27)

Consider now the nonlinear wave equation u t t  u D F(u)

with (p; q) wave admissible in dimension 3, while for the massive Dirac system the estimate takes the form ke i t(DCˇ ) f kL p L q . k f k

ku t kL 1 H k C kukL 1 H kC1

(28)

for some function F(s) sufficiently smooth. The classical approach to solve it is to look for a fixed point of the mapping u D ˚ (v), where u(t; x) solves the linearized equation u t t  u D F(v) :

(29)

169

170

Dispersion Phenomena in Partial Differential Equations

If we apply the energy estimate to the difference of two solutions u1 D ˚(v1 ) and u2 D ˚(v2 ) with the same initial data, we obtain k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. kF(v1 )  F(v2 )kL 1 H k : T

Using a standard Moser-type estimate (see e. g. [25,49]) k f (v1 )  f (v2 )kH k . C(kv1 kL 1 C kv2 kL 1 )kv1  v2 kH k we arrive at the inequality k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k L 1 H k : T L T T

k In particular, writing X T D L1 T H , and noticing that

kGkL 1 H k  kGk X T  T ; T

Actually, the two difficulties are related to each other. Indeed, from the above argument one can check that the lifespan of the local solution is a function of the H k norm of the initial data. Most equations with a physical origin possess conserved quantities, which are essentially H k norms of the solution with k low, e. g. k D 0 or 1. Assume now that we can prove the local existence for data say in H 1 , on some interval [0; T0 ] with T 0 depending on the H 1 norm of the data. Since this norm is conserved, we can take T 0 as a new initial time and construct a solution on [T0 ; 2T0 ], and so on. This is the essence of the continuation method. As a consequence, the local existence of low regularity solution with the addition of suitable conservation laws can lead to global existence. We shall now apply the dispersive information to improve on the preceding result and obtain the global existence of small solution, under suitable assumptions on the nonlinear term. Consider a free wave w(t; x), solution of w t t  w D 0 ;

w(0; x) D f (x) ;

w t (0; x) D g(x) :

we obtain Recalling the dispersive estimate (10), here the following simplified version

ku1  u2 k X T . C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k X T  T : T L T

jw(t; x)j  C(1 C t)

n1 2

L1

norm of vj , which can be We still have to handle the controlled by the H k norm provided k > n/2. In conclusion k>

n 2

H)

ku1  u2 k X T

. C(kv1 k X T C kv2 k X T )  kv1  v2 k X T  T : At this point the philosophy of the method should be clear: if we agree to differentiate n/2 times the equation, we can prove a nonlinear estimate, with a coefficient which can be made small for T small. Thus, for T  1 the mapping v 7! u is a contraction, and we obtain a local (in time) solution of the nonlinear equation. The energy method is quite elementary, yet it has the remarkable property that it can be adapted to a huge number of equations: it can be applied to equations with variable coefficients, and even to fully nonlinear equations. An outstanding example of its flexibility is the proof of the local solvability of Einstein’s equations ([14]). The main drawbacks of the energy method are: 1) A high regularity of the data and of the solution is required; 2) The nonlinear nature of the estimate leads to a local existence result, leaving completely open the question of global existence.

k f kW N;1 ;

N>n

will be sufficient. This suggests modification of the energy norm by adding a term that incorporates the decay information. To this end, we introduce the time-dependent quantity M(t) D sup

n

0t

kD

hni 2

ku(; )kH kC1 C (1 C )

n1 2

o ku(; )kL 1 ;

:

Consider now the nonlinear Eq. (28) with initial data u0 ; u1 . We already know that a local solution exists, provided the initial data are smooth enough; we shall now try to extend this solution to a global one. Using Duhamel’s principle we rewrite the equation as Z u(t; x) D w(t; x) C 0

t

sin((t  s)jDj) F(u(s; ))ds jDj

where we are using the symbolic notations of Sect. “Other Dispersive Equations”. F being the space Fourier transform. Note that the above dispersive estimate can be applied in particular to the last term; we obtain ˇ ˇ ˇ sin((t  s)jDj) ˇ ˇ ˇ  C(1 C t  s) n1 2 kFk N;1 : F(s) W ˇ ˇ jDj

Dispersion Phenomena in Partial Differential Equations

By this inequality and the energy estimate already proved, in a few steps we arrive at the estimate n1 2

M(T) . C0 C kFkL 1 H k C (1 C T) T Z T n1 (1 C T  s) 2 kF(u(s))kW N;1 ds 0

where C0 D ku0 kW N;1 C ku1 kW N;1 and the mute constant is independent of T. By suitable nonlinear estimates of Moser type, we can bound the derivatives of the nonlinear term using only the L1 and the H k norm of u, i. e., the quantity M(t) itself. If we assume that F(u) juj for small u, we obtain

Consider the Cauchy problem iu t  u D juj ;

u(0; x) D f (x)

(31)

for some  1. We want to study the local and global solvability in L2 of this Cauchy problem. Recall that this is a difficult question, still not completely solved; however, at least in the subcritical range, a combination of dispersive techniques and the contraction mapping principle gives the desired result easily. Instead of Eq. (31), we are actually going to solve the corresponding integral equation Z t e i(ts) ju(s)j ds : u(t; x) D e i t f C i 0

n1

M(T) . C0 C M(T) (1 C T) 2 Z T n1 n1 (1 C T  s) 2 (1 C s) 2 ( 2) ds : 0

Our standard approach is to look for a fixed point of the mapping u D ˚(v) defined by Z t ˚(v) D e i t f C i e i(ts) jv(s)j ds ; 0

Now we have Z

t

(1 C t  )

n1 2

 (1 C )

n1 2 ( 2)

d

0

 C(1 C t)

n1 2

k˚(v)kL p L q . k f kL 2 C kj˚(v)j kL p˜ 0 L q˜0

provided

> 2C

it is easy to check, using Strichartz estimates, that the mapping ˚ is well defined, provided the function v is in the suitable L p L q . Indeed we can write, for all admissible cou˜ ples (p; q) and ( p˜; q)

2 n1

so that, for such values of , we obtain M(T) . C0 CM(T) : For small initial data, which means small values of the constant C0 , (30) implies that M(T) is uniformly bounded as long as the smooth solution exists. By a simple continuation argument, this implies that the solution is in fact global, provided the initial data are small enough.

and by Hölder’s inequality this implies  0 0 L  p˜ L  q˜

k˚(v)kL p L q  C1 k f kL 2 C C1 kvk

:

(32)

Thus, we see that the mapping ˚ operates on the space L p L q , provided we can satisfy the following set of conditions: (30) p˜0 D p ; q˜0 D q; 2 n n n n 2˜ C D ; C D ; ˜ p q˜ p q 2 2

 2n p; p˜ 2 [2; 1] ; q˜ 2 2; : n2 Notice that from the first two identities we have

The Nonlinear Schrödinger Equation Strichartz estimates for the Schrödinger equation are expressed entirely in terms of Lp norms, without loss of derivatives, in contrast with the case of the wave equations. For this reason they represent a perfect tool to handle semilinear perturbations. Using Strichartz estimates, several important results can be proved with minimal effort. As an example, we chose to illustrate in this Section the critical and subcritical well posedness in L2 in the case of power nonlinearities (see [13] for a systematic treatment).

n 2

n

2 C D˛ Cn p˜ q˜ p q and by the admissibility conditions this implies n n

D2Cn ; 2 2 which forces the value of to be

D1C

4 : n

This value of is called L2 critical.

(33)

171

172

Dispersion Phenomena in Partial Differential Equations

Let us assume for the moment that is exactly the critical exponent (33). Then it is easy to check that there exist ˜ which satisfy all the above conchoices of (p; q) and ( p˜ ; q) straints. To prove that ˚ is a contraction on the space

p˜0 < p ;

we take any v1 ; v2 2 X and compute   k˚(v1 )  ˚(v2 )k X . jv1 j  jv2 j  L p˜ 0 L q˜0   . jv1  v2 j(jv1 j 1 C jv2 j 1 )

0

L p˜ L q˜

 1

k˚ (v1 )  ˚(v2 )k X  C2 kv1  v2 k X kjv1 j C jv2 jk X

0

;

(34)

for some constant C2 . Now assume that

 C2 T  kv1  v2 k X T kjv1 j C jv2 jkX1 : T

provided "; ı are so small that

k f kL 2 <

On the other hand, by (34) we get 1 kv1 v2 k X 2

provided " is so small that 1 : 2

In conclusion, if the initial data are small enough, the mapping ˚ is a contraction on a ball B(0; ") X and the unique fixed point is a global small solution of the critical L2 equation 4

iu t  u D juj1C n : Notice that in estimate (32) the first couple can also be taken equal to (1; 2), thus the solution we have constructed belongs to L1 L2 . A limit argument (dominated convergence) in the identity t

(35)

(36)

Now let M be so large and T so small that

" C1 " < : 2 

Z

and proceeding as above and using Hölder inequality in time, we obtain the inequality. Indeed, choosing the indices as above and applying Hölder’s inequality in time we have

k˚ (v1 )  ˚(v2 )k X T

k˚(v)k X  C1 (ı C " ) < "

u D e i t f C i

q˜0 D q

for some strictly positive  > 0. Analogously, estimate (34) will be replaced by

using (32) we can write

C2 (2") 1 <

p˜0 < p ;

k˚(v)k X T  C1 k f kL 2 C C1 T  kvk X T

k f kL 2 < ı ;

k˚(v1 )˚(v2 )k X  C2 kv1 v2 k X (2") 1 <

q˜0 < q :

˜ so that It is easy to check that we can shift the point ( p˜ ; q)

the same computation as above gives

" C1 ı < ; 2

p

X T D L T L q D L p (0; T; L q (Rn )) : If we follow the steps of the preceding proof and choose the same indices, we have now

X D L p Lq

kv i k X < " ;

We briefly sketch a proof of the local existence in the subcritical case 1 < < 1 C n4 . Consider the local space

e i(ts) ju(s)j ds ;

0

proves that we have actually u 2 C(R; L2 (Rn )).

M ; 2C1

C1 T  <

M 2

we obtain from (35) that ˚ takes the ball B(0; M) X T into itself; if in addition we have also, by possibly taking T smaller, C2 T  (2M) 1 <

1 2

we see by (36) that ˚ is a contraction, and in conclusion we obtain the existence of a unique local solution to the subcritical equation. By the same argument as above this solution is in C([0; T]; L2 (Rn )). We notice that in the above proof the quantity M is only determined by the L2 norm of f , and hence also the lifespan of the local solution is a function of k f kL 2 . If for some reason we know a priori that the L2 norm of the solution is conserved by the evolution, this remark implies immediately that the solution is in fact global, by an elementary continuation argument. This is the case for instance of the gauge invariant equation iu t  u D ˙juj 1 u for which any solution continuous with values in L2 satisfies ku(t)kL 2 D ku(0)kL 2 . The above computations apply without modification and we obtain the global well posedness for all subcritical exponents .

Dispersion Phenomena in Partial Differential Equations

Future Directions The dispersive properties of constant coefficient equations are now fairly well understood. In view of the wide range of applications, it would be very useful to extend the techniques to more general classes of equations, in particular for equations with variable coefficients and on manifolds. These problems have been the subject of intense interest in recent years and are actively pursued. Strichartz estimates have been extended to very general situations, including the fully variable coefficients case, under suitable smoothness, decay and spectral assumptions on the coefficients (see e. g. [36,37] and, for singular coefficients, [18]). The results concerning dispersive estimates are much less complete. They have been proved for perturbations of order zero iu t  u C V(x)u D 0 ;

u t t  u C V (x)u D 0

with a potential V(x) sufficiently smooth and decaying at infinity (see e. g. [2,19,28,53,54,55]) while for perturbation of order one iu t  u C a(x)  ru x j C V(x)u D 0 ; u t t  u C a(x)  ru x j C V(x)u D 0 very few results are available, (see [16] for the threedimensional wave and Dirac equations, and [1,17] for the one-dimensional case where quite general results are available). A closely connected problem is the study of the dispersive properties of equations on manifolds. This part of the theory is advancing rapidly and will probably be a very interesting direction of research in the next years (see e. g. [12] and related papers for the case of compact manifolds, and [5,24] for noncompact manifolds). Bibliography 1. Artbazar G, Yajima K (2000) The Lp -continuity of wave operators for one dimensional Schrödinger operators. J Math Sci Univ Tokyo 7(2):221–240 2. Beals M (1994) Optimal L1 decay for solutions to the wave equation with a potential. Comm Partial Diff Eq 19(7/8):1319– 1369 3. Ben-Artzi M, Trèves F (1994) Uniform estimates for a class of evolution equations. J Funct Anal 120(2):264–299 4. Bergh J, Löfström J (1976) Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften, No 223. Springer, Berlin 5. Bouclet J-M, Tzvetkov N (2006) On global strichartz estimates for non trapping metrics. 6. Bourgain J (1993) Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom Funct Anal 3(2):107–156

7. Bourgain J (1995) Some new estimates on oscillatory integrals. In: (1991) Essays on Fourier analysis in honor of Elias M. Stein. Princeton Math Ser vol. 42. Princeton Univ Press, Princeton, pp 83–112 8. Bourgain J (1995) Estimates for cone multipliers. In: Geometric aspects of functional analysis. Oper Theory Adv Appl, vol 77. Birkhäuser, Basel, pp 41–60 9. Bourgain J (1998) Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int Math Res Not 5(5):253–283 10. Brenner P (1975) On Lp  Lp0 estimates for the wave-equation. Math Z 145(3):251–254 11. Brenner P (1977) Lp  Lp0 -estimates for Fourier integral operators related to hyperbolic equations. Math Z 152(3):273–286 12. Burq N, Gérard P, Tzvetkov N (2003) The Cauchy problem for the nonlinear Schrödinger equation on a compact manifold. J Nonlinear Math Phys 10(1):12–27 13. Cazenave T (2003) Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol 10. New York University Courant Institute of Mathematical Sciences, New York 14. Choquet-Bruhat Y (1950) Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique. CR Acad Sci Paris 230:618–620 15. Christodoulou D, Klainerman S (1989) The nonlinear stability of the Minkowski metric in general relativity. In: Bordeaux (1988) Nonlinear hyperbolic problems. Lecture Notes in Math, vol 1402. Springer, Berlin, pp 128–145 16. D’Ancona P, Fanelli L (2006) Decay estimates for the wave and dirac equations with a magnetic potential. Comm Pure Appl Anal 29:309–323 17. D’Ancona P, Fanelli L (2006) Lp – boundedness of the wave operator for the one dimensional Schrödinger operator. Comm Math Phys 268:415–438 18. D’Ancona P, Fanelli L (2008) Strichartz and smoothing estimates for dispersive equations with magnetic potentials. Comm Partial Diff Eq 33(6):1082–1112 19. D’Ancona P, Pierfelice V (2005) On the wave equation with a large rough potential. J Funct Anal 227(1):30–77 20. D’Ancona P, Georgiev V, Kubo H (2001) Weighted decay estimates for the wave equation. J Diff Eq 177(1):146–208 21. Foschi D (2005) Inhomogeneous Strichartz estimates. J Hyperbolic Diff Eq 2(1):1–24 22. Ginibre J, Velo G (1985) The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann Inst Poincaré H Anal Non Linéaire 2(4):309–327 23. Ginibre J, Velo G (1995) Generalized Strichartz inequalities for the wave equation. J Funct Anal 133(1):50–68 24. Hassell A, Tao T, Wunsch J (2006) Sharp Strichartz estimates on nontrapping asymptotically conic manifolds. Am J Math 128(4):963–1024 25. Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations. Mathématiques & Applications (Mathematics & Applications), vol 26. Springer, Berlin 26. John F (1979) Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr Math 28(1–3):235– 268 27. John F, Klainerman S (1984) Almost global existence to nonlinear wave equations in three space dimensions. Comm Pure Appl Math 37(4):443–455 28. Journé J-L, Soffer A, Sogge CD (1991) Decay estimates for Schrödinger operators. Comm Pure Appl Math 44(5):573–604

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43. Shatah J, Struwe M (1998) Geometric wave equations. In: Courant Lecture Notes in Mathematics, vol 2. New York University Courant Institute of Mathematical Sciences, New York 44. Strichartz RS (1970) Convolutions with kernels having singularities on a sphere. Trans Am Math Soc 148:461–471 45. Strichartz RS (1970) A priori estimates for the wave equation and some applications. J Funct Anal 5:218–235 46. Strichartz RS (1977) Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. J Duke Math 44(3):705–714 47. Tao T (2000) Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm Part Diff Eq 25(7–8):1471–1485 48. Tao T (2003) Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J Diff Eq 189(2):366–382 49. Taylor ME (1991) Pseudodifferential operators and nonlinear PDE. Progr Math 100. Birkhäuser, Boston 50. von Wahl W (1970) Über die klassische Lösbarkeit des CauchyProblems für nichtlineare Wellengleichungen bei kleinen Anfangswerten und das asymptotische Verhalten der Lösungen. Math Z 114:281–299 (in German) 51. Wolff T (2001) A sharp bilinear cone restriction estimate. Ann Math (2) 153(3):661–698 52. Yajima K (1987) Existence of solutions for Schrödinger evolution equations. Comm Math Phys 110(3):415–426 53. Yajima K (1995) The W k;p -continuity of wave operators for Schrödinger operators. J Math Soc Japan 47(3):551–581 54. Yajima K (1995) The W k;p -continuity of wave operators for schrödinger operators. III. even-dimensional cases m  4. J Math Sci Univ Tokyo 2(2):311–346 55. Yajima K (1999) Lp -boundedness of wave operators for two-dimensional schrödinger operators. Comm Math Phys 208(1):125–152

Dynamics on Fractals

˜

Dynamics on Fractals RAYMOND L. ORBACH Department of Physics, University of California, Riverside, USA Article Outline Glossary Definition of the Subject Introduction Fractal and Spectral Dimensions Nature of Dynamics on Fractals – Localization Mapping of Physical Systems onto Fractal Structures Relaxation Dynamics on Fractal Structures Transport on Fractal Structures Future Directions Bibliography Glossary Fractal Fractal structures are of two types: deterministic fractals and random fractals. The former involves repeated applications of replacing a given structural element by the structure itself. The process proceeds indefinitely, leading to dilation symmetry: if we magnify part of the structure, the enlarged portion looks the same as the original. Examples are the Mandelbrot– Given fractal and the Sierpinski gasket. A random fractal obeys the same properties (e. g. dilation symmetry), but only in terms of an ensemble average. The stereotypical example is the percolating network. Fractals can be constructed in any dimension, d, with for example, d D 6 being the mean-field dimension for percolating networks. Fractal dimension The fractal dimension represents the “mass” dependence upon length scale (measuring length). It is symbolized by Df , with the number of sites on a fractal as a function of the measurement length L being proportional to L D f , in analogy to a homogeneous structure embedded in a dimension d having mass proportional to the volume spanned by a length L proportional to Ld . Spectral (or fracton) dimension The spectral (or fracton) dimension refers to the dynamical properties of fractal networks. It is symbolized by d˜s and can be most easily thought of in terms of the density of states of a dynamical fractal structure (e. g. vibrations of a fractal network). Thus, if the excitation spectrum is measured as a function of energy !, the density of states for excitations of a fractal network would be propor-

tional to ! (d s 1) , in analogy to a homogeneous structure embedded in a dimension d having a density of states proportional to ! (d1) . Localization exponent Excitations on a fractal network are in general strongly localized in the sense that wave functions fall off more rapidly than a simple exponential,  (r) exp[f[r/(!)]d  g] where (!) is an energy dependent localization length, and the exponent d is in general greater than unity. Definition of the Subject The dynamical properties of fractal networks are very different from homogeneous structures, dependent upon strongly localized excitations. The thermal properties of fractals depend upon a “spectral dimension” d˜s less than the “Euclidean” or embedding dimension d. Fractal randomness introduces statistical properties into relaxation dynamics. Transport takes place via hopping processes, reminiscent of Mott’s variable range-rate hopping transport for electronic impurity states in semi-conductors. Fractal dynamics serve as a guide for the behavior of random structures where the short length scale excitations are localized. Introduction The study of the dynamics on and of fractal networks [29, 30,31] is not an arcane investigation, with little application to real physical systems. Rather, there are many examples of real materials that exhibit fractal dynamics. Examples are the vibrational properties of silica aerogels [21,45], and magnetic excitations in diluted antiferromagnets [18,19,43]. Beyond these explicit physical realizations, one can learn from the very nature of the excitations on fractal structures how localized excitations in homogeneous materials behave. That is, the dynamics of fractal networks serve as a calculable model for transport of localized excitations in random structures that are certainly not mass fractals. Examples are thermal transport in glasses above the so-called “plateau” temperature [2,20,26], and the lifetime of high-energy lattice vibrations in a-Si [32,33,36,37]. As we shall show in subsequent sections, the former is an example of vibrational hopping-type transport (similar in nature to Mott’s variable-range hopping [22,23] for localized electronic states), while the latter exhibits counter-intuitive energy dependences for vibrational lifetimes. The following Sect. “Fractal and Spectral Dimensions” describes in brief terms the concept of the fractal (or mass) dimension, Df [7,8,9,10,15,40,41,46], and the spectral (or fracton) dimension d˜s [5,14,17,34,35]. The latter is re-

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lated to the anomalous diffusion characteristics of fractals (the so-called “ant in a labyrinth” introduced by de Gennes [8]), and is at the heart of the thermal properties of fractal structures. A conjecture introduced by Alexander and Orbach [5] suggested that the mean-field value (exact at d D 6) for d˜s for percolating networks, d˜s D 4/3, might be universal, independent of dimension. It is now generally regarded that this conjecture is only approximate, though a very good approximation for percolating networks in 2 D d < 6. Were it to be exact, it would mean that the dynamical properties of percolating networks could be expressed in terms of their geometrical properties. Section “Nature of Dynamics on Fractals–Localization” introduces the nature of the wave function for an excitation embedded in a fractal structure. In general, random self-similar structures generate “super-localized” wave functions, falling off faster than exponential. We will characterize the envelope of the wave function by the functional form [1]  (r) exp[f[r/(!)]d  g] ; where (!) is an energy-dependent localization length, and the exponent d is in general greater than unity. For excitations on fractal structures, the localization length (!) can be calculated analytically, allowing for explicit computation of physical phenomena. This allows analytic expressions for scattering lifetimes and excitation transport. Mapping of the diffusion problem onto the secular equation for scalar lattice vibrations allows the extraction of the density of vibrational states [5]. It is given ˜ by the simple relationship D(!) ! (d s 1) displaying the utility of the spectral dimension d˜s . A similar though slightly more complicated mapping can be applied to diluted Heisenberg antiferromagnets. The silica aerogels [13,16,42,44,45], and Mnx Zn1x F2 [43] and RbMnx Mg1x F3 [18,19] are realizations of such systems, respectively, and have been extensively investigated experimentally. The random nature of fractal structures introduces statistical complexity into relaxation processes. The direct (single vibration) [38] and Raman (scattering of two vibrations) [24,39] spin-lattice relaxation processes are discussed for random fractal networks in Sect. “Relaxation Dynamics on Fractal Structures”. The theoretical results are complex, related to the “Devil’s staircase” for Raman relaxation processes. Experiments probing these rather bizarre behaviors would be welcome. Given the “super-localization” of excitations on fractal networks [1,42], transport properties need to be ex-

pressed according to the same concepts as introduced by Mott [22,23] for the insulating phase of doped semiconductors—the so-called “variable range-rate hopping”. This is developed in Sect. “Transport on Fractal Structures”, and applied to the thermal transport of fractal structures [2,20,26]. The “lessons learned” from these applications suggest an interpretation of the thermal transport of glassy materials above the so-called “plateau” temperature range. This instance of taking the insights from fractal structures, and applying it to materials that are certainly not fractal yet possess localized vibrational states, is an example of the utility of the analysis of fractal dynamics. Finally, some suggestions, hinted at above, for future experimental investigations are briefly discussed in Sect. “Future Directions”. Fractal and Spectral Dimensions Fractals can be defined by the nature of the symmetry they exhibit: self-similar geometry or equivalently dilation symmetry [7,8,9,10,15,40,41,46]. This symmetry is most assuredly not translational, leading to a very different set of behaviors for physical realizations that shall be explored in subsequent Sections. Stated most simply, self-similar structures “look the same” on any length scale. Conversely, one cannot extract a length scale from observation of a self-similar structure. There are clearly limits to self-similarity for physical realizations. For example, at the atomic level, dilation symmetry must come to an end. And for finite structures, there will be a largest length scale. There are other length scales that bracket the self-similar regime. Thus, for percolating networks, an example of a random fractal [38,39], the percolation correlation length, (p), where p is the concentration of occupied sites or bonds, defines an upper limit to the regime of fractal geometry. In general, for percolating networks, the range of length scales ` over which the structure is fractal is sandwiched between the atomic length a, and the percolation correlation length (p) : a < ` < (p). Fractal structures can be deterministic or random. The former is a result of applying a structural rule indefinitely, replacing an element of a design by the design itself. Examples are the Mandelbrot–Given fractal and the Sierpinski gasket. The latter arises from a set of probabilistic rules. A simple example is the percolating network where a site or bond (e. g. in d D 2, a square grid) is occupied randomly with probability p. The resulting clusters contain a random number of sites or bonds, where the distribution function for the number of finite clusters of size s is denoted by ns (p). There exists a critical value p D pc where a connected cluster extends over all space (the “in-

Dynamics on Fractals

finite” cluster). For p < pc only finite clusters exist, with the largest cluster spanned by the length scale (p). (p) diverges at p D pc as jp  pc j . For p > pc finite clusters and the infinite cluster coexist, the number of finite clusters vanishing at p approaches unity. The probability that a site belongs to the infinite cluster varies with occupation probability p as (p  pc )ˇ . The characteristic length (p) for p > pc is a measure of the largest finite cluster size. Hence, for length scales ` > (p) the systems looks homogeneous. Thus, the fractal regime is sandwiched between the lattice constant a and the percolation correlation length (p). The remarkable utility of the percolating network becomes clear from these properties: there is a crossover length scale between fractal and homogeneous structural behavior that can be chosen at will by choosing a value for the site or bond occupation probability p > pc . Excitations of structures that map onto percolating networks can change their nature from fractal to continuous as a function of their length scale `, the transition occurring when `c (p). This property makes the percolating network the “fruit fly” of random structures. The flexibility of adjusting the crossover length `c enables a fit between those properties calculated for a percolating network with the properties of those physical systems that map onto percolating networks. The fractal dimension, Df , is defined from the dependence of the “mass” on length scale. For fractals, the number of points or bonds on the infinite cluster within a length scale ` (hence, the “mass”), for `  (p), depends upon ` as M(`) ` D f . From above it is straight forward to show Df D d  (ˇ/). For a percolation network [29] in d D 2; Df D 91/48. For d D 3; Df D 2:48 ˙ 0:09. In mean field, d D 6; Df D 4. The spectral dimension, d˜s , follows from the analysis of diffusion on a fractal network, as first postulated by de Gennes [14]: “An ant parachutes down onto an occupied site of the infinite cluster of a percolating network. At every time unit, the ant makes one attempt to jump to one of its adjacent sites. If that site is occupied, it moves there. If it is empty, the ant stays at its original site. What is the ensemble-averaged square distance that the ant travels in time t?” Gefen et al. [17] found that hr2 (t)i t 2C where depends upon the scaling exponent  (conductivity exponent), the probability that a site belongs to the infinite cluster scaling exponent ˇ, and the scaling exponent  for (p) : D (  ˇ)/. Alexander and Orbach [5] defined the spectral (or fracton) dimension as d˜s D 2Df /(2 C ). The importance of d˜s can be seen from the calculation of the probability of finding a diffusing particle at the starting point at time t, P0 (t), which for compact diffusion is the inverse of the number of visited sites in

˜

time t, V(t) t (d s /2) . For scalar elasticity, the vibrational density of states ˜

D(!) D (2!/)Im P˜0 (! 2 C i0C ) ! (d s 1) ; where P˜0 (!) is the Laplace transform of P0 (t). Noting that for homogeneous systems, D(!) ! (d1) , the name spectral dimension for d˜s becomes evident. Alexander and Orbach [5] named the vibrational excitations of fractal structures, fractons, in analogy with the vibrational excitations of homogeneous structures, phonons. Hence, they termed d˜s the fracton dimension. Alexander and Orbach noted that for percolation structures in the mean-field limit, d D 6, d˜s D 4/3 precisely. At that time (1982), values for d˜s for d < 6 appeared close to that value, and they conjectured that d˜s D 4/3 for 2  d  6. We now know that this is only approximate (but remarkably close [24]): d˜s D 1:325 ˙ 0:002 in d D 2, and 1:317 ˙ 0:003 in d D 3. Were the conjecture exact, a numerical relationship between structure and transport would be exact [ D (3Df /2)  2], dictated solely by fractal geometry. For percolating networks, the structure appears fractal for length scales `  (p) and homogeneous for `  (p). The vibrational excitations in the fractal regime are termed fractons, and in the homogeneous regime phonons. We define the crossover frequency !c as that for which the excitation length scale equals (p). Then !c (p  ˜ pc )( D f /d s ) . Continuity leads to a phonon velocity v(p) (p  pc ) [(Df /d˜s )  1]. This then leads to the dispersion relations: [`  (p); !  !c ; phonon regime]! v(p)k ; and ˜

[`  (p); !  !c ; fracton regime]! k (D f /d s ) : In the latter case, as we shall show in the next Section, vibrational excitations in the fractal regime are localized, so that k should not be thought of as a wave vector but rather the inverse of the localization length (!). Nature of Dynamics on Fractals – Localization The conductance of a d dimensional percolating network of size L; G(L), can be shown to be proportional to Lˇ where the exponent ˇ D (Df /d˜s )(d˜s  2). In the Anderson sense [6], localization occurs when ˇ  0 (marginal at ˇ D 0). The Alexander–Orbach conjecture [5], d˜s  4/3, leads to ˇ well less than zero for all embedding dimensions d. One can think of this form of localization as geometrical, as opposed to the scattering localization that traditionally is associated with Anderson localization. The lo-

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calization is strong, the wave function falling off faster than simple exponential. The wave function has the form [1]: h (r)i exp[ f[r/(!)]d  g]. The exponent d is a geometrical exponent describing the fact that an exponential decay along the fractal will be distorted when viewed in real space. In general, 1  d  dmin , where dmin is defined by ` R d min , with ` being the shortest path along the network between two points separated by a Pythagorean distance R. Bunde and Roman [11] have found that for percolating networks d D 1. The decay length (!) is the localization length ˜ from Sect. “Introduction”, (!) ! (d s /D f ) . For a random system such as a percolating network, a given realization of  (r) is badly behaved, depending upon the particular choice of origin taken for the position coordinate r. The ensemble average takes the average over all realizations of  (r), and is denoted by h (r)i. This has its dangers, as the calculation of physical properties should be performed using a specific realization for  (r), and then ensemble averaged. Calculation of physical properties utilizing an ensemble averaged wave function h (r)i will not yield the same result, and could be very misleading. Specific realizations for  (r) for two-dimensional bond percolation vibrational networks have been calculated by Nakayama and Yakubo [25,28,47]. The fracton “core” (or largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost steplike character and a long tail. The tail extends over large distances and the amplitude oscillates in sign. This is required for orthogonality to the uniform translational mode with ! D 0. Clearly these modes do not look like simple exponentials, hence the warning about the use of h (r)i as compared to  (r) for the calculation of matrix elements. Mapping of Physical Systems onto Fractal Structures The foregoing description of dynamics of fractal networks would be interesting in and of itself, but its applicability to physical systems makes it of practical importance. Further, the nature of localization, and computation of “hopping-type” transport on fractal structures, has application to materials that are not fractal. Thus, it is worthwhile to consider what systems are fractal, the nature of dynamics on such systems, and the lessons we can learn from them for structures that possess similar characteristics but are not fractal. Specifically, Mott’s [22,23] variable range-rate hopping for localized electronic impurity states has direct application to transport of vibrational energy via fracton hopping on fractal networks [2], and we shall argue to

thermal transport in glassy materials above the “plateau” temperature [20,26]. The fundamental equations for fractal dynamics spring from the equation for diffusion of a random walker P on a network: dPi /dt D j¤I w i j (P j  Pi ) where Pi is the probability that the ith site is occupied, wij is the probability per unit time for hopping from site i to site j, and w i j D w ji . Introducing new quantities Wi j D w i j for P i ¤ j, and Wi i D  j¤I w i j , one can rewrite the difP fusion equation as dPi /dt D j Wi j P j with the defined P relation j Wi j D 0. For scalar elasticity, the equation of motion for atomic P vibrations is d2 u i /dt 2 D j K i j u j , with ui the atomic displacement at position i, and K i j (i ¤ j) is given by K i j D k i j /m i with kij the force constant connecting two sites i and j, and mi the mass of the ith site. The diagonal element W ii is obtained from the balancing of forces at P the ith site: j K i j D 0. The only difference between these two equations is the order of the time derivatives, leading to the equivalence of the eigenvalue problems. We have used this equivalence to extract the density of vibrational states for scalar elasticity, and the definition of the spectral (or fracton) dimension in Sect. “Introduction”. The most direct physical example of a vibrating fractal network is the silica aerogel [13,16,42,44]. Different preparation conditions can change the density of these materials to very small values, indicative of the open porous structure of these materials. Two length scales are present: the fundamental building block of the structure (the silica), and the correlation length of the gel. In between, the clusters possess a fractal structure, while at length scales beyond the correlation length the gel behaves as a homogeneous porous glass. This crossover from fractal to homogeneous is the physical example of our previous discussion for percolating networks when the length scale of excitations passes through the percolation correlation length (p). The thermal, transport, and scattering properties of aerogels have been well studied, and are the classic example of fractal behavior. There are other examples of structures that map onto the eigenvalue spectrum of the diffusion equation. Ferromagnetic and antiferromagnetic materials can be diluted by non-magnetic impurities. This leads to random site dilution, mapping directly onto the site-diluted percolating network for magnetic interactions. This can be seen from the Heisenberg Hamiltonian for spin systems: P H D 1/2 i; j J i j S i S j , where S i is the vector spin at the ith site, and J ij the isotropic exchange interaction between sites i and j. The equation of motion for the spin operx x ator S C i D S i C iS i becomes for diluted ferromagnets

Dynamics on Fractals

P z C z C i„@S C j¤i J i j (S i S j  S j S i ). For low levels of i /@t D spin wave excitations, S zi  S zj  S, so that one obP tains the linear equation of motion i„@S C j¤I J i j i /@t D C C (S j  S i ). This is the same form as that for diffusion of a random walker on a network, so that, in analogy with the scalar vibrational network, the eigenvalue spectrum is the same. This mapping allows the spin wave spectrum to exhibit magnons and fracton waves obeying the same properties that phonons and fractons exhibit for scalar vibrations. For antiferromagnets, the change in sign of S zi for the two sublattices complicates the linearized equations of P C C motion. They become, i„@S C j¤I J i j (S j CS i ), i /@t D  i where  i D 1 for down spins and  i D 1 for up spins. This alters the dynamical equations for diluted antiferromagnets, and requires a separate calculation of the eigenvalue spectrum. Neutron diffraction experiments exhibit both conventional magnon and fracton excitations. In the case of the site-diluted antiferromagnet Mnx Zn1x F2 both excitations are observed simultaneously for length scales in the vicinity of (p) [43]. Relaxation Dynamics on Fractal Structures Electron spin resonance and non-radiative decay experiments can be performed in fractal materials. It is of interest to compare these dynamical properties with those found in translationally ordered structures. At first sight, for vibrational relaxation, it might be thought that the only difference would originate with the differences in the densities of vibrational states D(!). However, the localized nature of the vibrational states (fractons) and their random positions in the fractal network, introduces properties that are specific to fractal lattices. There are two cases of interest: one-fracton relaxation, and two-fracton inelastic scattering. The former is referred to as the “direct” relaxation process; the latter the “Raman” relaxation process. In both cases, the main effect of vibrational localization is on the relaxation time profile. Different spatial sites can have very different relaxation rates because of the randomness of their distance from the appropriate fracton sites. The result is a strongly non-exponential time decay. For one-fracton relaxation [1], a straightforward generalization of the direct relaxation process rate, W(!0 ; L), for relaxation energy ! 0 arising from interaction with a fracton of the same energy centered a distance L away, is proportional to 2q1

W(!0 ; L) !0

f[(!0 )](D f ) g

 [coth(ˇ!0 /2)](1/ıL ) expf[L/(!0 )]d  g :

Here, ˇ D 1/kB T and the factor ı L represents the energy width of the fracton state. In the homogeneous limit, where phonons represent the extended vibrational states, an energy conserving delta function would replace ı L . There are two limits for ı L , case (a) when the fracton relaxation rate ı caused by anharmonicity dominates; and case (b) when the electronic relaxation rate caused by the electron-fracton interaction is greater than ı. In the former case, ıL D ı; while in the latter, ıL D W(!0 ; L) itself, leading to a self-consistent determination of the relaxation rate. The latter case requires explicit consideration of the L dependence of ı L , further complicating the calculation of W(!0 ; L). The calculation of the time profile of the electronic state population and the average relaxation rate are complex, and the reader is referred to [32] for details. For case (a), the population of the initial electronic state is found to vary as h i [(D f /d  )1] g : (ln t)[(D f d  )/2d  ] t fc 1 (ln t) Here, c1 is a constant independent of time, but dependent upon ! 0 and ı. The population of the initial electronic state decays is thus found to be faster than a power law, but slower than exponential or stretched exponential. For case (b), the population of the initial electronic state is found to vary as n o (1/t) (ln t)[(D f /d  )1] : The decay is slower than in case (a), and is closer to a power law. The average relaxation rates for cases (a) and (b) have the same functional form: hWi D(!0 )[(!0 )(2q1) ] coth(ˇ!0 /2) where q D d˜s (d /Df ). The temperature dependence is the usual one found for the direct process, but the energy dependence differs, depending on the values of the dimensions and parameters for fractal networks. Two fracton relaxation is considerably more complex [3,4]. In summary, the time profile of the initial electronic state in the long time regime begins as a stretched exponential, then crosses over into a form similar to case (a) for the one fracton decay process. In the presence of rapid electronic cross relaxation, the time profile is exponential, with a low-temperature relaxation time h1/T1 i ˜ proportional to T f2d s [1C2(d  /D f )]1g for Kramers transitions (between half-integer time-reversed spin states), and ˜ T f2d s [1C2(d  /D f )]3g for non-Kramers transitions (between integer time-reversed spin states). The complexity of random systems adds an interesting consideration: the nature of the relaxation rate at a specific site as compared to the average relaxation rate calculated above. The decay profile of single-site spin-lattice

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relaxation is always exponential. However, the temperature dependence does not simply follow that of the average relaxation rate. Instead, it exhibits irregular statistical fluctuations reflecting the environment of the chosen site. As the temperature is raised, new relaxation channels, involving higher-frequency localized vibrations, will become activated, adding to the relaxation processes responsible for the relaxation at low temperatures. The temperature dependence of the relaxation rate W(T) at different sites will fluctuate with strong correlations in the temperature dependence between them, underlying large accumulative fluctuations. Using a step function for the vibrational Bose functions, the relaxation rate has a T 2 temperature dependence between randomly spaced steps that occur when new relaxation channels are opened: a “devil’s staircase” structure. The full Bose function smoothes out these steps, but the qualitative features remain. All of these features point to the complexity and richness of electronic relaxation associated with localized vibrational spectra. Their remarkable complexity is worthy of investigation. Transport on Fractal Structures The localization of excitations discussed in Sect. “Fractal and Spectral Dimensions” has profound influence on the transport properties of and on fractal structures. For example, in the case of the silica aerogels, the “conventional” form for glasses or amorphous materials for the thermal conductivity (T) is found [12]: a rapid rise at low temperatures leveling off to a “plateau” as a function of temperature T, and then a rise in (T) roughly as T 2 . At first sight, this is to be “expected” as the aerogels are random, and one could make an analogy with glassy systems. However, there is a fundamental inconsistency in such an analysis. The fracton excitations are localized, and therefore cannot contribute to transport. Thus, one would expect instead a rapid rise in (T) at low temperatures where the long length scale excitations are phonon-like (and hence delocalized), flattening off to a constant value when all the extended state density of states have reached their duLong– Petit value. The specific heat for these states is then a constant, and (T) would be a constant independent of temperature T. So far, this behavior is consistent with experiment. But what happens for temperatures T greater than the plateau temperature region? How can (T) increase when the only excitations are fractons and therefore localized? The solution to this conundrum lies in the anharmonic forces connecting the silica clusters in the aerogel network. The

presence of the anharmonicity allows for a process of fracton hopping [2,20,26], in precisely the same fashion as in Mott’s [22,23] variable range-rate hopping for localized electronic impurity states in semiconductors. The fracton can absorb/emit a lower frequency extended state phonon, and hop to another fracton location, thereby transmitting excitation energy spatially. This possibility is amusing, for it means that for fractal networks the anharmonicity contributes to thermal transport, whereas in translationally invariant systems the anharmonicity inhibits thermal transport. In fact, were there no anharmonicity in a fractal network, the thermal conductivity (T) would stay at the plateau value for all T greater than the temperature T p , the onset of the plateau, for all T > Tp . In this sense, anharmonicity stands on its head. It is essential for transport in the fracton regime, defined by T  Tp  „!c /kB . The localized vibrational excitation hopping contribution to the thermal conductivity is hop (T) D (kB /2V)˙0 R2 (!0 )/sl (!0 ; T). Here sl means strongly localized modes, R(!0 ) is the hopping distance associated with the sl mode, 0 ; sl (!0 ; T) the hopping lifetime of the sl mode 0 caused by anharmonic interactions at temperature T, and V the volume of the system. We have already derived the ensemble averaged wave function for fractons, h (r)i, in Sect. “Fractal and Spectral Dimensions”. We adopt the sl notation [26] because the hopping transport arguments apply to strongly localized states, independent of whether or not they are fractons. The dynamical process for hot (T) is an sl mode at one site coupling with a low energy extended state to hop to an sl mode at another site, thereby transferring excitation energy spatially. The evaluation of sl (!0 ; T) will depend upon the hopping distance for transport of the sl modes, R(!0 ). The argument of Mott [22,23], derived by him for localized electronic states, can be taken over for sl vibrational modes [2]. The volume that contains at least one sl mode is given by (4/3)D(!0 )!sl [R(!0 )]3 D 1, where D(!0 ) is the density of sl states per unit energy. This condition assures that, for an sl mode at the origin, a second sl mode can be found within a hopping distance R(!0 ). We find R(!0 )  (!0 /!sl )(!0 ). Thus, the most probable hopping distance is at least the sl localization length. The hopping rate 1/sl (!0 ; T) caused by anharmonicity, with anharmonic coupling constant Ceff can be found from the Golden Rule, leading to the hopping contribution to the thermal conductivity, with ( M )3 the volume for finding a single sl mode, hop (T) D [43 3 (Ceff )2 (kB )2 T]/[ 3 (vs )5 3 ( M )3 h(! )i2 ]. Here h(! )i is the average sl length scale, vs the velocity of sound, and the density. There are very few undetermined parameters in this expression for hop (T). The anharmonic coupling

Dynamics on Fractals

constant can be found from the shift of the “boson peak” in the Raman spectra with pressure [48,49], typically a factor of 25 times larger than the long length scale third-order elastic coupling constant for extended modes. Making use of experiments on amorphous a-GeS2 one finds [26,27] hop (T)  0:0065T (W/mK) using the appropriate values for and vs , and  M  h(! )i D 15 Å. This contribution is comparable to that observed for other networkforming glasses above the plateau value. There is a quantum phenomenon associated with sl mode hopping. The hopping vertex is associated with a low energy extended mode coupling with an sl mode to hop to another (spatially separated) sl mode. As the temperature increases, the lifetime of the sl mode will become shorter than 1/! of the low energy extended mode. This quantum effect leads to a breakdown of the Golden Rule expression, causing a leveling off of the linear-in-T thermal conductivity above the plateau value. The sl mode hopping contribution to (T) above the plateau temperature for glasses and amorphous materials appears to provide a quantitative explanation of the observed phenomena. It is a specific example of how dynamics on a fractal network can be used to determine the properties of materials that, while certainly not fractal, have properties that map onto the behavior of fractal structures. Future Directions As stated in the Introduction, studying the dynamics on fractal networks is a convenient structure for analyzing the properties of localized states, with all of the frequency and temperature-dependent physical properties determined without arbitrary parameters. For example, the frequency dependence of the localization length scale, (!), is known precisely on fractal networks, and is used for precise calculation of the hopping transport in Sect. “Relaxation Dynamics on Fractal Structures”, without adjustable parameters. If one extrapolates the properties of fractal networks to those random systems that are certainly not fractal, but which exhibit localization, the transport and relaxation properties can be understood. Examples are the thermal transport of glasses and amorphous materials above the plateau temperature, and the lifetime of high energy vibrational states in a-Si. Measurements of these properties may offer opportunities for practical devices in frequency regimes beyond extended state frequencies. For example, the lifetime of high energy vibrational states in a-Si has been shown to increase with increasing vibrational energies [36,37]. Such behavior is opposite to that expected from extended vibrational states interacting anharmoni-

cally. But it is entirely consistent if these states are localized and behave as shown for fractal networks [32,33]. The thesis presented here points to a much broader application of the consequences of fractal geometry than simply of those physical systems which exhibit such structures. The claim is made that using what has been learned from fractal dynamics can point to explication of physical phenomena in disordered systems that are certainly not fractal, but display properties analogous to fractal structures. In that sense, the future lies in application of the ideas contained in the study of fractal dynamics to that of random systems in general. This very broad class of materials may well have properties of great practical importance, predicted from the dynamics on fractals.

Bibliography Primary Literature 1. Alexander S, Entin-Wohlman O, Orbach R (1985) Relaxation and nonradiative decay in disordered systems, I. One-fracton emission. Phys Rev B 32:6447–6455 2. Alexander S, Entin-Wohlman O, Orbach R (1986) Phonon-Fracton anharmonic interaction: the thermal conductivity of amorphous materials. Phys Rev B 34:2726–2734 3. Alexander S, Entin-Wohlman O, Orbach R (1986) Relaxation and non-radiative decay in disordered systems, II. Two-fracton inelastic scattering. Phys Rev B 33:3935–3946 4. Alexander S, Entin-Wohlman O, Orbach R (1987) Relaxation and non-radiative decay in disordered systems, III. Statistical character of Raman (two-quanta) spin-lattice relaxation. Phys Rev B 35:1166–1173 5. Alexander S, Orbach R (1982) Density of states on fractals, “fractons”. J Phys Lett (Paris) 43:L625–L631 6. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492–1505 7. Avnir D (ed) (1989) The Fractal Approach to Heterogeneous Chemistry. Wiley, Chichester 8. Barabaási A-L, Stanley HE (1995) Fractal Concepts in Crystal Growth. Cambridge University Press, Cambridge 9. Bunde A, Havlin S (eds) (1991) Fractals and Disordered Systems. Springer, Berlin 10. Bunde A, Havlin S (eds) (1994) Fractals in Science. Springer-Verlag, Berlin 11. Bunde A, Roman HE (1992) Vibrations and random walks on random fractals: anomalous behavior and multifractality. Philos Magazine B 65:191–211 12. Calemczuk R, de Goer AM, Salce B, Maynard R, Zarembowitch A (1987) Low-temperature properties of silica aerogels. Europhys Lett 3:1205–1211 13. Courtens E, Pelous J, Phalippou J, Vacher R, Woignier T (1987) Brillouin-scattering measurements of phonon-fracton crossover in silica aerogels. Phys Rev Lett 58:128–131 14. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 15. Feder J (1988) Fractals. Plenum Press, New York

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16. Freltoft T, Kjems J, Richter D (1987) Density of states in fractal silica smoke-particle aggregates. Phys Rev Lett 59:1212–1215 17. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:77–80 18. Ikeda H, Fernandez-Baca JA, Nicklow RM, Takahashi M, Iwasa I (1994) Fracton excitations in a diluted Heisenberg antiferromagnet near the percolation threshold: RbMn0.39 Mg0.61 F3 . J Phys: Condens Matter 6:10543–10549 19. Ikeda H, Itoh S, Adams MA, Fernandez-Baca JA (1998) Crossover from Homogeneous to Fractal Excitations in the Near-Percolating Heisenberg Antiferromagnet RbMn0.39 Mg0.61 F3 . J Phys Soc Japan 67:3376–3379 20. Jagannathan A, Orbach R, Entin-Wohlman O (1989) Thermal conductivity of amorphous materials above the plateau. Phys Rev B 39:13465–13477 21. Kistler SS (1932) Coherent Expanded–Aerogels. J Phys Chem 36:52–64 22. Mott NF (1967) Electrons in disordered structures. Adv Phys 16:49–144 23. Mott NF (1969) Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands. Philos Mag 19:835–852 24. Nakayama T (1992) Dynamics of random fractals: large-scale simulations. Physica A 191:386–393 25. Nakayama T (1995) Elastic vibrations of fractal networks. Japan J Appl Phys 34:2519–2524 26. Nakayama T, Orbach R (1999) Anharmonicity and thermal transport in network glasses. Europhys Lett 47:468–473 27. Nakayama T, Orbach RL (1999) On the increase of thermal conductivity in glasses above the plateau region. Physica B 263– 264:261–263 28. Nakayama T, Yakubo K (2001) The forced oscillator method: eigenvalue analysis and computing linear response functions. Phys Rep 349:239–299 29. Nakayama T, Yakubo K (2003) Fractal Concepts in Condensed Matter Physics. Solid-State Sciences. Springer, Berlin 30. Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 31. Orbach R (1986) Dynamics of Fractal Networks. Science 231:814–819 32. Orbach R (1996) Transport and vibrational lifetimes in amorphous structures. Physica B 219–220:231–234 33. Orbach R, Jagannathan A (1994) High energy vibrational lifetimes in a-Si:H. J Phys Chem 98:7411–7413 34. Rammal R (1984) Spectrum of harmonic excitations on fractals. J Phys (Paris) 45:191–206 35. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett (Paris) 44:L13–L22

36. Scholten AJ, Dijkhuis JI (1996) Decay of high-frequency phonons in amorphous silicon. Phys Rev B 53:3837–3840 37. Scholten AJ, Verleg PAWE, Dijkjuis JI, Akimov AV, Meltzer RS, Orbach R (1995) The lifetimes of high-frequency phonons in amorphous silicon: evidence for phonon localization. Solid State Phenom 44–46:289–296 38. Stanley HE (1977) Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents. J Phys A 10:L211–L220 39. Stauffer D, Aharony A (1992) Introduction to Percolation Theory, 2nd edn. Taylor and Francis, London/Philadelphia 40. Stauffer D, Stanley HE (1995) From Newton to Mandelbrot: A primer in Theoretical Physics, 2nd edn. Springer, Berlin 41. Takayasu H (1990) Fractals in the Physical Sciences. Manchester University Press, Manchester 42. Tsujimi Y, Courtens E, Pelous J, Vacher R (1988) Raman-scattering measurements of acoustic superlocalization in silica aerogels. Phys Rev Lett 60:2757–2760 43. Uemura YJ, Birgeneau RJ (1987) Magnons and fractons in the diluted antiferromagnet Mnx Zn1x F2 . Phys Rev B 36:7024– 7035 44. Vacher R, Courtens E, Coddens G, Pelous J, Woignier T (1989) Neutron-spectroscopy measurement of a fracton density of states. Phys Rev B 39:7384–7387 45. Vacher R, Woignier T, Pelous J, Courtens E (1988) Structure and self-similarity of silica aerogels. Phys Rev B 37:6500–6503 46. Vicsek T (1993) Fractal Growth Phenomena. World Scientific, Singapore 47. Yakubo K, Nakayama T (1989) Direct observation of localized fractons excited on percolating nets. J Phys Soc Japan 58:1504–1507 48. Yamaguchi M, Nakayama T, Yagi T (1999) Effects of high pressure on the Bose peak in a-GeS2 studied by light scattering. Physica B 263–264:258–260 49. Yamaguchi M, Yagi T (1999) Anharmonicity of low-frequency vibrations in a-GeS2 studied by light scattering. Europhys Lett 47:462–467

Recommended Reading While many of the original articles are somewhat difficult to access, the reader is nevertheless recommended to read them as there are subtle issues that tend to disappear as references are quoted and re-quoted over time. For a very helpful introduction to percolation theory, Ref. 39 is highly recommended. A thorough review of dynamics of/on fractal structures can be found in Ref. 30. Finally, a comprehensive treatment of localization and multifractals, and their relevance to dynamics of/on fractal networks, can be found in Ref. 29.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation ALAN CHAMPNEYS Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom Article Outline Glossary Definition of the Subject Introduction Linear Resonance or Nonlinear Instability? Multibody Systems Continuous Systems Future Directions Bibliography Glossary Parametric excitation Explicit time-dependent variation of a parameter of a dynamical system. Parametric resonance An instability that is caused by a rational relationship between the frequency of parametric excitation and the natural frequency of free oscillation in the absence of the excitation. If ! is the excitation frequency, and ! 0 the natural frequency, then parametric resonance occurs when ! D (n/2)!0 for any positive integer n. The case n D 1 is usually the most prominent form of parametric resonance, and is sometimes called the principle subharmonic resonance. Autoparametric resonance A virtual parametric resonance that occurs due to the coupling between two independent degrees of freedom within a system. The output of one degree of freedom acts like the parametric excitation of the other. Ince–Strutt diagram A two-parameter bifurcation diagram indicating stable and unstable regions, specifically plotting the instability boundaries as the required amplitude of parametric excitation against the square of the ratio of natural to excitation frequency. Floquet theory The determination of the eigenvalue spectrum that governs the stability of periodic solutions to systems of ordinary differential equations. Bifurcation A qualitative change in a system’s dynamics as a parameter is varied. One-parameter bifurcation diagrams often depict invariant sets of the dynamics against a single parameter, indicating stability and any bifurcation points. Two-parameter bifurcation diagrams depict curves in a parameter plane on which one-parameter bifurcations occur.

Modal analysis The study of continuum models by decomposing their spatial parts into eigenmodes of the dynamic operator. The projection of the full system onto each mode gives an infinite system of differential equations in time, one for each mode. Monodromy matrix The matrix used in Floquet theory, whose eigenvalues (also known as Floquet multipliers) determine stability of a periodic solution. Definition of the Subject Parametric excitation of a system differs from direct forcing in that fluctuations appear as temporal modulation of a parameter rather than as a direct additive term. A common paradigm is that of a pendulum hanging under gravity whose support is subjected to a vertical sinusoidal displacement. In the absence of any dissipative effects, instabilities occur to the trivial equilibrium whenever the natural frequency is a multiple of half the excitation frequency. At amplitude levels beyond the instability, further bifurcations (dynamical transitions) can lead to more complex quasi-periodic or chaotic dynamics. In multibody mechanical systems, one mode of vibration can effectively act as the parametric excitation of another mode through the presence of multiplicative nonlinearity. Thus autoparametric resonance occurs when one mode’s frequency is a multiple of half of the other. Other effects include combination resonance, where the excitation is at a sum or difference of two modal frequencies. Parametric excitation is important in continuous systems too and can lead to counterintuitive effects such as stabilization “upside-down” of slender structures, complex wave patterns on a fluid free surface, stable pulses of light that overcome optical loss, and fluid-induced motion from parallel rather than transverse flow. Introduction Have you observed a child on a playground swing? In either a standing or a sitting position, generations of children have learned how to “pump” with their legs to set the swing in motion without making contact with the ground, and without external forcing. Note that pumping occurs at twice the natural frequency of swing of the pendulum, since the child extends her legs maximally at the two extremities of the swing’s cycle, that is, two times per period. What is happening here? The simplest case to analyze is where the child is standing. Then, it has been argued by Curry [17] that the child plus swing is effectively a simple pendulum system, where the child’s pumping has the effect of periodic variation of the position of the center of the mass along the arm of the pendulum (see Fig. 1a,b).

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Dynamics of Parametric Excitation, Figure 1 a A child standing on a swing modeled as b a parametrically excited pendulum c a directly forced pendulum

We shall return to the self-propelled swing problem in Sect. “Future Directions” below, where we shall find out that all is not what it seems with this motivating example, but for the time being lets make the simple hypothesis that the child’s pumping causes the center of mass to move up and down. Using the standard model of a pendulum that consists of a lumped mass suspended by a rigid massless rod of length l from a pivot, the equations of motion for the angular displacement (t) of a pendulum hanging under gravity can be written in the form g ¨ (t) C sin[ (t)] D 0 : l Here g is the acceleration due to gravity, a dot denotes differentiation with respect to time t and, for the time being, we have ignored any damping that must inevitably be present. Now, the moving center of mass means that the effective length of the rigid rod oscillates, and let us suppose for simplicity that this oscillation is sinusoidal about some mean length l0 : l(t) D l0 [1  " cos(! t)] : Here " > 0 assumed to be small and as yet we haven’t specified the frequency !. Then, upon Taylor expansion in ", we find to leading order ¨ C !02 [1 C " cos(! t)] sin D 0 ; p where !0 D (g/l0 ) is the natural frequency of small amplitude swings in the absence of the pump. Finally, upon rescaling time ˜t D ! t and dropping the tildes we find ¨ C [˛ C ˇ cos t] sin D 0 ;

(1)

where ˛D

!02 !2

and ˇ D "

!02 !2

(2)

are dimensionless parameters that respectively describe the square of the ratio of natural frequency to excitation frequency, and the amplitude of the excitation. Equation (1) is a canonical example of a parametrically excited system. Note that (1) also describes, at least to leading order, the case of a pendulum hanging under gravity that is excited by a vertical force of size "!02 cos t which we can think of as a periodic modulation of the gravity. This would be in contrast to a directly forced simple pendulum where the forcing would act in the direction of swing for small amplitude motion; see Fig. 1c. For the forced pendulum, the equivalent equations of motion would look more like ¨ C ˛ sin D ˇ cos t ;

(3)

where the periodic forcing occurs as an additive input into the differential equation. In contrast, for (1) the periodic excitation occurs as a modulation of a parameter (which you can think of as gravity), hence the name ‘parametric’. Another feature of (1) that makes it a good paradigm for this article is that it is nonlinear and, as we shall see, a full appreciation of the dynamics resulting from parametric excitation requires treatment of nonlinear terms. An interesting contrast between parametrically excited and directly forced systems is in the nature of the trivial response to a non-zero input. In particular, for any ˛ and ˇ, Eq. (1) admits the trivial solution (t)  0, whereas there is no such solution to (3) for non-zero ˇ. The simplest “steady” solution to the directly forced system for small ˇ is a small-amplitude periodic oscillation of period 2 (in our dimensionless time units). So any analysis of a parametrically excited system should start with an analysis of the steady state D 0, which corresponds to the pendulum just hanging under gravity. To analyze stability of this equilibrium state let D 0C x where x is small, then x sat-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 2 Possible behavior of Floquet multipliers  (eigenvalues of the monodromy matrix M) for a a conservative Hill equation and b in the presence of small damping

isfies linearization (1) x¨ C (˛ C ˇ cos t)x D 0 :

(4)

The equilibrium position is stable if all solutions x(t) of this equation, remain bounded as t ! 1 and it is unstable if there are solutions x(t) ! 1 as t ! 1.

Equation (4) is known as the Mathieu equation [42], and is a specific example of Hill’s equation [29,40] x¨ C [˛ C V(t)]x D 0 ;

(5)

where V(t) is any periodic function with period 2. Hill’s equation can be solved using classical methods for linear

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time-dependent ordinary differential equations (ODEs), see e. g. [31,33]. In particular, by writing y1 D x and y2 D x˙ , Hill’s equation (5) is just a two-dimensional example of a linear first-order time-periodic equation y˙ D P(t)y ;

y 2 Rn :

(6)

The general solutions to (6) for any initial condition can be expressed in terms of a fundamental matrix ˚ (t), with ˚(0) D I n (the identity matrix of dimension n): y(t) D ˚(t)y(0) : Note that just because P(t) has period 2 it does not follow that ˚(t) does. Also ˚(t) cannot be computed exactly, except in highly specialized cases, and one has to rely on approximate methods as described in the next Sect. “Floquet Analysis” and in the companion article to this one  Perturbation Analysis of Parametric Resonance. However, in order to study stability we don’t actually need to construct the full solution ˚ (t), it is sufficient to consider M D ˚(2), which is called the monodromy matrix associated with Hill’s equation. Stability is determined by

studying the eigenvalues of M: solutions to (5) with small initial conditions remain small for all time if all eigenvalues of M lie on or within the unit circle in the complex plane; whereas a general initial condition will lead to unbounded motion if there are eigenvalues outside this circle. See Fig. 2a. Eigenvalues of the Monodromy matrix are also known as Floquet multipliers of the system, and the general study of the stability of periodic systems is called Floquet theory. In the case of the Mathieu equation, where V (t) D ˇ cos(t), we have

 0 1 P(t) D : (7) 0 ˛ C ˇ cos(t) Figure 3 depicts graphically the behavior of Floquet multipliers for this case. Here, in what is known as an Ince– Strutt diagram [31,63], the shaded regions represent the values of parameters at which the origin x D 0 is stable (with Floquet multipliers on the unit circle) and the white regions are where the origin is unstable (at least one multiplier outside the unit circle). We describe how to calculate such a diagram in the next section. For the time be-

Dynamics of Parametric Excitation, Figure 3 Ince–Strutt diagram showing resonance tongues of the Mathieu equation (4). Regions of instability of the trivial solution x D 0 are shaded and a distinction is drawn between instability curves corresponding to a period-T (T D 2) and to a period-2T orbit. After (2nd edn. in [33]), reproduced with permission from Oxford University Press

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 4 Similar to Fig. 3 but for the damped Mathieu equation (8) with ı D 0:1. After http://monet.unibas.ch/~elmer/pendulum/parres.htm, reproduced with permission

ing we note that there are resonance tongues that emanate from ˇ D 0 whenever ˛ is the square of a half-integer: ˛ D 1/4, 1, 9/4, 4, 25/4, etc. The width (in ˛) of the tongue for small ˇ turns out to be proportional to ˇ to the power of twice the half-integer that is being squared. That is, the ˛ D 1/4 tongue has width O(ˇ), the ˛ D 1 tongue has width O(ˇ)2 , the ˛ D 9/4 tongue has width O(ˇ 3 ) etc. On the tongue boundaries (between the shaded and the unshaded regions of Fig. 3), the Mathieu equation admits non-trivial periodic solutions. For the tongues that are rooted at square integers, ˛ D 1, 9, 16, etc., these solutions have period 2 (the same period as that of the excitation) and the instability that occurs upon crossing such a tongue boundary is referred to as a harmonic instability. On the other boundaries, those of tongues rooted at ˛ D 1/4, 9/4, 25/4 etc., the periodic solution has period 4 (twice that of the excitation). Hence such instabilities are characterized by frequencies that are half that of the excitation and are hence referred to as subharmonic instabilities. So far, we have ignored any dissipative effects, so that Eq. (5) can be regarded as a conservative or Hamiltonian system. Damping can be introduced via a dissipative force that is proportional to velocity, hence (4) becomes x¨ C ı x˙ C (˛ C ˇ cos t)x D 0 ;

(8)

where ı is a dimensionless damping coefficient. One can also apply Floquet theory to this damped Mathieu equation; see Fig. 2b. The results for Eq. (8) are plotted in Fig. 4, where we can see the intuitive effect that damping

increases the areas of stability. In particular, in comparison with Fig. 3, note that the resonance tongues have been lifted from the ˛-axis, with the amount of the raise being inversely proportional to the width of the undamped tongue. Thus, the resonance tongue corresponding to ˛ D 1/4 is easily the most prominent. Not only for fixed forcing amplitude ˇ does this tongue occupy for the greatest interval of square frequency ratio ˛, but it is accessible with the least forcing amplitude ˇ. Thus, practicing engineers often think of this particular subharmonic instability close to ˛ D 1/4 (i. e. where the forcing frequency is approximately half the natural frequency) as being the hallmark of parametric resonance. Such an instability is sometimes called the principal parametric resonance. Nevertheless, we shall see that other instability tongues, in particular the fundamental or first harmonic parametric resonance near to ˛ D 1 can also sometimes be of significance. Returning briefly to the problem of how children swing, it would seem that the choice ˛  1/4 is in some sense preferable. This is the largest instability tongue for small ˇ. That is, the smallest effort is required in order to cause an exponential instability of the downwards hanging solution, and set the swing in motion. And, according to (2), ˛ D 1/4 corresponds to ! D 2!0 . Thus generations of children seem to have hit upon an optimal strategy [48], by pumping their legs at precisely twice the frequency of the swing. Finally, looking at Fig. 4, note the curious feature that we have continued the diagrams into negative ˛, despite the fact that ˛ was defined as a square term. In the case of

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a pendulum though, ˛ D l/(g! 2 ), and so we can think of negative ˛ as representing negative g. That is, a pendulum with negative ˛ is defying gravity by ‘hanging upwards’. Clearly without any excitation, such a situation would be violently unstable. The big surprise then is that there is a thin wedge of stability (which is present either with or without damping) for ˛ < 0. Note though that the wedge is thickest for small ˇ and small j˛j, which limit corresponds to small amplitude, high-frequency oscillation. This remarkable stabilization of an upside-down pendulum by tiny, rapid vertical vibration was first described by Stephenson [53] in 1908, re-discovered by Kapitza [34] in 1951 (and hence often called the Kapitza pendulum problem) and has been experimentally verified many times; see e. g. [3]. We shall return to this upside down stability in Sect. “Upside-Down Stability”. The rest of this article is outlined as follows. In Sect. “Linear Resonance or Nonlinear Instability?” we shall examine how the Ince–Strutt diagram is constructed and see that this is just an example of the linearized analysis around any periodic state of a nonlinear system. Further quasi-periodic or chaotic motion can ensue as we push beyond this initial instability. Section “Multibody Systems” concerns mechanical systems with multiple degrees of freedom. We consider autoparametric resonance where the motion of one degree of freedom acts as the parametric excitation of another. We also look at conditions in multibody systems where a combination of two natural frequencies can be excited parametrically, focusing in particular on the somewhat illusive phenomenon of difference combination resonances. Section “Continuous Systems” then looks at parametric excitation in continuous systems, looking at phenomena as diverse as pattern formation on the surface of a liquid, flow-induced oscillation of pipes, a clever way of overcoming loss in optical fibers and the oscillation-induced stiffening of structures. Finally, Sect. “Future Directions” draws conclusions by looking at future and emerging ideas. Linear Resonance or Nonlinear Instability? Let us now consider a more detailed analysis of parametric resonance of the Mathieu equation and of the implications for a nonlinear system, using the parametrically excited pendulum as a canonical example.

behave as parameters are varied. Essentially, we explain the behavior of Fig. 2a. Our treatment follows the elementary account in [33], to which the interested reader is referred for more details. For a general linear system of the form (6), a lot of insight can be gained by considering the so-called Wronskian W(t) D det(˚(t)). From the elementary theory of differential equations (e. g. [33]) we have Z t  trace[P(s)] ds ; (9) W(t) D W(t0 ) exp t0

where the trace of a matrix is the sum of its diagonal elements. Note that the specific form of P(t) for the Mathieu equation (7) has trace[P(t)] D 0, hence, from (9) that that det(M) D det[W(2)] D 1. But from elementary algebra we know that the trace of the matrix is the product of its eigenvalues. So we know that the two Floquet multipliers 1;2 satisfy 1 2 D 1 and must therefore solve a characteristic equation of the form 2  2(˛; ˇ) C 1 D 0 ; for some unknown real function . There are two generic cases jj < 1 and jj > 1. If  > 1, then the Floquet multipliers are complex and lie on the unit circle. This corresponds to stability of the zero-solution of the Mathieu equation. Conversely if jj < 1, then the Floquet multipliers are real and satisfy j1 j < 1 and j2 j D 1/j1 j > 1. The larger multiplier represents an exponentially growing solution and hence corresponds to an instability. The two boundary cases are:  D 1, in which case we have a double Floquet multiplier at  D 1; or  D 1, which corresponds to a double multiplier at  D 1. These represent respectively the harmonic and the subharmonic stability boundaries in the Ince–Strutt diagram. But, how do we compute ? The answer is that we don’t. We simply recognize that  D 1 corresponds to the existence of a pure 2-periodic solution of the fundamental matrix M D ˚ (2), and that  D 1 represents a periodic solution of ˚(2)2 D ˚(4) and hence a 4periodic solution of the fundamental matrix. To look for a 2-periodic solution of the Mathieu equation, it is convenient to use Fourier series. That is, we seek a solution (4) in the form x(t) D

nDC1 X

c n e i nt :

nD1

Floquet Analysis We begin by giving a reasoning as to why the Ince–Strutt diagram Fig. 3 for the Mathieu equation (ignoring damping for the time being) looks the way it does, by making a more careful consideration of how Floquet multipliers

Substitution into (4), using cos(t) D (1/2)(e i t Cei t ) leads to infinitely many equations of the form (1/2)ˇc nC1 C (˛  n2 )c n C (1/2)ˇc n1 D 0 ; n D 1; : : : ; 1 ;

with cn D c n ;

(10)

Dynamics of Parametric Excitation

where an asterisk represents complex conjugation. Note that the final condition implies that x is real. Clearly, for ˇ D 0, a nontrivial solution can be found for any integer p which has c n D 0 for all n ¤ p, provided ˛ 2 D p2 . That is, solution curves bifurcate from ˇ D 0 for each ˛-value that is the square of a whole integer. Furthermore, nonlinear analysis can be used to show that for small positive ˇ, there are two solution branches that emerge from each such point (provided p ¤ 0); one corresponding to cp being real, which corresponds to motion x(t) cos(pt) in phase with the excitation; and another corresponding to cp imaginary, which corresponds to motion x(t) sin(pt) out of phase with the excitation. Similarly, if we look for solutions which are 4-periodic by making the substitution x(t) D

mDC1 X

d m e i mt/2 ;

(11)

mD1

we arrive at an infinite system of equations (1/2)ˇd mC2 C (˛  (1/4)m2 )d m C (1/2)ˇd m1 D 0 ; m D 1; : : : ; 1 ;

 with dm D d m :

(12)

Note that the equations for m even and m odd in (12) are decoupled. The equations for even m become, on setting m D 2n, precisely the system (10) with d2m D c n . For m D 2n C 1, however, we arrive at new points of bifurcation, by exactly the same logic as we used to analyze (10). Namely whenever ˛ 2 D (p/2)2 for any odd integer p we have the bifurcation from ˇ D 0 of two 4-periodic solutions (one like cos(pt/2) and one like sin(pt/2)). Perturbation analysis can be used to obtain asymptotic expressions for the transition curves that bifurcate from ˇ D 0 at these special ˛-values; see the companion article  Perturbation Analysis of Parametric Resonance for the details. In particular it can be shown that the width of the stability tongue (separation in ˛-values from the sine and the cosine instability curves) originating at ˛ D (p/2)2 (for p being odd or even) scales like tongue width p ˇ p C h.o.t : Hence the principal subharmonic instability arising from ˛ D 1/4 has width O(ˇ) (actually the transition curves are given by ˛ D 1/4 ˙ (1/2)ˇ C O(ˇ 2 )), and the fundamental harmonic instability arising from ˛ D 1 has width O(ˇ 2 ), etc. Hence the principal subharmonic instability is by far the most prevalent as it occupies the greatest frequency width for small ˇ, and is often thought to be the hallmark of parametric resonance in applications.

Note that we have not as yet mentioned the transition curve emerging from ˛ D ˇ D 0, which corresponds to n D 0 in the above analysis. This case can also be analyzed asymptotically, and the single transition curve can be shown to scale like ˛ D (1/2)ˇ 2 C O(ˇ 3 ) for small ˇ. It is this bending back into ˛ < 0 that leads to the Kapitza pendulum effect, which we shall return to in Sect. “Upside-Down Stability” below. Finally, the addition of small damping can be shown to narrow the tongues and lift them off from the ˇ D 0 axis as shown in Fig. 4. The amount of lift is in proportion to the width of the tongue. This property further enhances the pre-eminence of the principal subharmonic instability for parametrically excited systems in practice, as they thus require the least amplitude of vibration in order to overcome the damping forces. Bifurcation Theory We have just analyzed the Mathieu equation, which is purely linear. But, for example as shown in the Introduction, such an equation is often derived in the course of analyzing the stability of a trivial solution to a nonlinear parametrically excited system. It is worthwhile therefore to consider the nonlinear implications of the harmonic and subharmonic instabilities. The time-periodic Mathieu equation can be written as an example of a three-dimensional autonomous system of ODEs x˙ D y ;

y˙ D f (x; y; s) ;

s˙ D 1 (mod 2) ;

for (x; y; s) 2 R2  S 1 , where f (x; y; s) D ı y  (˛ C ˇ cos(s))x. As such, the trivial solution x D 0 should actually be seen as the periodic solution x D y D 0, s D t (mod 2). Hence we can understand instability of the trivial solution as points of bifurcation of periodic solutions, for which there is a mature theory, see for example [27,37,65,67]. In particular, the subharmonic instability with Floquet multiplier  1 is a period doubling bifurcation, and the harmonic bifurcation corresponding to multiplier + 1 is a pitchfork bifurcation. Note that both these bifurcations come in super- and sub-critical forms. Hence, depending on the nature of the nonlinearity we may find either stable bounded motion close to the instability inside the resonant tongue close to the boundary edge, or motion that is not of small amplitude. See Fig. 5. The case of the Mathieu or Hill equations without damping have received special attention in the literature. This is because of the extra mathematical structure they posses, namely that they can be expressed as reversible and Hamiltonian systems. Specifically, Hill’s equation of the form (5) can be shown to conserve the energy function

189

190

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 5 Contrasting a super-critical and b sub-critical bifurcations close to the resonance tongue boundary of a subharmonic (period-doubling) instability, here represented by the abbreviation P.D. Solid lines represent stable solutions, dashed lines represent unstable solutions. c A one-parameter bifurcation diagram close to the super/sub-critical boundary. d A two parameter unfolding of the codimension-two point P where the super/sub-critical transition occurs. Shading is used to represent where a stable period-two cycle exists. Note in this case that for higher ˇ-values the right-hand boundary of this existence region is therefore the fold curve and not the period-doubling point. Between the period-doubling and the fold there is hysteresis (coexistence) between the stable trivial solution and the stable period-two cycle

H(x; y; t) D (1/2)y 2 C (1/2)(˛ C V(t))x 2 ; y D x˙ , and to be expressible in Hamiltonian form x˙ D

@H ; @y

y˙ D 

@H : @x

Reversibility of Hill’s equation is expressed in the fact that the model is invariant under the transformation t ! t, y ! y. Broer and Levi [7] used Hamiltonian and reversible theory to look at Hill’s equations for general periodic functions V(t). They showed that under certain conditions, the resonance tongues may become of zero width for nonzero ˇ. Effectively, the left and right-hand boundaries of the tongue pass through each other. This is a generic phenomenon, in that a small perturbation within the class of Hill’s equations will lead not destroy such an intersection. However, the Mathieu equation for which V(t) D cos(t) does not posses this property. Many nonlinear models that undergo parametric resonance also preserve the properties of being Hamiltonian and reversible. Special tools from reversible or Hamiltonian systems theory can sometimes be used to analyze the bifurcations that occur under parametric resonance in nonlinear systems, see for example [8,9]. Points outside

the resonance tongues correspond to where the trivial solution is an elliptic (weakly stable) fixed point of the time2 map, and those inside the resonance tongue to where it is a hyperbolic (unstable) fixed point. One therefore does not generally see asymptotically stable non-trivial solutions being born at a resonance tongue boundary, rather one sees elliptic nontrivial dynamics. In fact, for a general periodic solution in a Hamiltonian system, pitchfork and period-doubling bifurcations are not the only kind of bifurcations that produce non-trivial elliptic periodic orbits. If the Floquet multipliers pass through any nth root of unity, there is a period multiplying (subharmonic) bifurcation at which a periodic orbit of period 2n is born (see, e. g. [62]). A period-doubling is just the case n D 2. Such bifurcations can also occur for the non-trivial elliptic periodic orbits within a resonance tongue, but they are all destroyed by the presence of even a small amount of damping. Beyond the Initial Instability For a nonlinear system, the pitchfork or period-doubling bifurcation that occurs upon crossing a resonance tongue might just be the first in a sequence of bifurcations that is

Dynamics of Parametric Excitation

encountered as a parameter is varied. As a general rule, the more ˇ is increased for fixed ˛, the more irregular the stable dynamics becomes. For example, the period-doubling bifurcation that is encountered upon increasing ˇ for ˛ close to 1/4 is typically just the first in a sequence of perioddoublings, that accumulate in the creation of a chaotic attractor, obeying the famous Feigenbaum scaling of this route to chaos (see e. g. [18]). Van Noort [64] produced a comprehensive numerical and analytical study into the nonlinear dynamics of the parametrically excited pendulum (1) without damping. Of interest in that study was to complete the nonlinear dynamics of one of the simplest non-integrable Hamiltonian and reversible dynamical systems. The parameter ˇ acts like a perturbation to integrability. Certain structures that can be argued to exist in the integrable system, like closed curves of quasiperiodic motion in a Poincaré section around an elliptic fixed point, serve as approximate organizers of the dynamics for non-zero ˇ. However, KAM theory (see e. g. [55]) shows that these closed curves become invariant tori that typically breakup as ˇ is increased. At the heart of the regions of chaotic dynamics lie the resonance tongues associated with a period-multiplying bifurcation. Inside the resonance tongues are nontrivial fixed points, whose surrounding invariant tori break up into heteroclinic chains, which necessarily imply chaos. Acheson [2] studied the dynamics of the parametrically forced pendulum in the presence of damping (i. e. Eq. (1) with an additional term x˙ on the right-hand side) via computer simulation. He found parameter regions inside the principal resonance tongue of the pendulum where it undergoes what he referred to as multiple nodding oscillations. The subharmonic motion inside the tongue should be such that during one cycle of the excitation, the pendulum swings to the left and back, and during the next cycle it swings to the right and back. That is, one complete oscillation per two cycles. Multiple nodding motion by contrast is asymmetric and involves two swings to the left, say, (one swing per cycle of the excitation), followed by one swing to the right, with the motion repeating not every two but every three cycles. He also found similar four-cycle and five-cycle asymmetric motion. This behavior can be explained as pockets of stable behavior that survives from the period-multiplying bifurcations that are only present in the case of zero damping [21]. Clearly, beyond the initial instability the universality of the Ince–Strutt diagram for the Mathieu equation, or for any other Hill equation for that matter, disappears and the precise features of the nonlinear dynamics depends crucially on the details of the particular nonlinear system being investigated.

Multibody Systems Of course, there is little in the real world that can be accurately described by deterministic, parametrically excited single degree-of-freedom ODEs. In this section we shall consider slightly more realistic situations that are governed by higher-dimensional systems, albeit still deterministic, and still described by finitely many dynamic variables. For simplicity, we shall restrict attention to rigidbody mechanisms that can be described by a system of (finitely many) coupled modes of vibration. Furthermore, we shall adopt a Lagrangian framework where the dynamics of mode i can be represented by a generalized coordinate qi and its associated generalized velocity q˙i , such that we arrive at a system of (generally nonlinear) coupled second-order ODEs; one equation for each acceleration q¨i . Autoparametric Resonance Autoparametric resonance occurs when there is asymmetric nonlinear multiplicative coupling between modes of a multibody system. Specifically, suppose that a mode (q1 ; q˙1 ) is coupled to a mode (q2 ; q˙2 ) through a term proportional to q1 q2 in the equation of motion for the 2nd mode (i. e. the q¨2 equation), yet there is no corresponding q2 q1 term in the q¨1 equation. For example, consider q¨1 C ı1 q˙1 C ˛1 q1 D f1 (q1 ; t)

(13)

q¨2 C ı2 q˙2 C ˛2 q2 C ˇq1 q2 D f2 (q2 ) ;

(14)

where f 1 and f 2 are nonlinear functions of their arguments and the dependence of f 1 on t is periodic. Furthermore suppose that mode 1 undergoes a periodic motion (either as a result of direct forcing, its own parametric resonance, or self-excitation). For example, if f2 D 0 and f 1 were simply cos(! t) for some ! 2 sufficiently detuned from ˛ 1 , then the solution q1 (t) to (13) would, after allowing sufficient time for transients to decay, be simply proportional to cos(! t C ) for some phase . Then, the q1 q2 term in (14) would be effectively act like ˇ cos(! t C )q2 and so (14) would be precisely the Mathieu equation. Values of ! 2 D 4˛2 /n for any positive integer n correspond to the root points of the resonance tongues of the Ince–Strutt diagram. Such points are referred to as internal or autoparametric resonances. A canonical example of an autoparametrically resonant system is the simple spring-mass-pendulum device depicted in Fig. 6, which we shall use to discuss the some key generic features of autoparametric resonance; see the book by Tondl et al. [58] for more details. In the figure, a mass M is mounted on a linear spring with stiffness k,

191

192

Dynamics of Parametric Excitation

x(t) D R0 cos(t C where R0 D q and tan

0

D

0)

; A

ı12 C (˛1  1)2

ı1 : ˛1  1

(17)

Note that the solution (17) exists for all values of the physical parameters, in particular for all positive ˛ 1 provided ı1 > 0. Substitution of the semi-trivial solution (17) into the pendulum Eq. (16), leads to ¨ C ı2 ˙ C [˛2 C R0 cos(t C

0 )] sin 

D0;

which, after a time shift t ! t  0 is precisely the Eq. (1) for a parametrically excited pendulum in the presence of damping. Thus, the Ince–Struttqdiagram of the damped

Dynamics of Parametric Excitation, Figure 6 A mass-spring-pendulum system

is forced by an applied displacement P cos(! ) and is attached to the pivot of a pendulum of length l and lumped mass m that hangs under gravity. The motion of the mass and the pendulum are assumed to be subject to proportional viscous damping with coefficients b and c respectively. Using standard techniques, the equations of motion can of such a device can be written in the dimensionless form x¨ C ı1 x˙ C ˛1 C (¨ sin() C ˙ 2 cos ) D A cos t (15) ¨ C ı2 ˙ C ˛2 sin  C x¨ sin  D 0 ;

(16)

where x D y/l, time t has been scaled so that t D !, and the dimensionless parameters are b k m ; ˛1 D 2 ; D ; !(M C m) ! (M C m) MCm c g P ; ˛2 D : ; ı2 D AD 2 l! (M C m) !ml 2 l! 2

ı1 D

The simplest form of output of such a model is the socalled semi-trivial solution in which the pendulum is at rest, ˙ D   0, and the mass-spring system alone satisfies the simple equation x¨ C ı1 x˙ C 1 x D A cos t : After transients have died out, this single equation has the attracting solution

Mathieu equation, with ˇ D A/( ı12 C (˛1  1)2 ), ı D ı2 and ˛ D ˛2 tell us precisely the parameter values for which the semi-trivial solution is stable. Inside the resonance tongues we have an instability of the semi-trivial solution, which leads to stable non-trivial motion with the pendulum swinging. To determine precisely what happens away from the boundaries of the stability tongue, one again has to perform fully nonlinear analysis, as outlined in Sect. “Linear Resonance or Nonlinear Instability?”. Note now though that for nonzero pendulum motion,  6 0, the Eqs. (15) and (16) become a fully coupled two-degreeof-freedom system, whose dynamics can be even more complex than that of the one-degree-of-freedom parametrically excited pendulum. Indeed in [58], a further destabilizing secondary-Hopf (Neimark–Sacker) bifurcation is identified within the principal (˛2 D 1/4) subharmonic resonance tongue, and numerical evidence is found for several different kinds of quasiperiodic motion. There is one difference though between autoparametric resonance and pure parametric resonance. The paradigm for parametric resonance is that the external parametric excitation is held constant (its amplitude and frequency are parameters of the system). Here, however, for nonzero , the system is fully coupled, so that the parametric term (17) can no longer be thought of as being independent of , effectively the excitation becomes a dynamic variable itself. Thus, motion of the pendulum can affect the motion of the mass-spring component of the system. In particular if there is an autoparametric instability where the pendulum is set in motion, then energy must be transferred from the mass-spring component to the pendulum, a phenomenon known as quenching. But the pendulum is only set in motion close to certain resonant frequency ratios ˛ 2 . Thus, we have a tuned damper that is designed to

Dynamics of Parametric Excitation

take energy out of the motion of the mass at certain input frequencies ! frequencies. To see how such a damper might work in practice, suppose we build a civil engineering structure such as a bridge or a building, that has a natural frequency ! 1 in some mode, whose amplitude we call x(t). Suppose the system is subjected to an external dynamic loading (perhaps from traffic, wind or earthquakes) that is rich in a frequency ! which is close to ! 1 . Then, if the damping is small (as is typical in large structures) using (15) with  D 0, we find that the response given by (17) is large since ˛ 1 is close to 1 and ı 1 is small. This is the fundamental resonant response of the structure close to its natural frequency. Now suppose we add the pendulum as a tuned damper. In particular we design the pendulum so that ˛2  ˛1 . The pendulum now sees a large amplitude parametric excitation within its main subharmonic resonance tongue. It becomes violently unstable, thus sucking energy from the structure. To see just how effective this quenching effect can be, consider the following asymptotic calculation [58]. Supˆ for some pose the external forcing is very small, A D "2 A, small parameter ", and that the damping and frequency detunings scale with ": ı1 D "ıˆ1 , ı2 D "ıˆ2 , ˛1 D 1 C "ˆ 1 , ˛2 D (1/4) C "ˆ 2 , where the hatted quantities are all assumed to be O(1) as " ! 0. Upon making these substitutions, and dropping all hats, Eqs. (15) and (16) become 1 x¨ C x C "[ı1 x˙ C 1   2 C ˙ 2 ] 4 D "A cos(t) C O("2 ) ; 1 1 ¨ C  C "[ı2 ˙ C 2   x] D O("2 ) : 4 2

(18) (19)

We then look for solutions of the form  x D R1 (t) cos t C 1 (t) ;

 t  D R2 (t) cos C 2 (t) : 2 Making this substitution into (18), (19), and keeping only O(") terms we find

 1 2R˙ 1 D " ı1 R1  R22 sin( 1  2 2 ) A sin( 1 ) ; 4

 1 2R1 ˙ 1 D " 1 R1  R22 cos( 1  2 2 )  A cos 1 ; 4  2R˙ 2 D "  ı2 R2 C R1 R2 sin( 1  2 2 ) ;  2R2 ˙ 2 D " 2 R2  R1 R2 cos( 1  2 2 ) : Equilibrium solutions of these equations correspond to 4-periodic solutions that are valid close to the instability

tongue. Upon setting the left-hand sides to zero, a straightforward combination of the final two equations yields q R1 D 22 C ı22 ; (20) and a slightly longer calculation reveals that this leads to a unique (stable) solution for R2 provided A2 > (ı12 C 12 )(ı22 C 22 ). Equation (20) gives the amplitude of the quenched solution for the structure. Compare this with the amplitude R0 of the semi-trivial solution (17), which is the forced response of the structure without the pendulum-tuneddamper added. Written in the rescaled coordinates, this solution is R0 D

A2 : 12 C ı12

Here we make the important observation that, unlike the simple forced response R0 , the quenched amplitude R1 is independent of the forcing amplitude A, and does not blow up at the fundamental resonance 1 ! 0. In fact, at least to leading order in this asymptotic approximation, the amplitude of the quenched solution is independent of the frequency detuning and damping constant of the structure, and depends only on the frequency detuning and proportional damping of the pendulum! Tondl et al. [58] give a number of applications of autoparametric resonance. First, the above quenching effect means that analogues of the mass-spring-pendulum system can be used as tuned dampers, that are capable of suppressing certain undesirable response frequencies from structures or machines. Another important example of parametric resonance they cite is in flow-induced oscillation, where periodic flow features such as vortex shedding can cause a parametric response of a structure. The galloping of cables in strong winds is an example of just such an effect. Cable-stayed bridges are engineering structures that are particularly susceptible to autoparametric effects. Not only is there potential for resonant interaction between the cables and the wind, but, since the cables are all of different lengths, there is a good chance that there is at least one cable whose natural frequency is either half or twice that of a global vibration mode of the entire bridge. Models of deck-cable interaction have shown the propensity for undesirable effects, see e. g. [23,24]. For example, vibrations caused by traffic could cause particular cables to undergo large-amplitude vibration leading to potential problems with wear. Alternatively, large-amplitude cable vibrations (caused for example by fluid-structure interaction) could cause significant deck vibration leading to an uncomfortable ride for those crossing the bridge.

193

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Dynamics of Parametric Excitation

Combination Parametric Resonances Mailybayev and Seyranian [41] consider general systems of second-order equations of the form M y¨ C ıD y˙ C (A C ˇB(˝ t))y D 0 ;

y 2 Rn ;

(21)

where M, D and A are positive definite symmetric matrices, and B is a periodic matrix with period 2/˝. They derive conditions under which parametric resonance may occur. Suppose that when ı D 0 D ˇ D 0, the linear system has non-degenerate natural frequencies, !1 ; !2 ; : : : ! n . Then they showed that the only kinds of parametric resonance that are possible are when ˝D

2! j ; k

j D 1; : : : m ; k D 1; 2; : : : ;

(simple resonance) ;

!i ˙ ! j or ˝ D ; i; j D 1; : : : m ; i > j ; k k D 1; 2; : : : ; (combination resonance) :(22)

that the system with ˇ D 0 is written in diagonal form, and finally that time has been rescaled so that ˝ D 1. Hence we obtain a system of the form        x¨1 ˛1 0 b11 b12 C Cˇ cos(t) 0 ˛2 x¨2 b21 b22   x1 D 0 ; (23) x2 where ˛1 D !12 /˝ 2 , ˛2 D !22 /˝ 2 and the matrix Bˆ D fb i j g is the original constant matrix within B0 written in the transformed coordinates. Recalling, how in Subsect. “Floquet Analysis” we found the transition curves for the Mathieu equation using Floquet theory, we can look for solutions to (23) in the form x1 D c n

1 X

p 2

e i nt/

nD1 1 X

x2 D ˙c n

e

;

p si nt/ 2

(24) ;

nD1

The sign ‘+’ or ‘’ in (22) gives what are called sum or difference combination resonances respectively. There are many examples of such combination resonances in the literature. For example, the book by Nayfeh [45] considers many such cases, especially of mechanical systems where the two modes are derived from a Galerkin reduction of a continuum system such as a plate or beam. In all these examples, though, it is a sum combination resonance that is excited. That is, by forcing the system at the sum of the two individual frequencies ˝ D !1 C !2 , individual responses are found at frequency ! 1 or ! 2 . However, there do not seem to be any concrete examples of difference resonances in the literature. In fact, such a mechanical device might be somewhat strange. If we had !1  !2 , then ˝ D !1  !2 would be small, perhaps several orders of magnitude smaller. So a difference combination resonance would give a response at a high frequency from low frequency excitation. This would be like making a drum vibrate by sending it up and down in an elevator! In fact, for the case that B D B0 (t) is a constant matrix times a function of time, then it is shown in (Theorem 1 in [41]) that a given system can only excite either sum or difference combination resonances, but not both. To explain this result, it is sufficient to consider a simplification of (21) to the case n D 2. Consider (21) in the case ı D 0 and where B(t) is a constant matrix times a pure function of time with period ˝. Without loss of generality, let us suppose this function to be a pure cosine B D B0 cos(˝ t). Suppose that we change coordinates so

where s D ˙1. We find an infinite system of algebraic equations analogous to (10). It is straightforward to see that in the case ˇ D 0, there is a non-trivial solution with c n D 0 for all n ¤ k and c k ¤ 0, whenever ˛1 C s˛2 D k 2 . This would suggest that both sum and difference combination resonances are possible. However, when looking at the conditions for the bifurcation equations to have a real solution for nonzero ˇ, one finds the following condition sign(b12 b21 ) D s :

(25)

That is, to excite a sum resonance, the diagonal entries of B0 must be of the same sign, and to excite a difference resonance these diagonal entries must be of opposite sign. In particular, if B0 is a symmetric matrix (as is the case in many mechanical applications) then only sum combinations can be excited. In fact, it is argued in [68] that pure Hamiltonian systems cannot ever excite difference combination resonance; in other words, if difference parametric resonance is possible at all, then the matrix B(˝ t) must contain dissipative terms. One might wonder whether difference combination resonance can ever be excited in any physically derived system. To show that they can, consider the following example with a non-conservative parametric excitation force, a detailed analysis of which will appear elsewhere [59]. Figure 7 depicts a double pendulum (which we think of as lying in a horizontal plane so that gravity does not affect it) with stiff, damped joints and is loaded by an end

Dynamics of Parametric Excitation

Now, (26) is not in diagonal form, but a simple coordinate transformation can it in the form (21) with M the identity matrix,  DDAD  and

B0 D

0:1492 0 0:0466 0:1288

0 3:3508



0:3788 1:0466

 :

Hence the system has two natural frequencies !1 D p p 0:1492 D 0:3863 and !2 D 3:3508 D 1:8305, and the quantities b12 b21 D 0:3788  0:1288 D 0:0488 < 0. Hence, according to (25), a difference combination resonance should be possible by exciting the system at ˝ D !2  !1 D 1:4442, as verified in [59]. Upside-Down Stability Dynamics of Parametric Excitation, Figure 7 The partially follower-loaded elastic double pendulum, depicting a definition of the various physical quantities. In this study we take m1 D m2 D m and ˛ D 1

force – a so-called follower load – that is maintained in the direction of the axis of the second pendulum. Note that such a load is by its nature non-conservative because work (either positive or negative) has to be done to maintain the direction of the follower. In practice, such a force might be produced by a jet of fluid emerging from the end of the outer pendulum, for example if the pendula were actually pipes, although the equations of motion (and the consequent nonlinear dynamics) are rather different for that case [5,12,13]. Non-dimensional equations of motion for this system are derived in [56], where it was shown that the system can undergo quite violent chaotic dynamics for large deflections. We shall restrict ourselves here to small deflection theory and allow the follower load to be a harmonic function of time P D A cos ˝ t. After p nondimensionalization using ˇ D jAjl/k;pı D C/ kml 2 , and rescaling time according to tnew D k/m2 l 2 ) told , we obtain the equations  ¨  1 ¨2     ˙  2 1 1 2 1 1 Cı 1 1 2 1 1 ˙2    1 1 1 C p cos ˝ t D 0 : (26) 2 0 0



2 1

1 1



C

As mentioned in the Introduction, it has been known for 100 years [53] that the ‘negative gravity’ (˛ < 0) region of the Ince–Strutt diagram for the Mathieu equation implies that a simple pendulum can be stabilized in the inverted position by application of parametric excitation. More recently Acheson and Mullin [3] demonstrated experimentally that a double and even a triple pendulum can be similarly stabilized when started in the inverted position. Moreover, the stabilized equilibrium is quite robust, such that the system relaxes asymptotically back to its inverted state when given a moderate tap. At first sight, these observations seem remarkable. A multiple pendulum has several independent normal modes of oscillation and naively one might expect that single frequency excitation could at best stabilize only one of these modes at a time. However, Acheson [1] (see also the earlier analysis of Otterbein [46]) provided a simple one-line proof that, in fact, for sufficiently high frequency and sufficiently small amplitude sinusoidal excitation, in theory any ideal finite chain of N pendulum can be stabilized in the inverted position. To see why this is true, consider the multiple pendulum system represented in Fig. 8 but without the springs at the joints. Then, upon performing a modal analysis, we find that the equations of motion reduce to the study of N uncoupled Mathieu equations each with different parameters ˛ i > 0 and ˇ i > 0, for i D 1; : : : ; N. Hence the frequency can be chosen to be sufficiently high and the amplitude chosen to be sufficiently small for each ˛ i , ˇ i to lie in the thin wedge of stability for ˛ < 0 in Fig. 4. Thus, each mode is stable, and so the whole finite chain has been stabilized by parametric excitation.

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Dynamics of Parametric Excitation, Figure 8 a A multiple pendulum with stiff joints. b,c Ince–Strutt diagrams of dimensionless amplitude of vibration " against scaled elastic stiffness B for eight identical jointed pendulums with weak damping. b For dimensionless applied frequency ! D 10; c ! D 20. Shading represents the stability region of the upright equilibrium and digits in other regions give the number of unstable Floquet multipliers. After [22], to which the reader is referred to for the precise details; reproduced with permission from Elsevier

It has been suggested that the limit of such a multiple pendulum, in which the total length and mass of the system stays constant while the number of pendulums in the chain becomes infinite, could be used as a possible explanation of the so called ‘Indian rope trick’, a classic conjuring trick in which a rope is ‘charmed’ by a magician to appear vertically out of a basket like a snake. Unfortunately, as noted [1], in this limit, which is that of a piece of string, the stability region becomes vanishingly small and so this potential scientific explanation of the magic trick fails. However, a further experiment by Mullin announced in [4], demonstrated that a piece of ‘bendy curtain wire’, clamped at the bottom and free at the top, that is just too long to support its own weight can be stabilized by parametric excitation (see Fig. 9). In order to explain this result, Galán et al. [22] introduced the model depicted in Fig. 8 and studied the stability of the vertical equilibrium

position in the presence of both damping and stiffness in each of the joints. The normal modes of this problem lead to fully coupled equations and so Acheson’s argument does not apply. Using numerical bifurcation analysis Galán et al. were able to find stability boundaries and also to find an appropriate continuum limit; providing a good motivation for us to turn our attention to parametric resonance in continuous systems. Continuous Systems Parametric resonance is also of importance in continuum systems. Canonical effects we shall discuss here include the wire stiffening we have just introduced, the excitation of patterns at fluid interfaces, more general theories of flowinduced oscillations of structures, and the stabilizing effects of periodic modulation of optical fibers.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 9 Experimental results on a piece of ‘bendy curtain wire’ which is just long enough to buckle under its own weight (a). For tiny vertical oscillations between about 15 and 35 Hz, the wire can be made to stand upright (b), and to be stable against even quite large perturbations (c). Beyond the upper frequency threshold, a violent dynamic stability sets in which involves the third vibration mode excited harmonically (d). After [44], reproduced with permission from the Royal Society

Structural Stiffening; the ‘Indian Rod Trick’ Thomsen [57] has shown that parametric excitation generally speaking has a stiffening effect on a structure. This is essentially because of the same effect that causes the upside down stability of the simple pendulum for high-enough frequencies and small enough amplitude. However, detailed experiments by Mullin and co-workers [44], reproduced in Fig. 9, showed that there can also be a destabilizing effect of parametric resonance. In particular, a piece of curtain wire (a thin helical spring surrounded by plastic) that was just too long (about 50 cm) to support its own weight, was clamped vertically upright and subjected to rapid, small-amplitude oscillations of about 1 cm peakto-peak. At around 15 Hz the rod was found to be stabilized, such that when released it remains erect (see Fig. 9b). This effect continues for higher frequencies, until about 30 Hz at which point a violent dynamic instability sets in that is commensurate with the driving frequency (harmonic, rather than subharmonic). The experimentally determined parameter values in which the upside-down stabilization can be achieved are represented in terms of dimensionless parameters in Fig. 10b. Champneys and Fraser [14,15,20], introduced a continuum theory to explain these experimental results, which produced the shaded region in Fig. 10b, that we now explain. Consider an intrinsically straight column of length `, with a uniform circular cross-section of radius a  ` and mass linear density m per unit length (see Fig. 10a). The column is assumed to be inextensible, unshearable and linearly elastic with bending stiffness Bˆ and material damping coefficient ˆ (where hatted variables are

dimensional). The lower end is clamped upright and is to vertical displacement  cos ! ˆt , whereas the upper end is free. In [15], it is shown that the stability of the vertical equilibrium position can be studied by analyzing solutions to the PDE Bussss C [(1  s)us ]s C  (u t t C usssst  "[(1  s)us ]s cos t) D 0 ;

(27)

for the scaled lateral displacement u(s; t) assumed without loss of generality to be in a fixed coordinate direction, where the scaled arc length s 2 (0; 1), the scaled time variable t D ! ˆt, and the boundary conditions are u D us D 0 ;

at s D 0 ;

(28)

uss D ussss D 0 ; at s D 1 : The dimensionless parameters appearing in (27) are

D

 `! ; mg

BD

Bˆ ; mg`3

D

! 2` ; g

"D

 ; (29) `

which represent respectively material damping, bending stiffness, the ratio of forcing frequency to the equivalent pendulum natural frequency, and forcing amplitude. If we momentarily consider the problem with no forcing or damping, " D D 0, then the peigenmodes  n and corresponding natural frequencies n / of free vibration of the rod satisfy B( n )ssss C [(1  s)( n )s ]s  n  n  n D 0 ; and form a complete basis of functions that satisfy the boundary conditions (28). Now the eigenmodes  n (s; B)

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Dynamics of Parametric Excitation, Figure 10 a Definition sketch of the parametrically excited flexible rod model. b Comparison between theory (solid line and shading) and the experimental results of [44] (dashed line), for the stability region as a function of the dimensionless parameters B and  using the experimental value " D 0:02, and the representative damping value 1 D 0:01 (which was not measured in the experiment) The theory is based on calculations close to the resonance tongue interaction between when B0;1 D B1;3 when  D c D 460:7. After [15], reproduced with permission from SIAM

are in general not expressible in closed form except at the special values of B at which n (B) D 0, in which case [20] there is a solution in terms of the Bessel function J1/3 . The same analysis shows that there are infinitely many such B-values, B0;n , for n D 1; 2; 3; 4; : : :, accumulating at B D 0, the first few values of which are B0;1 D 0:127594, B0;2 D 0:017864, B0;3 D 0:0067336 and B0;4 D 0:0003503. These correspond respectively to where each eigenvalue locus n (B), for n D 1; 2; 3; 4, crosses the B-axis, with the corresponding eigenmodes there having n  1 internal zeros. The lowering of B through each B0;n -value implies that the nth mode becomes linearly unstable. Hence for B > B c :D B0;1 the unforced rod is stable to self-weight buckling (a result known to Greenhill [26]). Moreover, it can be shown [14,15] that each eigenmode  n (s) retains a qualitatively similar mode shape for B > B0;n with the nth mode having n D 1 internal zeros, and that the corresponding loci n (B) are approximately straight lines as shown in the upper part of Fig. 11. The lower part of Fig. 11 shows the results of a parametric analysis of the Eq. (27) for D 0. Basically, each time one of the eigenvalue loci passes through  times the square of a half-integer ( j/2)2 we reach a B-value for which there is the root point of an instability tongue. Essentially we have a Mathieu-equation-like Ince–Strutt diagram in the (B; ") plane for each mode n. These diagrams are overlaid on top of each other to produce the overall stability

plot. However, note from Fig. 11 that the slope of each eigenvalue locus varies with the frequency ratio . Hence as  increases, the values of all the B j/2;n , j D 0; : : : ; 1 for a given n increase with the same fixed speed. But this speed varies with the mode number n. Hence the instability tongues slide over each other as the frequency varies. Note the schematic diagram in the lower panel of Fig. 11 indicates a region of the stability for B < B0;1 for small positive ". Such a region would indicate the existence of rod of (marginally) lower bending stiffness than that required for static stability, that can be stabilized by the introduction of the parametric excitation. Essentially, the critical bending stiffness for the buckling instability is reduced (or, equivalently since B / `3 , the critical length for buckling is increased). This effect has been dubbed the ‘Indian rod trick’ because the need for bending stiffness in the wire means that mathematically this is a rod, not a rope (or string); see, e. g. the account in [4]. To determine in theory the precise parameter values at which the rod trick works turns out to be an involved process. That the instability boundary emerging from B0;1 in the lower panel of Fig. 11 should usually bend back to the left can be argued geometrically [7]. However, for -values at which B0;1  B1;n for some n, it has been shown that there is actually a singularity in the coefficient of the quadratic part of this buckling curve causing the instability boundary to bend to the right [20]. A complete un-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 11 Summarizing schematically the results of the parametric resonance analysis of the parametric excited vertical column in the absence of damping. The upper part shows eigenvalue loci n (B) and the definition of the points Bj/2;n , j D 0; 1; 2; 3; : : : In the lower part, instability tongues in the (B; ")-plane are shown to occur with root points Bj/2;n and to have width "j . The shaded regions correspond to the where the vertical solution is stable. Solid lines represent neutral stability curves with Floquet multiplier + 1 (where ˛ is an integer) and dashed lines to multipliers  1 (where ˛ is half an odd integer). Reproduced from [14] with permission from the Royal Society

folding of this so-called resonance tongue interaction was performed in [15], also allowing for the presence of material damping. The resulting asymptotic analysis needs to be carried out in the presence of four small parameters; ",

, B  B0;1 and  c where c is the frequency-ratio value at which the resonance-tongue interaction occurs. The results produce the shaded region shown in Fig. 10b, which can be seen to agree well with the experimental results. Faraday Waves and Pattern Formation When a vessel containing a liquid is vibrated vertically, it is well known that certain standing wave patterns form on the free surface. It was Faraday in 1831 [19] who first noted that the frequency of the liquid motion is half that of the vessel. These results were confirmed by Rayleigh [50] who suggested that what we now know as parametric resonance was the cause. This explanation was shown to be

correct by Benjamin and Ursell [6] who derived a system of uncoupled Mathieu equations from the Navier–Stokes equations describing the motion of an ideal (non-viscous) fluid. There is one Mathieu equation (4), for each wave number i that fits into the domain, each with its own parameters ˛ i and ˇ i . They argue that viscous effects would effectively introduce a linear damping term, as in (8) and they found good agreement between the broad principal (subharmonic) instability tongue and the results of a simple experiment. Later, Kumer and Tuckerman [36] produced a more accurate Ince–Strutt diagram (also known as a dispersion relation in this context) for the instabilities of free surfaces between two viscous fluids. Through numerical solution of appropriate coupled Mathieu equations that correctly take into account the boundary conditions, they showed that viscosity acts in a more complicated way than the simple velocity proportional damping term in (8). They produced the diagram shown in Fig. 12b, which showed a good match with the experimental results. Note that, compared with the idealized theory, Fig. 12a, the correct treatment of viscosity causes a small shift in the critical wavelength of each instability, a broadening of the higher-harmonic resonance tongues and a lowering of the amplitude required for the onset of the principal subharmonic resonance if damping is introduced to the idealized theory (see the inset to Fig. 12b). More interest comes when Faraday waves are excited in a vessel where two different wave modes have their subharmonic resonance at nearly the same applied frequency, or when the excitation contains two (or more) frequencies each being close to twice the natural frequency of a standing wave. See for example the book by Hoyle [30] and references therein, for a glimpse at the amazing complexity of the patterns of vibration that can result and for their explanation in terms of dynamical systems with symmetry. Recently a new phenomenon has also been established, namely that of localized spots of pattern that can form under parametric resonance, so-called oscillons see e. g. [61] for a review and [39] for the first steps at an explanation. Fluid-Structure Interaction Faraday waves are parametric effects induced by the gross mechanical excitation of a fluid. In Subsect. “Autoparametric Resonance” we briefly mentioned that fluid-structure interaction can go the other way, namely that external fluid flow effects such as vortex shedding can excite (auto)parametric resonances in simple mechanical systems. Such flow-induced parametric resonance can also occur for continuous structures, in which waves can be excited in the structure due to external parametric loading of

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Dynamics of Parametric Excitation, Figure 12 Stability boundaries for Faraday waves from: a Benjamin and Ursell’s theory for ideal fluids; b the results of full hydrodynamic stability analysis by Kumar and Tuckerman. The horizontal axis is the frequency and the vertical scale the amplitude of the excitation. The inset to panel b shows the difference between the true instability boundary for the principal subharmonic resonance (lower curve) and the idealized boundary in the presence of simple proportional damping. After [36], to which the reader is referred for the relevant parameter values; reproduced with permission from Cambridge University Press

the fluid; see for example the extensive review by Païdousis and Li [47]. They also treat cases where the flow is internal to the structure: we are all familiar with the noisy ducts and pipes, for example in central heating or rumbling digestive systems, and it could well be that many of these effects are a result of parametric resonance. Semler and Païdousis [52] consider in detail parametric resonance effects in the simple paradigm of a cantilevered (free at one end, clamped at the other) flexible pipe conveying fluid. Think of this as a hosepipe resting on a garden lawn. Many authors have considered such an apparatus as a paradigm for internal flow-induced oscillation

problems (see the extensive references in [47,52]), starting from Benjamin’s [5] pioneering derivation of equations of motion from the limit of a segmented rigid pipe. The equations of motion for such a situation are similar to those of the rod (27), but with significant nonlinear terms arising from inertial and fluid-structure interaction effects. In the absence of periodic fluctuations in the flow, then it is well known that there is a critical flow rate beyond which the pipe will become oscillatory, through a Hopf bifurcation (flutter instability). This effect can be seen in the way that hosepipes writhe around on a lawn, and indeed forms the basis for a popular garden toy. Another form of instability can occur when the mean flow rate is beneath the critical value for flutter, but the flow rate is subject to a periodic variation, see [52] and references therein. There, a parametric instability can occur when the pulse rate is twice the natural frequency of small amplitude vibrations of the pipe. In [52] this effect was observed in theory, in numerical simulation and in a careful experimental study, with good agreement between the three. Parametric resonance could also be found for super-critical flow rates. In this case, the effect was that the periodic oscillation born at the Hopf bifurcation was excited into a quasi-periodic response. Another form of fluid structure interaction can occur when the properties of an elastic body immersed in a fluid are subject to a temporal periodic variation. With a view to potential biological applications such as active hearing processes and heart muscles immersed in blood, Cortez et al. [16] consider a two-dimensional patch of viscous fluid containing a ring filament that is excited via periodic variation of its elastic modulus. Using a mixture of immersed boundary numerical methods and analytic reductions, they are able to find an effective Ince–Strutt diagram for this system and demonstrate the existence of parametric instability in which the excitation of the structure causes oscillations in the fluid. Taking a fixed wave number p of the ring in space, they find a sequence of harmonic and subharmonic instabilities as a parameter effectively like ˛ in the Mathieu equation is increased. Interestingly though, even for low viscosity, the principal subharmonic resonance tongue (corresponding to ˛ D 1/4) does not occur for low amplitude forcing, and the most easily excited instability is actually the fundamental, harmonic resonance (corresponding to ˛ D 1). Dispersion Managed Optical Solitons The stabilizing effect of high-frequency parametric excitation has in recent years seen a completely different application, in an area of high technological importance, namely

Dynamics of Parametric Excitation

optical communications. Pulses of light sent along optical fibers are a crucial part of global communication systems. A pulse represents a packet of linear waves, with a continuum of different frequencies. However, optical fibers are fundamentally dispersive, which means that waves with different wavelengths travel at different group velocities. In addition, nonlinearity, such as the Kerr effect of intensity-dependent refractive index change, and linear losses mean that there is a limit to how far a single optical signal can be sent without the need for some kind of amplifier. Depending on the construction of the fiber, dispersion can be either normal which means that lower frequencies travel faster, or anomalous in which case higher frequencies travel faster. One idea to overcome dispersive effects, to enable pulses to be travel over longer distances without breaking up, is to periodically vary the medium with which the fiber is constructed so that the dispersion is alternately normally and anomalously dispersive [38]. Another idea, first proposed by Hasegawa and Tappert [28], is to balance dispersion with nonlinearity. This works because the complex wave envelope A of the electric field of the pulse can be shown to leading order to satisfy the nonlinear Schrödinger (NLS) equation i

dA 1 d2 A C jAj2 A D 0 ; C dz 2 dt 2

(30)

in which time and space have been rescaled so that the anomalous dispersion coefficient (multiplying the secondderivative term) and the coefficient of the Kerr nonlinearity are both unity. Note the peculiarity in optics compared to other wave-bearing system in that time t plays the role of the spatial coordinate, and the propagation distance z is the evolution variable. Thus we think of pulses as being time-traces that propagate along the fiber. Now, the NLS equation is completely integrable and has the well-known soliton solution 2

A(z; t) D  sech(t  cz)e i z ;

(31)

which represents a smooth pulse with amplitude , which propagates at ‘speed’ c. Moreover, Eq. (30) is completely integrable [69] and so arbitrary initial conditions break up into solitons of different speed and amplitude. Optical solitons were first realized experimentally by Mollenauer et al. [43]. However, true optical fibers still suffer linear losses, and the necessary periodic amplification of the optical solitons to enable them to survive over long distances can cause them to jitter [25] and to interact with each other as they move about in time. The idea of combining nonlinearity with dispersion management in order to stabilize soliton propagation was

first introduced by Knox, Forysiak and Doran [35], was demonstrated experimentally by Suzuki et al. [54] and has shown surprisingly advantageous performance that is now being exploited technologically (see [60] for a review). Essentially, dispersion management introduces a periodic variation of the dispersion coefficient d(z), that can be used to compensate the effects of periodic amplification, which with the advent of modern erbium-doped optical amplifiers can be modeled as a periodic variation of the nonlinearity. As an envelope equation we arrive at the modified form of (30), i

dA d(z) d2 A C  (z)jAj2 A D 0 ; C dz 2 dt 2

(32)

where the period L of the dispersion map d(z) and the amplifier spacing (the period of  (z)) Z may not necessarily be the same. Two particular regimes are of interest. One, of application to long-haul, sub-sea transmission systems where the cable is already laid with fixed amplifiers, is to have L  Z. Here, one modifies an existing cable by applying bits of extra fiber with large dispersion of the opposite sign only occasionally. In this case, one can effectively look only at the long scale and average out the effect of the variation of  . Then the parametric term only multiplies the second derivative. Another interesting regime, of relevance perhaps when designing new terrestrial communication lines, is to have L D Z so that one adds loops of extra opposite-dispersion fiber at each amplification site. In both of these limits we have an evolution equation whose coefficients are periodic functions of the evolution variable z with period L. In either regime, the periodic (in z) parameter variation has a profound stabilizing influence, such that the effects of linear losses and jitter are substantially reduced. A particular observation is that soliton solutions appear to (32) breathe during each period; that is, their amplitude and phase undergo significant periodic modulation. Also, with increase in the amplitude of the oscillatory part of d, so the average shape of the breathing soliton becomes less like a sech-pulse and more like a Gaussian. An advantage is that the Gaussian has much greater energy the regular NLS soliton (31) for the same average values of d and  , hence there is less of a requirement for amplification. Surprisingly, it has been found that the dispersion managed solitons can propagate stably even if the average of d is slightly positive, i. e. in the normal dispersion region where there is no corresponding NLS soliton. This has a hugely advantageous consequence. The fact that the local dispersion can be chosen to be high, but the average dispersion can be chosen to be close to zero, is the key to enabling the jitter introduced by the amplifiers to be greatly suppressed

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compared with the regular solitons. Finally the large local modulation of phase that is a consequence of the breathing, means that neighboring solitons are less likely to interact with each other. A mathematical analysis of why these surprising effects arise is beyond the scope of this article, see e. g. [32,60] for reviews, but broadly-speaking the stable propagation of dispersion-managed solitons is due to the stabilizing effects of high frequency parametric excitation. Future Directions This article has represented a whistle-stop tour of a wide areas of science. One of the main simplifications we have taken is that we have assumed the applied excitation to the system to be periodic; in fact, in most examples it has been pure sinusoidal. There is a growing literature on quasiperiodic forcing, see e. g. [51], and it would be interesting to look at those effects that are unique to parametrically excited systems that contain two or more frequencies. The inclusion of noise in the excitation also needs to be investigated. What effect does small additive or multiplicative noise have on the shape of the resonance tongues? And what is the connection with the phenomenon of stochastic resonance [66]? There is a mature theory of nonlinear resonance in systems with two degrees of freedom which, when in an integrable limit, undergoes quasiperiodic motion with two independent frequencies. Delicate results, such as the KAM theorem, see e. g. [55], determine precisely which quasiperiodic motions (invariant tori) survive as a function of two parameters, the size of the perturbation from integrability and the frequency detuning between the two uncoupled periodic motions. For the tori that break up (inside so-called Arnol’d tongues) we get heteroclinic tangles and bands of chaotic motion. Parametric resonance tends to be slightly different though, since in the uncoupled limit, one of the two modes typically has zero amplitude (what we referred to earlier as a semi-trivial solution). It would seem that there is scope for a deeper connection to be established between the Arnol’d tongues of KAM theory and the resonance tongues that occur under parametric excitation. In terms of applications, a large area that remains to be explored further is the potential exploitation of parametric effects in fluid-structure interaction problems. As I write, the heating pipes in my office are buzzing at an extremely annoying frequency that seems to arise on warm days when many radiators have been switched off. Is this a parametric effect? It would also seem likely that nature uses parametric resonance in motion. Muscles contract

longitudinally, but motion can be in a transverse direction. Could phenomena such as the swishing of bacterial flagella, fish swimming mechanisms and the pumping of fluids through vessels, be understood in terms of exploitation of natural parametric resonances? How about motion of vocal chords, the Basilar membrane in the inner ear, or the global scale pumping of the heart? Does nature naturally try to tune tissue to find the primary subharmonic resonance tongue? Perhaps other effects of parametric excitation that we have uncovered here, such as the quenching of resonant vibrations, structural stiffening, and frequency tuned motion, are already being usefully exploited in nature. If so, there could be interesting consequences for nature inspired design in the engineered world. Finally, let us return to our opening paradigm of how children effectively excite a parametric instability in order to induce motion in a playground swing. In fact, it was carefully shown by Case and Swanson [11] that the mechanism the child uses when in the sitting position is far more accurately described as an example of direct forcing rather than parametric excitation. By pushing on the ropes of the swing during the backwards flight, and pulling on them during the forward flight, the child is predominately shifting his center of gravity to and fro, rather than up and down; effectively as in Fig. 1c rather than Fig. 1b. Later, Case [10] argued that, even in the standing position, the energy gained by direct forcing from leaning backwards and forwards is likely to greatly outweigh any gained from parametric up-and-down motion of the child. The popular myth that children swinging is a suitable playground science demonstrator of parametric resonance was finally put to bed by Post et al. [49]. They carried out experiments on human patients and analyzed the mechanisms they use to move, reaching the conclusion that even in the standing position, typically 95% of the energy comes from direct forcing, although parametric effects do make a slightly more significant contribution for larger amplitudes of swing.

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31. Ince E (1956) Ordinary Differential Equations. Dover, New York 32. Jones C (2003) Creating stability from instability. In: Nonlinear dynamics and chaos; where do we go from here? IOP Publishing, Bristol, chap 4, pp 73–90 33. Jordan DW, Smith P (1998) Nonlinear Ordinary Differential Equations, 3rd edn. Oxford University Press, Oxford 34. Kapitza P (1951) Dinamicheskaya ustoichivost mayatnika pri koleblyushcheisya tochke podvesa. Z Eksperimentalnoi I Teoreticheskoi Fiziki 21:588–597, in Russian 35. Knoz F, Forysiak W, Doran N (1995) 10 gbit/s soliton communication systems over standard fiber at 1.55m and the use of dispersion compensation. IEEE J Lightwave Technol 13:1960–1995 36. Kumar K, Tuckerman L (1994) Parametric instability of the interface between two fluids. J Fluid Mech 279:49–68 37. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York 38. Lin C, Kogelnik H, Cohen L (1980) Optical-pulse equilization of low-dispersion transmission in single fibers in the 1.3–1.7m spectral region. Opt Lett 5:476–480 39. Lloyd D, Sandstede B, Avitabilie D, Champneys A (2008) Localized hexagon patterns in the planar swift-hohenberg equation. To appear in SIAM Appl Dyn Sys 40. Magnus W, Winkler S (1979) Hill’s Equation. Dover, New York 41. Mailybayev A, Seyranian AP (2001) Parametric resonance in systems with small dissipation. PMM J App Math Mech 65:755–767 42. Mathieu E (1868) Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. J Math Pure Appl 13:137–203 43. Mollenauer L, Stolen R, Gordon J (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45:1095–1097 44. Mullin A, Champneys T, Fraser W, Galan J, Acheson D (2003) The ‘Indian wire trick’ via parametric excitation: a comparison between theory and experiment. Proc Roy Soc Lond A 459:539–546 45. Nayfeh A (2000) Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley Interscience, New York 46. Otterbein S (1982) Stabilization of the N-pendulum and the Indian link trick. Arch Rat Mech Anal 78:381–393 47. Païdoussis MP, Li G (1993) Pipes conveying fluid: a model dynamical problem. J Fluids Structures 7:137–204 48. Piccoli B, Kulkarni J (2005) Pumping a swing by standing and squatting: Do children pump time optimally? IEEE Control Systems Magazine 25:48–56 49. Post A, de Groot G, Daffertshofer A, Beek P (2007) Pumping a playground swing. Motor Control 11:136–150 50. Rayliegh L (1883) On the crispations of fluid resting upon a vibrating support. Phil Mag 16:50 51. Romeiras F, Romeiras F, Bondeson A, Ott E, Antonsen TM, Grebogi C (1989) Quasiperiodic forcing and the observability of strange nonchaotic attractors. Phys Scr 40:442–444 52. Semler C, Païdoussis M (1996) Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe. J Fluids Structures 10:787–825 53. Stephenson A (1908) On a new type of dynamical stability. Mem Proc Manch Lit Phil Soc 52:1–10 54. Suzuki M, Morita N, Edagawa I, Yamamoto S, Taga H, Akiba S (1995) Reduction of gordon-haas timing jitter by periodic

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Entropy in Ergodic Theory

Entropy in Ergodic Theory JONATHAN L. F. KING University of Florida, Gainesville, USA Article Outline Glossary Definition of the Subject Entropy example: How many questions? Distribution Entropy A Gander at Shannon’s Noisy Channel Theorem The Information Function Entropy of a Process Entropy of a Transformation Determinism and Zero-Entropy The Pinsker–Field and K-Automorphisms Ornstein Theory Topological Entropy Three Recent Results Exodos Bibliography Glossary Some of the following definitions refer to the “Notation” paragraph immediately below. Use mpt for ‘measure-preserving transformation’. Measure space A measure space (X; X ; ) is a set X, a field (that is, a -algebra) X of subsets of X, and a countably-additive measure  : X ! [0; 1]. (We often just write (X; ), with the field implicit.) For a collection C X , use Fld(C)  C for the smallest field including C. The number (B) is the “-mass of B”. Measure-preserving map A measure-preserving map : (X; X ; ) ! (Y; Y; ) is a map : X ! Y such that the inverse image of each B 2 Y is in X , and ( 1 (B)) D (B). A (measure-preserving) transformation is a measure-preserving map T : (X; X ; ) ! (X; X ; ). Condense this notation to (T : X; X ; ) or (T : X; ). Probability space A probability space is a measure space (X; ) with (X) D 1; this  is a probability measure. All our maps/transformations in this article are on probability spaces. Factor map A factor map : (T : X; X ; ) ! (S : Y; Y; ) is a measure-preserving map : X ! Y which intertwines the transformations, ıT D S ı . And is an

isomorphism if – after deleting a nullset (a mass-zero set) in each space – this is a bijection and 1 is also a factor map. Almost everywhere (a.e.) A measure-theoretic statement holds almost everywhere, abbreviated a.e., if it holds off of a nullset. (Eugene Gutkin once remarked to me that the problem with Measure Theory is . . . that you have to say “almost everywhere”, almost everywhere.) a:e: For example, B  A means that (B X A) is zero. The a.e. will usually be implicit. Probability vector A probability vector vE D (v1 ; v2 ; : : : ) is a list of non-negative reals whose sum is 1. We generally assume that probability vectors and partitions (see below) have finitely many components. Write “countable probability vector/partition”, when finitely or denumerably many components are considered. Partition A partition P D (A1 ; A2 ; : : : ) splits X into pairwise disjoint subsets A i 2 X so that the disjoint union F # i A i is all of X. Each Ai is an atom of P. Use jPj or P for the number of atoms. When P partitions a probability space, then it yields a probability vector vE, where v j :D (A j ). Lastly, use Phxi to denote the P-atom that owns x. Fonts We use the font H ; E ; I for distribution-entropy, entropy and the information function. In contrast, the script font ABC : : : will be used for collections of sets; usually subfields of X . Use E() for the (conditional) expectation operator. Notation Z D integers, ZC D positive integers, and N D natural numbers D 0; 1; 2; : : :. (Some wellmeaning folk use N for ZC , saying ‘Nothing could be more natural than the positive integers’. And this is why 0 2 N. Use de and bc for the ceiling and floor functions; bc is also called the “greatest-integer function”. For an interval J :D [a; b) [1; C1], let [a : : : b) denote the interval of integers J \ Z (with a similar convention for closed and open intervals). E. g., (e : : : ] D (e : : : ) D f3g. For subsets A and B of the same space, ˝, use A B for inclusion and A ¦ B for proper inclusion. The difference set B X A is f! 2 B j ! … Ag. Employ Ac for the complement ˝ X A. Since we work in a probability space, if we let x :D (A), then a convenient convention is to have xc

denote 1  x ;

since then (Ac ) equals xc . Use A4B for the symmetric difference [AXB][[BXA]. For a collection C D fE j g j of sets in ˝, let the disjoint F F S union j E j or (C) represent the union j E j and also assert that the sets are pairwise disjoint.

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Use “8large n” to mean: “9n0 such that 8n > n0 ”. To refer to left hand side of an Eq. (20), use LhS(20); do analogously for RhS(20), the right hand side.

Definition of the Subject The word ‘entropy’ (originally German, Entropie) was coined by Rudolf Julius Emanuel Clausius circa 1865 [2,3], taken from the Greek  o ˛, ‘a turning towards’. This article thus begins (Prolegomenon, “introduction”) and ends (Exodos1 , “the path out”) in Greek. Clausius, in his coinage, was referring to the thermodynamic notion in physics. Our focus in this article, however, will be the concept in measurable and topological dynamics. (Entropy in differentiable dynamics2 would require an article by itself.) Shannon’s 1948 paper [6] on Information Theory, then Kolmogorov’s [4] and Sinai’s [7] generalization to dynamical systems, will be our starting point. Our discussion will be of the one-dimensional case, where the acting-group is Z. “My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. John von Neumann had a better idea, he told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function goes by that name in statistical mechanics. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.’ ” (Shannon as quoted in [59]) Entropy example: How many questions? Imagine a dartboard, Fig. 1, split in five regions A; : : : ; E with known probabilities. Blindfolded, you throw a dart 1 This is

the Greek spelling. 2 For instance, see [15,18,24,25].

Entropy in Ergodic Theory, Figure 1 This dartboard is a probability space with a 5-set partition. The atoms have probabilities 12 ; 18 ; 18 ; 18 ; 18 . This probability distribution will be used later in Meshalkin’s example

at the board. What is the expected number V of Yes/No questions needed to ascertain the region in which the dart landed? Solve this by always dividing the remaining probability in half. ‘Is it A?’ – if Yes, then V D 1. Else: ‘Is it B or C?’ – if Yes, then ‘Is it B?’ – if No, then the dart landed in C, and V D 3 was the number of questions. Evidently V D 3 also for regions B; D; E. Using “log” to denote base-2 logarithm3 , the expected number of questions4 is thus 1 1 1 1 1 1C 3C 3C 3C 3 2 8 8 8 8   4 X 1 note D p j log D 2: pj

E(V ) D

(1)

jD0

  Letting vE :D 12 ; 18 ; 18 ; 18 ; 18 be the probability vector, we can write this expectation as X (x) : E(V ) D x2E v

Here,  : [0; 1] ! [0; 1) is the important function5 (x) :D x log(1/x) ;

so extending by continuity gives

(0) D 0 : (2) An interpretation of “(x)” is the number of questions needed to winnow down to an event of probability x. Distribution Entropy Given a probability vector vE, define its distribution entropy as X H (E v ) :D (x) : (3) x2E v

This article will use the term distropy for ‘distribution entropy’, reserving the word entropy for the corresponding 3 In this paper, unmarked logs will be to base-2. In entropy theory, it does not matter much what base is used, but base-2 is convenient for computing entropy for messages described in bits. When using the natural logarithm, some people refer to the unit of information as a nat. In this paper, I have picked bits rather than nats. 4 This is holds when each probability p is a reciprocal power of two. For general probabilities, the “expected number of questions” interpretation holds in a weaker sense: Throw N darts independently at N copies of the dartboard. Efficiently ask Yes/No questions to determine where all N darts  landed. Dividing by N, then sending N ! 1, will be the p log 1p sum of Eq. (1). 5 There does not seem to be a standard name for this function. I use , since an uppercase  looks like an H, which is the letter that Shannon used to denote what I am calling distribution-entropy.

Entropy in Ergodic Theory

dynamical concept, when there is a notion of time involved. Getting ahead of ourselves, the entropy of a stationary process is the asymptotic average value that its distropy decays to, as we look at larger and larger finite portions of the process.   An equi-probable vector vE :D K1 ; : K: :; K1 evidently has H (E v ) D log(K). On a probability space, the “distropy of partition P”, written H (P) or H (A1 ; A2 ; : : :) shall mean the distropy of probability vector j 7! (A j ). A (finite) partition necessarily has finite distropy. A countable partition can have finite distropy, e. g. H (1/2; 1/4; 1/8; : : : ) D 2. One could also have infinite distropy: Consider a piece B X of mass 1/2 N . Splitting B into 2 k many equal-mass atoms gives an -sum of 2 k (k C N)/ (2 k 2 N ). Setting k D k N :D 2 N  N makes this -sum F1 equal 1; so splitting the pieces of X D ND1 B N , with 1 (B N ) D 2N , yields an 1-distropy partition.

Entropy in Ergodic Theory, Figure 2 Using natural log, here are the graphs of: (x) in solid red, H (x; x c ) in dashed green, 1  x in dotted blue. Both (x) and H (x; x c ) are strictly convex-down. The 1  x line is tangent to (x) at x D 1

Function  The (x) D x log(1/x) function6 has vertical tangent at x D 0, maximum at 1/e and, when graphed in nats slope 1 at x D 1. Consider partitions P and Q on the same space (X; ). Their join, written P _ Q, has atoms A \ B, for each pair A 2 P and B 2 Q. They are independent, written P?Q if (A\ B) D (A)(B) for each A; B pair. We write P < Q, and say that “P refines Q”, if each P-atom is a subset of some Q-atom. Consequently, each Q-atom is a union of Patoms. Recall, for ı a real number, our convention that ı c means 1  ı, in analogy with (B c ) equaling 1  (B) on a probability space. Distropy Fact

Entropy in Ergodic Theory, Figure 3 Using natural log: The graph of H (x1 ; x2 ; x3 ) in barycentric coordinates; a slice has been removed, between z D 0:745 and z D 0:821. The three arches are copies of the distropy curve from Fig. 2.

For partitions P; Q; R on probability space (X; ): (a) H (P)  log(# P), with equality IFF P is an equi-mass partition. (b) H (Q _ R)   H (Q) C H (R), with equality IFF Q?R. (c) For ı 2 0; 12 , the function ı 7! H (ı; ı c ) is strictly increasing. a:e: (d) R 4 P implies H (R)  H (P), with equality IFF R D P. 6 Curiosity:

Just in this paragraph we compute distropy in nats, that is, using natural logarithm. Given a small probability p 2 [0; 1] and setting x :D 1/p, note that (p) D log(x)/x 1/(x), where (x) denotes the number of prime numbers less-equal x. (This approximation is a weak form of the Prime Number Theorem.) Is there any actual connection between the ‘approximate distropy’ function P H (E p) :D p2Ep 1/(1/p) and Number Theory, other than a coincidence of growth rate?

Proof Use the strict concavity of (), together with Jensen’s inequality.  Remark 1 Although we will not discuss it in this paper, most distropy statements remain true with ‘partition’ replaced by ‘countable partition of finite distropy’. Binomial Coefficients The dartboard gave an example where distropy arises in a natural way. Here is a second example. For a small   ı > 0, one might guess that the binomial coefficient ınn grows asymptotically (as n ! 1) like 2"n , for some small ". But what is the correct relation between " and ı?

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Well, Stirling’s formula n!  [n/e]n gives nn

n!  c [ın]![ı c n]! [ın][ı n] [ı c n][ı n] 1 (recall ı c D 1  ı) : D ın c ıc n [ı [ı ] ]   Thus n1 log ınn  H (ı; ı c ). But by means of the above distropy inequalities, we get an inequality true for all n, not just asymptotically.  Lemma 2 (Binomial Lemma) Fix a ı 2 0; 12 and let H :D H (ı; ı c ). Then for each n 2 ZC : ! X n  2Hn : (4) j j2[0:::ı n]

Proof Let X f0; 1gn be the set of xE with #

Noisy Channel Shannon’s theorem says that a noisy channel has a channel capacity. Transmitting above this speed, there is a minimum error-rate (depending how much “above”) that no error-correcting code can fix. Conversely, one can transmit below – but arbitrarily close to – the channel capacity, and encode the data so as to make the error-rate less than any given ". We use Corollary 3 to show the existence of such codes, in the simplest case where the noise7 is a binary independent-process (a “Bernoulli” process, in the language later in this article). We have a channel which can pass one bit per second. Alas, there is a fixed noise-probability  2 [0; 12 ) so that a bit in the channel is perturbed into the other value. Each perturbation is independent of all others. Let H :D H (;  c ). The value [1  H] bits-per-second is the channel capacity of this noise-afflicted channel.

fi 2 [1 : : : n] j x i D 1g  ın :

On X, let P1 ; P2 ; : : : be the coordinate partitions; e. g. P7 D (A7 ; Ac7 ), where A7 :D fE x j x7 D 1g. Weighting each point 1 , the uniform distribution on X, gives that (A7 )  by jXj ı. So H (P7 )  H, by (c) in Sect. “Distropy Fact”. Finally, the join P1 _    _ Pn separates the points of X. So log(# X) D H (P1 _    _ Pn )  H (P1 ) C : : : C H (P n )  Hn ; making use of (a),(b) in “Distropy Fact”. And # X equals LhS (4).  A Gander at Shannon’s Noisy Channel Theorem We can restate the Binomial lemma using the Hamming metric on f0; 1gn , Dist(E x ; yE) :D # fi 2 [1 : : : n] j x i ¤ y i g :

Encoding/Decoding Encode using an “k; n–blockcode”; an injective map F : f0; 1g k ! f0; 1gn . The source text is split into consecutive k-bit blocks. A block xE 2 f0; 1g k is encoded to F(E x ) 2 f0; 1gn and then sent through the channel, where it comes out perturbed to ˛E 2 f0; 1gn . The transmission rate is thus k/n bits per second. For this example, we fix a radius r > 0 to determine the decoding map, Dr : f0; 1gn ! fOopsg t f0; 1g k : We set Dr (E ˛) to zE if there is a unique zE with F(Ez) 2 Bal(E ˛; r); else, set Dr (E ˛) :D Oops. One can think of the noise as a f0; 1g-independentprocess, with Prob(1) D , which is added mod-2 to the signal-process. Suppose we can arrange that the set fF(E x )) j xE 2 f0; 1g k g of codewords, is a strongly r-separated-set. Then

Use Bal(E x ; r) for the open radius-r ball centered at xE, and Bal(E x ; r) :D f yE j Dist (E x ; yE)  rg for the closed ball. The above lemma can be interpreted as saying that x ; ın)j  2H (ı;ı jBal(E

c )n

for each xE 2 f0; 1gn : (5)  Corollary 3 Fix n 2 ZC and ı 2 0; 12 , and let ;

H :D H (ı; ı c ) : Then there is a set C f0; 1gn , with # C  2[1H]n , that is strongly ın-separated. I. e., Dist (E x ; yE) > ın for each distinct pair xE; yE 2 C.

The probability that a block is mis-decoded is the probability, flipping a -coin n times that we get more than r many Heads:

(6)  1 Theorem 4 (Shannon) Fix a noise-probability  2 0; 2 and let H :D H (;  c ). Consider a rate R < [1  H] and an " > 0. Then 8large n there exists a k and a code F : f0; 1g k ! f0; 1gn so that: The F-code transmits bits at faster than R bits-per-second, and with error-rate < ". 7 The noise-process is assumed to be independent of the signalprocess. In contrast, when the perturbation is highly dependent on the signal, then it is sometimes called distortion.

Entropy in Ergodic Theory

Proof Let H0 :D H (ı; ı c ), where ı >  was chosen so close to  that ı<

1 2

and 1  H0 > R :

(7)

The information function has been defined so that its expectation is the distropy of P, E(IP ) D

where

k :D b[1  H0 ]nc :

(8)

By Corollary 3, there is a strongly ın-separated-set C

0 f0; 1gn with # C  2[1H ]n . So C is big enough to permit an injection F : f0; 1g k ! C. Courtesy Eq. (6), the probability of a decoding error is that of getting more than ın many Heads in flipping a -coin n times. Since ı > , the Weak Law of Large Numbers guarantees – once n is large enough – that this probability is less than the given ".  The Information Function Agree to use P D (A1 ; : : :), Q D (B1 ; : : :), R D (C1 ; : : :) for partitions, and F ; G for fields. With C a (finite or infinite) family of subfields of X , W their join G2C G is the smallest field F such that G F , for each G 2 C. A partition Q can be interpreted also as a field; namely, the field of unions of its atoms. A join of denumerably many partitions will be interpreted as a field, but a join of finitely many, P1 _    _ PN , will be viewed as a partition or as a field, depending on context. Conditioning a partition P on a positive-mass set B, let PjB be the probability vector A 7! (A\B) (B) . Its distropy is   X 1 (A \ B) H (PjB) D log : (A \ B)/(B) (B) A2P So conditioning P on a partition Q gives conditional distropy X H (PjQ) D H (PjB)(B) B2Q

D

X A2P;B2Q

log

1 (A\B) (B)

Conditioning on a Field For a subfield F X , recall that each function g 2 L1 () has a conditional expectation E(gjF ) 2 L1 (). It is the unique F -measurable function with Z

8B 2 F :

E(gjF )d D B

Z gd : B

Returning to distropy ideas, use (AjF ) for the conditional probability function; it is the conditional expectation E(1A jF ). So the conditional information function is IPjF (x) :D

X A2P

 log

 1 1A (x) : (AjF (x))

(11)

Its integral H (PjF ) :D

Z

IPjF d ;

is the conditional distropy of P on F . When F is the field of unions of atoms from some partition Q, then the number H (PjF ) equals the H (PjQ) from Eq. (9). Write G j % F to indicate  fields G1 G2 : : : S1 that are nested, and that Fld 1 G j D F , a.e. The Martingale Convergence Theorem (p. 103 in [20]) gives (c) below. Conditional-Distropy Fact Consider partitions P; Q; R and fields F and G j . Then a:e:

! (A \ B) :

(9)

A “dartboard” interpretation of H (PjQ) is The expected number of questions to ascertain the P-atom that a random dart x 2 X fell in, given that we are told its Q-atom. For a set A X, use 1A : X ! f0; 1g for its indicator function; 1A (x) D 1 IFF x 2 A. The information function of partition P, a map IP : X ! [0; 1), is   X 1 IP :D log (10) 1A () : (A) A2P

IP ()d D H (P) : X

Pick a large n for which k >R; n

Z

(a) 0  H (PjF )  H (P), with equality IFF P F , respectively, P?F . (b) H (Q _ RjF )  H (QjF ) C H (RjF ). (c) Suppose G j % F . Then H (PjG j ) & H (PjF ). (d) H (Q _ R) D H (QjR) C H (R). (d’) H (Q _ R1 jR0 ) D H (QjR1 _ R0 ) C H (R1 jR0 ). Imagining our dartboard (Fig. 1) divided by superimposed partitions Q and R, equality (d) can interpreted as saying: ‘You can efficiently discover where the dart landed in both partitions, by first asking efficient questions about R, then – based on where it landed in R – asking intelligent questions about Q.’

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Entropy in Ergodic Theory

Entropy of a Process Consider a transformation (T : X; ) and partition P D (A1 ; A2 ; : : :). Each “time” n determines a partition Pn :D T n P, whose jth-atom is T n (A j ). The process T; P refers W to how T acts on the subfield 1 0 P n X . (An alternative view of a process is as a stationary sequence V0 ; V1 ; : : : of random variables Vn : X ! ZC , where Vn (x) :D j because x is in the jth-atom of Pn .) Write E (T; P) or E T (P) for the “entropy” of the T; P process”. It is the limit of the conditional-distropy-numbers c n :D H (P0 jP1 _ P2 _    _ Pn1 ) : This limit exists since H (P) D c1  c2      0. Define the average-distropy-number n1 h n , where h n :D H (P0 _ P1 _    _ Pn1 ) : Certainly h n D c n C H (P1 _    _ Pn1 ) D c n C h n1 , since T is measure preserving. Induction gives h n D Pn 1 jD1 c j . So the Cesàro averages n h n converge to the entropy. Theorem 5 The entropy of process (T; P : X; X ; ) equals

T(x) :D [n 7! xnC1 ] : The time-zero partition P separates points, under the action of the shift. This L-atom time-zero partition has Phxi D Phyi IFF x0 D y0 . So no matter what shift-invariant measure is put on X, the time-zero partition will generate under the action of T. Time Reversibility A transformation need not be isomorphic to its inverse. Nonetheless, the average-distropy-numbers show that E(T 1 ; P) D E(T; P); although this is not obvious from the conditioning-definition of entropy. Alternatively, ! ˇ_ ˇ n H P0 ˇˇ Pj 1

D H (P0 _    _ Pn )  H (P1 _    _ Pn ) D H (Pn _    _ P0 )  H (Pn _    _ P1 ) ! ˇ_ ˇ n (12) D H P0 ˇˇ P j :

1 H (P0 _    _ Pn1 ) n!1 n ! ˇ_ ˇ n ˇ D lim H P0 ˇ P j n!1 1 ˇ1 ! ˇ_ D H P0 ˇˇ P j : lim

1

Bernoulli Processes

1

Both limits are non-increasing. The entropy E T (P)  0, W T with equality IFF P 1 1 P j . And E (P)  H (P), with equality IFF T; P is an independent process. Generators We henceforth only discuss invertible mpts, that is, when T 1 is itself an mpt. Viewing the atoms of P as “letters”, then, each x 2 X has a T; P-name : : : x2 x1 x0 x1 x2 x3 : : : ; PhT n (x)i,

IFF P separates points. That is, after deleting a (T-invariant) nullset, distinct points of X have distinct T; P-names. A finite set [1 : : : L] of integers, interpreted as an alphabet, yields the shift space X :D [1 : : : L]Z of doubly-infinite sequences x D (: : : x1 x0 x1 : : : ). The shift T : X ! X acts on X by

T n (x).

where xn is the P-letter owning A partition P generates (the whole field) under (T : W n 8 X; ), if 1 1 T P D X . It turns out that P generates 8 I am now at liberty to reveal that our X has always been a Lebesgue space, that is, measure-isomorphic to an interval of R together with countably many point-atoms (points with positive mass).

A probability vector vE :D (v1 ; : : : ; v L ) can be viewed as a measure on alphabet [1 : : : L]. Let vE be the resulting product measure on X :D [1 : : : L]Z , with T the shift on X and P the time-zero partition. The independent process (T; P : X; vE ) is called, by ergodic theorists, a Bernoulli process. Not necessarily consistently, we tend to refer to the underlying  transformation as a Bernoulli  shift.  The 12 ; 12 -Bernoulli and the 13 ; 13 ; 13 -Bernoulli have different process-entropies, but perhaps their underlying transformations are isomorphic? Prior to the Kolmogorov–Sinai definition of entropy9 of a transformation, this question remained unanswered. The equivalence of generating and separating is a technical theorem, due to Rokhlin. Assuming  to be Lebesgue is not much of a limitation. For instance, if  is a finite measure on any Polish space, then  extends to a Lebesgue measure on the -completion of the Borel sets. To not mince words: All spaces are Lebesgue spaces unless you are actively looking for trouble. 9 This is sometimes called measure(-theoretic) entropy or (perhaps unfortunately) metric entropy, to distinguish it from topological

Entropy in Ergodic Theory

Entropy of a Transformation The Kolmogorov–Sinai definition of the entropy of an mpt is E(T) :D supfE T (Q) j Q a partition on Xg :

Certainly entropy is an isomorphism invariant – but is it useful? After all, the supremum of distropies of partitions is always infinite (on non-atomic spaces) and one might fear that the same holds for entropies. The key observation (restated in Lemma 8c and proved below) was this, from [4] and [7]. Theorem 6 (Kolmogorov–Sinai Theorem) If P generates under T, then E(T) D E(T; P).     Thereupon the 12 ; 12 and 13 ; 13 ; 13 Bernoulli-shifts are not isomorphic, since their respective entropies are log(2) ¤ log(3). Wolfgang Krieger later proved a converse to the Kolmogorov–Sinai theorem. Theorem 7 (Krieger Generator Theorem, 1970) Suppose T ergodic. If E(T) < 1, then T has a generating partition. Indeed, letting K be the smallest integer K > E(T), there is a K-atom generator.10 Proof See Rudolph [21], or § 5.1 in  Joinings in Ergodic Theory, where Krieger’s theorem is stated in terms of joinings.  Entropy Is Continuous

Then Lemma 8b says that process-entropy varies continuously with varying the partition. Lemma 8 Fix a mpt (T : X; ). For partitions P; Q; Q0 , define R :D Q4Q0 and let ı :D H (R). Then (a) jH (Q)  H (Q0 )j  ı. (Distropy varies continuously with the partition.) (b) jE T (Q)  E T (Q0 )j  ı. (Process-entropy varies continuously with the partition.) (c) For all partitions Q Fld(T; P) : E T (Q)  E T (P). Proof (of (a)) Evidently Q0 _ R D Q0 _ Q D Q _ R. So H (Q0 )  H (Q _ R)  H (Q) C ı.  Proof (of (b)) As above, H

N _

! Q0j

N _

H

1

! Qj C H

1

N _

! :

Rj

1

Sending N ! 1 gives E T (Q0 )  E T (Q) C E T (R). Finally, E T (R)  H (R) and so E T (Q0 )  E T (Q) C ı.  Proof (of (c)) Let K :D jQj. Then there is a sequence of WL P` . By above, K-set partitions Q(L) ! Q with Q(L) 4 L E T (Q(L) ) ! E T (Q), so showing that E

T

L _

!

?

P`  E T (P)

L

will suffice. Note that 0

(B01 ; : : :), 0

Given ordered partitions Q D (B1 ; : : :) and Q D extend the shorter by null-atoms until jQj D jQ j. Let F Fat :D j [B j \ B0j ]; this set should have mass close to 1 if Q and Q0 are almost the same partition. Define a new partition Q4Q0 :D fFatg t fB i \ B0j j with i ¤ jg : (In other words, take Q _ Q0 and coalesce, into a single atom, all the B k \ B0k sets.) Topologize the space of parti  tions by saying11 that Q(L) ! Q when H Q4Q(L) ! 0. entropy. Tools known prior to entropy, such as spectral properties, did not distinguish the two Bernoulli-shifts; see  Spectral Theory of Dynamical Systems for the definitions. 10 It is an easier result, undoubtedly known much earlier, that every ergodic T has a countable generating partition – possibly of 1-distropy. 11 On the set of ordered K-set partitions (with K fixed) this conver   gence is the same as: Q(L) ! Q when  Fat Q(L) ; Q ! 1. An alternative approach is the Rokhlin metric, Dist(P; Q) :D H (PjQ) C H (QjP), which has the advantage of working for unordered partitions.

h N :D H

N1 _

L _

Tn

1 NH

W

N1 Pj 0

0 D H@

P`

L

nD0

So N1 h N  N ! 1.

!!

N1CL _

1 PjA :

jDL



C

1 N 2LH (P).

Now send 

Entropy Is Not Continuous The most common topology placed on the space ˝ of mpts is the coarse topology12 that Halmos discusses in his “little red book” [14]. The Rokhlin lemma (see p. 33 in [21]) implies that the isomorphism-class of each ergodic mpt is dense in ˝, (e. g., see p. 77 in [14]) disclosing that the S 7! E(S) map is exorbitantly discontinuous. 12 i. e, S ! T IFF 8A 2 X : (S 1 (A)4T 1 (A)) ! 0; this is n n a metric-topology, since our probability space is countably generated. This can be restated in terms of the unitary operator U T on L2 (), where U T ( f ) :D f ı T. Namely, S n ! T in the coarse topology IFF U S n ! U T in the strong operator topology.

211

212

Entropy in Ergodic Theory

Indeed, the failure happens already for process-entropy with respect to a fixed partition. A Bernoulli process T; P has positive entropy. Take mpts S n ! T, each isomorphic to an irrational rotation. Then each E(S n ; P) is zero, as shown in the later section "Determinism and ZeroEntropy". Further Results When F is a T-invariant subfield, agree to use T F for “T restricted to F ”, which is a factor (see Glossary) of T. Transformations T and S are weakly isomorphic if each is isomorphic to a factor of the other. The foregoing entropy tools make short shrift of the following. Lemma 9 (Entropy Lemma) Consider T-invariant subfields G j and F . (a) Suppose G j % F . Then E T G j % E(T F ). In particular, G F implies that E(T G )  E(T F ), so entropy is an invariant of weak-isomorphism. P (b) E(T G1 _G2 _::: )  j E(T G j ). P And E(T; Q1 _ Q2 _ : : :)  j E(T; Q j ). P (c) For mpts (S j : Yj ;  j ): E(S1  S2    ) D j E(S j ). (d) E(T 1 ) D E(T). More generally, E(T n ) D jnj  E(T). Meshalkin’s Map In the wake of Kolmogorov’s 1958 entropy paper, for two Bernoulli-shifts to be isomorphic one now knew that they had to have equal entropies. Meshalkin provided the first non-trivial example in 1959 [45]. Let S : Y ! Y be the Bernoulli-shift over the “letter”  alphabet  fE; D; P; Ng, with probability distribution 14 ; 14 ; 14 ; 14 . The letters E; D; P; N stand for Even, oDd, Positive, Negative, and will be used to describe the code (isomorphism) between the processes. Use T : X ! X for the Bernoulli-shift over “digit” alphabet 1;  1 1f0;1 C1;  C2; 2g, with probability  1 1 1 distribu 1 1 1 tion ; ; ; ; . Both distributions 4 ; 4 ; 4 ; 4 and  1 1 12 18 18  8 8 2 ; 8 ; 8 ; 8 ; 8 have distropy log(4). After deleting invariant nullsets from X and Y, the construction will produce a measure-preserving isomorphism : X ! Y so that T ı D ı S. The Code

The leftmost 0 is linked to the rightmost +1, as indicated by the longest-overbar. The left/right-parentheses form a 12 ; 12 -random-walk. Since this random walk is recurrent, every position in x will be linked (except for a nullset of points x). Below each 0, write “P” or “N” as the 0 is linked to a positive or negative digit. And below the other digits, write “E” or “D” as the digit is even or odd. So the upper name in X is mapped to the lower name, a point y 2 Y. map : X !  Y carries  1 This   the upstairs 1 1 1 1 1 1 1 1 distribution to ; ; ; ; ; ; ; 2 8 8 8 8 4 4 4 4 , downstairs. It takes some arguing to show that independence is preserved. The inverse map 1 views D and E as right-parentheses, and P and N as left. Above D, write the odd digit C1 or 1, as this D is linked to Positive or Negative. Markov Shifts A Bernoulli process T; P has independence P(1:::0] ?P1 whereas a Markov process is a bit less aloof: The infinite Past P(1:::0] doesn’t provide any more information about Tomorrow than Today did. That is, the conditional distribution P1 jP(1:::0] equals P1 jP0 . Equivalently,   note H (P1 jP0 ) D H P1 jP(1:::0] D E(T; P) : (13) The simplest non-trivial Markov process (T; P : X; ) is over a two-letter alphabet fa; bg, and has transition graph Fig. 4, for some choice of transition probabilities s and c. The graph’s Markov matrix is

 s c ; M D [m i; j ] i; j D 1 0 where c D 1  s, and m i; j denotes the probability of going from state i to state j. If Today’s distribution on the two states is the probability-vector vE :D [pa pb ], then Tomorrow’s is the product vEM. So a stationary process needs vEM D vE. This equa1 c tion has the unique solution pa D 1Cc and pb D 1Cc .

In X, consider this point x:

: : : 0 0 0 1 0 0 +1 +2 1 +1 0 : : : Regard each 0 as a left-parenthesis, and each non-zero as a right-parenthesis. Link them according to the legal way of matching parentheses, as shown in the top row, below: 0 0 0 1 0 0 +1 +2 1 +1 0 P N N D P P D

E

D

D ?

Entropy in Ergodic Theory, Figure 4 Call the transition probabilities s :D Prob(a!a) for stay, and c :D Prob a!b for change. These are non-negative reals, and sCcD1

Entropy in Ergodic Theory

An example of computing the probability of a word or cylinder set; (see Sect. “The Carathéodory Construction” in  Measure Preserving Systems) in the process, is

approximately 2E(T;P)n ; this, after discarding small mass from the space. But the growth of n 7! 2n is sub-exponential and so, for our rotation, E(T; P) must be zero. Theorem 10 (Shannon–McMillan–Breiman Theorem (SMB-Theorem)) Set E :D E(T; P), where the tuple (T; P : X; ) is an ergodic process. Then the average information function

s (baaaba) D pb mba maa maa mab mba c D 1ssc1: 1Cc The subscript on s indicates the dependence on the transition probabilities; let’s also mark the mpt and call it T s . Using Eq. (13), the entropy of our Markov map is

1 n!1 IP (x) ! E ; n [0:::n)

for a.e.

x2X:

(15)

The functions f n :D IP[0:::n) converge to the constant function E both in the L1 -norm and in probability. 14

E(Ts ) D pa  H (s; c) C pb  H (1; 0)

„ ƒ‚ … D0

1 D  [s log(s) C c log(c)] : 1Cc

(14)

Determinism and Zero-Entropy Irrational rotations have zero-entropy; let’s reveal this in two different ways. Equip X :D [0; 1) with “length” (Lebesgue) measure and wrap it into a circle. With “˚” denoting addition mod-1, have T : X ! X be the rotation T(x) :D x ˚ ˛, where the rotation number ˛ is irrational. Pick distinct points y0 ; z0 2 X, and let P be the partition whose two atoms are the intervals [y0 ; z0 ) and [z0 ; y0 ), wrapping around the circle. The T-orbit of each point x is dense13 in X. In particular, y0 has dense orbit, so P separates points – hence ?

generates – under T. Our goal, thus, is E(T; P) D 0.

Proof See the texts of Karl Petersen (p. 261 in [20]), or Dan Rudolph (p. 77 in [21]).  Consequences Recall that P[0:::n) means P0 _ P1 _    _ Pn1 , where P j :D T j P. As usual, P[0:::n) hxi denotes the P[0:::n) -atom owning x. Having deleted a nullset, we can restate Eq. (15) to now say that 8"; 8x; 8large n:   1/2[EC"]n   P[0:::n) hxi  1/2[E"]n :

(16)

This has the following consequence. Fixing a number ı > 0, we consider any set with (B)  ı and count the number of n-names of points in B. The SMB-Thm implies 

8"; 8large n; 8B  ı : ˇ ˇ ˇfn-names in Bgˇ  2[E"]n :

(17)

Rotations are Deterministic

Rank-1 Has Zero-Entropy

The forward T-orbit of each point is dense. This is true for y0 , and so the backward T; P-name of each x actually W n tells us which point x is. I. e., P 1 1 T P, which is our definition of “process T; P is deterministic”. Our P being finite, this determinism implies that E(T; P) is zero, by Theorem 5.

There are several equivalent definitions for “rank-1 transformation”, several of which are discussed in the introduction of [28]. (See Chap. 6 in [13] as well as [51] and [27] for examples of stacking constructions.) A rank-1 transformation (T : X; ) admits a generating partition P and a sequence of Rokhlin stacks S n X, with heights going to 1, and with (S n ) ! 1. Moreover, each of these Rokhlin stacks is P-monochromatic, that is, each level of the stack lies entirely in some atom of P. Taking a stack of some height 2n, let B D B n be the union of the bottom n levels of the stack. There are at most n many length-n names starting in Bn , by monochromaticity. Finally, (B n ) is almost 12 , so is certainly larger than ı :D 13 . Thus Eq. (17) shows that our rank-1 T has zero entropy.

Counting Names in a Rotation The P0 _    _ Pn1 partition places n translates of points y0 and of z0 , cutting the circle into at most 2n intervals. Thus H (P0 _ : : : _ Pn1 )  log(2n). And n1 log(2n) ! 0. Alternatively, the below SMB-theorem implies, for an ergodic process T; P, that the number of length-n names is an " > 0 and an N > 1/". Points x; T(x); : : : ; T N (x) have some two at distance less than N1 ; say, Dist(T i (x); T j (x)) < ", for some 0  i < j  N. Since T is an isometry, " > Dist(x; T k (x)) > 0, where k :D j  i. So the T k -orbit of x is "-dense. 13 Fix

14 In engineering circles, this is called the Almost-everywhere equipartition theorem.

213

214

Entropy in Ergodic Theory

Cautions on Determinism’s Relation to Zero-Entropy

K-Processes

A finite-valued process T; P has zero-entropy iff P

W1 1 P j . Iterating gives

Kolmogorov introduced the notion of a K-process or Kolmogorov process, in which the present becomes asymptotically independent of the distant past. The asymptotic past of the T; P process is called its tail field, where

1 _ 0

Pj

1 _

Pj ;

1

i. e., the future is measurable with respect to the past. This was the case with the rotation, where a point’s past uniquely identified the point, thus telling us its future. While determinism and zero-entropy mean the same thing for finite-valued processes, this fails catastrophically for real-valued (i. e., continuum-valued) processes, as shown by an example of the author’s. A stationary realvalued process V D : : : V1 V0 V1 V2 : : : is constructed in [40] which is simultaneously strongly deterministic: The two values V0 ; V1 determine all of V, future and past. and non-consecutively independent. This latter means that for each bi-infinite increasing integer sequence fn j g1 jD1 with no consecutive pair (always 1 C n j < n jC1 ), then the list of random variables : : : Vn 1 Vn 0 Vn 1 Vn 2 : : : is an independent process. Restricting the random variables to be countablyvalued, how much of the example survives? Joint work with Kalikow [39] produced a countably-valued stationary V which is non-consecutively independent as well as deterministic. (Strong determinism is ruled out, due to cardinality considerations.) A side-effect of the construction is that V’s time-reversal n 7! Vn is not deterministic. The Pinsker–Field and K-Automorphisms Consider the collection of zero-entropy sets, Z D Z T :D fA 2 X j E(T; (A; Ac )) D 0g :

(18)

Courtesy of Lemma 9b, Z is a T-invariant field, and 8Q Z : E(T; Q) D 0 :

(19)

The Pinsker field15 of T is this Z. It is maximal with respect to Eq. (19). Unsurprisingly, the Pinsker factor T Z has zero entropy, that is, E(T Z ) D 0. A transformation T is said to have completely-positive entropy if it has no (nontrivial) zero-entropy factors. That is, its Pinsker field Z T is the trivial field, ; :D f¿; Xg. 15 Traditionally, this called the Pinsker algebra where, in this context, “algebra” is understood to mean “ -algebra”.

Tail(T; P) :D

1 \

M _

Pj :

MD1 jD1

This T; P is a K-process if Tail(T; P) D ;. This turns out to be equivalent to what we might a call a strong form of “sensitive dependence on initial conditions”: For each fixed length L, the distant future _

Pj

j2(G:::GCL]

becomes more and more independent of _

Pj ;

j2(1:::0]

as the gap G ! 1. A transformation T is a Kolmogorov automorphism if it possesses a generating partition P for which T; P is a K-process. Here is a theorem that relates the “asymptotic forgetfulness” of Tail(T; P) D ;, to the lack of determinism implied by having no zero-entropy factors (See Walters, [23], p. 113. Related results appear in Berg [44]). Theorem 11 (Pinsker-Algebra Theorem) Suppose P is a generating partition for an ergodic T. Then Tail(T; P) equals Z T . Since Z T does not depend on P, this means that all generating partitions have the same tail field, and therefore K-ness of T can be detected from any generator. Another non-evident fact follows from the above. The future field of T; P is defined to be Tail (T 1 P). It is not obvious that if the present is independent of the distant past, then it is automatically independent of the distant future. (Indeed, the precise definitions are important; witness the Cautions on determinism section.) But since the entropy of a process, E(T; (A; Ac )), equals the entropy of the time-reversed process E(T 1 ; (A; Ac )), it follows that Z T equals Z T 1 . Ornstein Theory In 1970, Don Ornstein [46] solved the long-standing problem of showing that entropy was a complete isomorphism-

Entropy in Ergodic Theory

invariant of Bernoulli transformations; that is, that two independent processes with same entropy necessarily have the same underlying transformation. (Earlier, Sinai [52]) had shown that two such Bernoulli maps were weakly isomorphic, that is, each isomorphic to a factor of the other.) Ornstein introduced the notion of a process being finitely determined, see [46] for a definition, proved that a transformation T was Bernoulli IFF it had a finitely-determined generator IFF every partition was finitely-determined with respect to T, and showed that entropy completely classified the finitely-determined processes up to isomorphism. This seminal result led to a vast machinery for proving transformations to be Bernoulli, as well as classification and structure theorems [47,51,53]. Showing that the class of K-automorphisms far exceeds the Bernoulli maps, Ornstein and Shields produced in [48] an uncountable family of non-isomorphic K-automorphisms all with the same entropy.

And the topological entropy of T is Etop (T) :D supV Etop (T; V ) ;

(21)

taken over all open covers V. Thus Etop counts, in some sense, the growth rate in the number of T-orbits of length n. Evidently, topological entropy is an isomorphism invariant. Two continuous maps T : X ! X and S : Y ! Y are topologically conjugate (as isomorphism is called in this category) if there exists a homeomorphism : X ! Y with T D S . Lemma 12 (Subadditive Lemma) Consider a sequence s D (s l )1 1 [1; 1] satisfying s kCl  s k C s l , for all k; l 2 Z. Then the following limit exists in [1; 1], and lim n!1 snn D inf n snn . Topological entropy, or “top ent” for short, satisfies many of the relations of measure-entropy. Lemma 13

Topological Entropy

(a) V 4 W implies

Adler, Konheim and McAndrew, in 1965, published the first definition of topological entropy in the eponymous article [33]. Here, T : X ! X is a continuous self-map of a compact topological space. The role of atoms is played by open sets. Instead of a finite partition, one uses a finite16 open-cover V D fU j gLjD1 , i. e. each patch U j is open, and S their union (V) D X. (Henceforth, ‘cover’ means “open cover”.) Let Card(V) be the minimum cardinality over all subcovers. [ ˚ (V0 ) D X ; Card(V) :D Min ] V0 j V0 V and and let H (V ) D Htop (V ) :D log(Card(V )) :

Analogous to the definitions for partitions, define V _ W :D fV \ W j V 2 V and W 2 W g ;

T V :D fT 1 (U) j U 2 Vg and V[0:::n) :D V0 _ V1 _    _ V n1 ; W < V ; if each W -patch is a subset of some V -patch:

The T; V-entropy is E T (V ) D E(T; V ) D Etop (T; V )

  1 :D lim sup Htop V[0:::n) : n!1 n

(20)

16 Because we only work on a compact space, we can omit “finite”. Some generalizations of topological entropy to non-compact spaces require that only finite open-covers be used [37].

H (V )  H (W )

and E(T; V )  E(T; W ) :

(b) H (V _ W )  H (V) C H (W ). (c) H (T(V))  H (V), with equality if T is surjective. Also, E(T; V )  H (V).   (d) In Eq. (20), the limn!1 n1 H V[0:::n) exists. (e) Suppose T is a homeomorphism. Then E(T 1 ; V ) D E(T; V ) ;

for each cover V. Consequently, Etop (T 1 ) D Etop (T). (f) Suppose C is a collection of covers such that: For each cover W , there exists a V 2 C with V < W . Then Etop (T) equals the supremum of Etop (T; V ), just taken over those V 2 C. (g) For all ` 2 N:   Etop T ` D `Etop (T) : Proof (of (c)) Let C 4 V be a min-cardinality subcover. Then T C is a subcover of T V. So Card T V  jT Cj D jCj. As for entropy, inequality (b) and the foregoing give H (V [0:::n) )  H (V )n.  Proof (of (d)) Set s n :D H (V[0:::n) ). Then    s kCl  s k C H T k V[0:::l )  s k C s l ; by (b) and (c), and so the Subadditive Lemma 12, applies. 

215

216

Entropy in Ergodic Theory

Proof (of (g)) WLOG, ` D 3. Given V a cover, triple it to b V :D V \ T V \ T 2 V ; so _  j   _ T3 b V D T i (V) : j2[0:::N)

i2[0:::3N)

  Thus H T 3 ; b V ; N D H (T; V ; 3N), extending notation. Part (d) and sending N ! 1, gives E(T 3 ; b V) D 3H (T; V). Lastly, take covers such that 









E T 3 ; C(k) ! Etop T 3



and

E T; D(k) ! Etop (T) ;

as k!1. Define V(k) :D C(k) _ D(k) . Apply the above to V (k) , then send k!1. 

You Take the High Road and I’ll Take the Low Road There are several routes to computing top-ent, some via maximization, others, minimization. Our foregoing discussion computed Etop (T) by a family of sizes f k (n) D f kT (n), depending on a parameter k which specifies the fineness of scale. (In Sect. “Metric Preliminaries”, this k is an integer; in the original definition, an open cover.) Define two numbers: 1 b L f (k) :D lim sup log f k (n) and n!1 n 1 L f (k) :D lim inf log f k (n) : (22) n!1 n L f (k). If the limit exists Finally, let E f (T) :D sup k b f in Eq. (22) then agree to write L (k) for the common value. The A-K-M definition used the size fV (n) :D Card(V[0:::n) ), where Card(W ) :DMinimum cardinality of a subcover from W :

Using a Metric From now on, our space is a compact metric space (X; d). Dinaburg [36] and Bowen [34,35], gave alternative, equivalent, definitions of topological entropy, in the compact metric-space case, that are often easier to work with than covers. Bowen gave a definition also when X is not compact17 (see [35] and Chap. 7 in [23]). Metric Preliminaries An "-ball-cover comprises finitely many balls, all of radius ". Since our space is compact, every cover V has a Lebesgue number " > 0. I. e., for each z 2 X, the Bal(z; ") lies entirely inside at least one V-patch. (In particular, there is an "-ball-cover which refines V.) Let LEB(V) be the supremum of the Lebesgue numbers. Courtesy of Lemma 13f we can Fix a “universal” list V(1) 4 V(2) 4 : : : , with V(k) a 1k -ball-cover. For every T : X ! X, then, the lim k E(T; V(k) ) computes Etop (T). An "-Microscope Three notions are useful in examining a metric space (X; m) at scale ". Subset A X is an "separated-set, if m(z; z0 )  " for all distinct z; z0 2 A. Subset F X is "-spanning if 8x 2 X; 9z 2 F with m(x; z) < ". Lastly, a cover V is "-small if Diam(U) < ", for each U 2 V. 17 When X is not compact, the definitions need not coincide; e. g. [37]. And topologically-equivalent metrics, but which are not uniformly equivalent, may give the same T different entropies (see p. 171 in [23]).

Here are three metric-space sizes f" (n): Sep(n; ") :DMaximum cardinality of a dn –"-separated set: Spn(n; ") :DMinimum cardinality of a dn –"-spanning set: Cov(n; ") :DMinimum cardinality of a dn –"-small cover: These use a list (d n )1 nD1 of progressively finer metrics on X, where   d N (x; y) :D Max j2[0:::N) d T j (x); T j (y) : Theorem 14 (All-Roads-Lead-to-Rome Theorem) Fix " and let W be any d-"-small cover. Then 8n : Cov(n; 2")  Spn(n; ")  Sep(n; ")  Card(W [0:::n) ) : (ii) Take a cover V and a ı < LEB(V). Then 8n: Card(V[0:::n) )  Cov(n; ı) : (iii) The limit LCov (") D lim n n1 log(Cov(n; ")) exists in [0 : : : 1) : (iv)

(i)

ESep (T) D ESpn (T) D ECov (T) D ECard (T) by defn

D Etop (T) :

Proof (of (i)) Take F X, a min-cardinality dn -"-spanS ning set. So z2F Dz D X, where note

Dz :D dn -Bal(z; ") D

n1 \ jD0

   T  j Bal T j z; " :

Entropy in Ergodic Theory

This D :D fDz gz is a cover, and it is dn –2"–small. Thus Cov(n; 2")  jDj D jFj. For any metric, a maximal "-separated-set is automatically "-spanning; adjoin a putative unspanned point to get a larger separated set. Let A be a max-cardinality dn -"-separated set. Take C, a min-cardinality subcover of W [0:::n) . For each z 2 A, pick a C-patch Cz 3 z. Could some pair x; y 2 A pick T j the same C? Well, write C D n1 jD0 T (Wj ), with each Wj 2 W . For every j 2 [0 : : : n), then, d(T j (x); T j (y))  Diam(Wj ) < " :

Proof (of (ii)) Choose a min-cardinality d n -ı-small cover C. For each C 2 C and j 2 [0 : : : n), the d-Diam(T j C) < ı. So there is a V-patch VC; j  T j (C). Hence note

n1 \

  T  j VC; j  C :

jD0

Thus V[0:::n) 4 C. So   Card V[0:::n)  Card(C)  jCj D Cov(n; ı) :



Proof (of (iii)) To upper-bound Cov(k C l; ") let V and W be min-cardinality "-small covers, respectively, for metrics d k and d l . Then V \ T l (W ) is a "-small for d kCl . Consequently Cov(kCl; ")  Cov(k; ")  Cov(l; "). Thus n 7! log(Cov(n; ")) is subadditive.  Proof (of (iv)) Pick a V from the list in Sect. “Metric Preliminaries”, choose some 2" < LEB(V) followed by an "-small W from Sect. “Metric Preliminaries”. Pushing n ! 1 gives LCard (V)  LCov (2") 

b LSep (") LSpn (")  b LSpn (")  LSep (")

1 log(Sep(n; ")) (24) n limit exists, in arguments that subsequently send "&0. Ditto for LSpn ("). LSep (") D lim

n!1

This will be used during the proof of the Variational Principle. But first, here are two entropy computations which illustrate the efficacy in having several characterizations of topological entropy. Etop (Isometry) D 0

Hence d n (x; y) < "; so x D y. Accordingly, the z 7! Cz map is injective, whence jAj  jCj. 

V [0:::n) 3

zero as "&0. Consequently, we can pretend that the

 LCard (W ) : (23)

Now send V and W along the list in Sect. “Metric Preliminaries”.  Pretension Topological entropy takes its values in [0; 1]. A useful corollary of Eq. (23) can be stated in terms of any Distance(; ) which topologizes [0; 1] as a compact interval. For each continuous T : X  ! X on a compact metric-space, the Distance b LSep ("); LSep (") goes to

Suppose (T : X; d) is a distancepreserving map of a compact metric-space. Fixing ", a set is dn -"-separated IFF it is d-"-separated. Thus Sep(n; ") does not grow with n. So each b LSep (") is zero. Topological Markov Shifts Imagine ourselves back in the days when computer data is stored on large reels of fast-moving magnetic tape. One strategy to maximize the density of binary data stored is to not put timing-marks (which take up space) on the tape. This has the defect that when the tape-writer writes, say, 577 consecutive 1-bits, then the tape-reader may erroneously count 578 copies of 1. We sidestep this flaw by first encoding our data so : :1 word, then writing to tape. as to avoid the 11:577 Generalize this to a finite alphabet Q and a finite list F of disallowed Q-words. Extend each word to a common length K C 1; now F QKC1 . The resulting “K-step TMS (topological Markov shift)” is the shift on the set of doubly-1 Q-names having no substring in F . In the above magnetic-tape example, K D 576. Making it more realistic, suppose that some string of zeros, say 00:574 : :0, is also forbidden18 . Extending to length 577, we get 23 D8 new disallowed words of form 00:574 : :0b1 b2 b3 . We recode to a 1-step TMS (just called a TMS or a subshift of finite type) over the alphabet P :D QK . Each outlawed Q-word w0 w1    w K engenders a length-2 forbidden P-word (w0 ; : : : ; w K1 )(w1 ; : : : ; w K ). The resulting TMS is topologically conjugate to the original K-step. The allowed length-2 words can be viewed as the edges in a directed-graph and the set of points x 2 X is the set of doubly-1 paths through the graph. Once trivialities removed, this X is a Cantor set and the shift T : X ! X is a homeomorphism. The Golden Shift As the simplest example, suppose our magnetic-tape is constrained by the Markov graph, Fig. 5 that we studied measure-theoretically in Fig. 4. 18 Perhaps the ;-bad-length, 574, is shorter than the 1-bad-length because, say, ;s take less tape-space than 1s and so – being written more densely – cause ambiguity sooner.

217

218

Entropy in Ergodic Theory

Entropy in Ergodic Theory, Figure 5 Ignoring the labels on the edges, for the moment, the Golden shift, T, acts on the space of doubly-infinite paths through this graph. The space can be represented as a subset XGold  fa; bgZ , namely, the set of sequences with no two consecutive b letters.

We want to store the text of The Declaration of Independence on our magnetic tape. Imagining that English is a stationary process, we’d like to encode English into this Golden TMS as efficiently as possible. We seek a shift-invariant measure  on XGold of maximum entropy, should such exist. View PDfa; bg as the time-zero partition on X Gold ; that is, name xD: : : x1 x0 x1 x2 : : :, is in atom b IFF letter x0 is “b”. Any measure  gives conditional probabilities (aja) D: s ; note

note

1 H (s; 1  s) 2s 1  slog(s) C (1  s)log(1  s) : D 2s

f (s) :D

(25)

Certainly f (0) D f (1) D 0, so f ’s maximum occurs at 0 s) the (it turns out) unique pointb ps where the derivative f (b equals zero. Thisb s D (1 C 5)/2. Plugging in, the maximum entropy supportable by the Golden Shift is p



 p 1 C 5 p  # 2 3 5 C log p : (26) 2 3 5 1 C 2

5

log

2

Exponentiating, the number of -typical n-names grows like Gn , where 2 G :D 4

2 1 C

p

p 1C p 5 5 5

5

1 ; 1 C jmj

(28)

0 . for the smallest jmj with x m ¤ x m

Lemma 15 Consider a subshift X. Then the 1 log(Names X (n)) n

exists in [0; 1], and equals Etop (X).

But recall, E(T) D H (P1 jP[1:::0) )  H (P1 jP0 ). So among all measures that make the conditional distribution Pja equal (s; c), the unique one maximizing entropy is the (s; c)-Markov-process. Its entropy, derived in Eq. (14), is

"

d(x; x 0 ) :D

n!1

(bjb) D 0 :

2 p MaxEnt D 5 5

Top-ent of the Golden Shift For a moment, let’s work more generally on an arbitrary subshift (a closed, shiftinvariant subset) X QZ , where Q is a finite alphabet. Here, the transformation is always the shift – but the space is varying – so agree to refer to the top-ent as Etop (X). Let ˚ Names X (n) be the number of distinct words in the set x [0:::n) j x 2 X . Note that a metric inducing the product-topology on QZ is

lim

(bja) D: c ;

(ajb) D 1 ;

This expression19 looks unpleasant to simplify – it isn’t even obviously an algebraic number – and yet topological entropy will reveal its familiar nature. This, because the Variational Principle (proved in the next section) says that the top-ent of a system is the supremum of measure-entropies supportable by the system.

32

p 3 3p5 2 5 5 54 5: p 3 5

(27)

Proof With " 2 (0; 1) fixed, two n-names are d n -"-separated IFF they are not the same name. Hence Sep(n; ") D  Names X (n). To compute Etop (X Gold ), declare that a word is “golden” if it appears in some x 2 XGold . Each [n C 1]-golden word ending in a has form wa, where w is n-golden. An [n C 1]golden word ending in b, must end in ab and so has form wab, where w is [n  1]-golden. Summing up, NamesXGold (n C 1) D NamesXGold (n) C Names XGold (n  1) : This is the Fibonacci recurrence, and indeed, these are the Fibonacci numbers, since NamesXGold (0) D 1 and Names XGold (1) D 2. Consequently, we have that NamesXGold (n) Const  n ; p

where  D 1C2 5 is the Golden Ratio. So the sesquipedalian number G from Eq. (27) is simply , and Etop (X Gold ) D log(). Since log()  0:694, each thousand bits written on tape (subject to the “no bb substrings” constraint) can carry at most 694 bits of information. 19 A popular computer-algebra-system was not, at least under my inexpert tutelage, able to simplify this. However, once top-ent gave the correct answer, the software was able to detect the equality.

Entropy in Ergodic Theory

Top-ent of a General TMS A (finite) digraph G engenders a TMS T : XG ! XG , as well as a f0; 1g-valued adjacency matrix ADAG , where a i; j is the number of directed-edges from state i to j. (Here, each a i; j is 0 or 1.) The (i; j)-entry in power An is automatically the number of length-n paths from i to j. Employing the matrix-norm P kMk :D i; j jm i; j j, then, kAkn D NamesX (n) : Happily Gelfand’s formula (see 10.13 in [58] or Spectral_ radius in [60]) applies: For an arbitrary (square) complex matrix, 1

lim kAn k n D SpecRad(A) :

n!1

(29)

This right hand side, the spectral radius of A, means the maximum of the absolute values of A’s eigenvalues. So the top-ent of a TMS is thus the Etop (X G ) D SpecRad(AG )

˚ ˇ :D Max jej ˇ e is an eigenvalue of AG : (30)

The (a; b)-adjacency matrix of Fig. 5 is

 1 1 ; 1 0 whose eigenvalues are  and

1  .

Labeling Edges Interpret (s; c; 1) simply as edge-labels in Fig. 5. The set of doubly-1 paths can also be viewed as a subset YGold fs; c; 1gZ , and it too is a TMS. The shift on YGold is conjugate (topologically isomorphic) to the shift on XGold , so they a fortiori have the same topent, log(). The (s; c; 1)-adjacency matrix is 2 3 1 1 0 40 0 15 : 1 1 0 Its j  j-largest eigenvalue is still , as it must. Now we make a new graph. We modify Fig. 5 by manufacturing a total of two s-edges, seven c-edges, and three edges 11 ; 12 ; 13 . Give these 2 C 7 C 3 edges twelve distinct labels. We could compute the resulting TMS-entropy from the corresponding 12 12 adjacency matrix. Alternatively, look at the (a; b)-adjacency matrix

 2 7 A :D : 3 0 p The roots of its characteristic polynomial arep1 ˙ 22. Hence Etop of this 12-symbol TMS is log(1 C 22).

The Variational Principle Let M :D M (X; d) be the set of Borel probability measures, and M (T) :D M (T : X; d) the set of T-invariant  2 M . Assign EntSup(T) :D sup fE (T) j  2 M (T)g : Theorem 16 (Variational EntSup(T) D Etop (T).

principle

(Goodson))

This says that top-ent is the top entropy – if there is a measure  which realizes the supremum. There doesn’t have to be. Choose a sequence of metric-systems (S k : Yk ; m k ) whose entropies strictly increase Etop (S k ) % L to some limit in (0; 1]. Let (S1 : Y1 ; m1 ) be the identity-map on a 1-point space. Define a new system (T : X; d), where F X :D k2[1:::1] Yk . Have T(x) :D S k (x), for the unique k with Yk 3 x. As for the metric, on Yk let d be a scaled version of m k , so that the d-Diam(Yk ) is less than 1/2 k . Finally, for points in distinct ˇcomponents, ˇ x 2 Yk and z 2 Y` , decree that d(x; z) :D ˇ2k  2` ˇ. Our T is continuous, and is a homeomorphism if each of the S k is. Certainly Etop (T) D L > Etop (S k ), for every k 2 [1 : : : 1]. If L is finite then there is no measure  of maximal entropy; for  must give mass to some Yk ; this pulls the entropy below L, since there are no compensatory components with entropy exceeding L. In contrast, when L D 1 then there is a maximalentropy measure (put mass 1/2 j on some component Yk j , where k j %1 swiftly); indeed, there are continuum-many maximal-entropy measures. But there is no 20 ergodic measure of maximal entropy. For a concrete L D 1 example, let S k be the shift on [1 : : : k]Z . Topology on M Let’s arrange our tools for establishing the Variational Principle. The argument will follow Misiurewicz’s proof, adapted from the presentations in [23] and [11]. Equip M with the weak- topology. 21 An A X is nice if its topological boundary @(A) is -null. And a partition is -nice if each atom is. 20 The ergodic measures are the extreme points of M(T); call them MErg (T). This M(T) is the set of barycenters obtained from Borel probability measures on MErg (T) (see Krein-Milman_ theorem, Choquet_theory in [60]). In this instance, what explains the failure to have an ergodic maximal-entropy measure? Let k be an invariant ergodic measure on Yk . These measures do converge to the one-point (ergodic) probability measure 1 on Y1 . But the map  7! E (T) is not continuousR at 1 . R 21 Measures ˛ !  IFF f d˛L ! f d, for each continuL ous f : X ! R. This metrizable topology makes M compact. Always, M(T) is a non-void compact subset (see  Measure Preserving Systems.)

219

220

Entropy in Ergodic Theory

Proposition 17 If ˛L !  and A X is -nice, then ˛L (A) ! (A). Proof Define operator U (D) :D lim supL ˛L (D). It suffices to show that U (A)  (A). For since Ac is -nice too, then U (Ac )  (Ac ). Thus limL ˛L (A) exists, and equals (A). Because C :D A is closed, the continuous functions f N & 1C pointwise, where f N (x) :D 1  Min(N  d(x; C); 1) : By the Monotone Convergence theorem, then, Z N f N d ! (C) : And (C) D (A), since AR is nice. Fixing N, then, it suffices to establish U (A)  f N d. But f N is continuous, so Z Z f N d D lim sup f N d˛L L!1 Z  lim sup 1A d˛L D U (A) :  L!1

Corollary 18 Suppose ˛L ! , and partition P is -nice. Then H˛L (P) ! H (P). The diameter of partition P is MaxA2P Diam(A). Proposition 19 Take  2 M and " > 0. Then there exists a -nice partition with Diam(P) < ". Proof Centered at an x, the uncountably many balls fBal(x; r) j r 2 (0; ")g have disjoint boundaries. So all but countably many are -nice; pick one and call it B x . Compactness gives a finite nice cover, say, fB1 ; : : : ; B7 g, at different centers. Then the partition P :D (A1 ; : : : ; A7 ) is S  nice, 22 where A k :D B k X k1 jD1 B j . Here is a consequence of Jensen’s inequality. Lemma 20 (Distropy-Averaging Lemma) For ;  2 M , a partition R, and a number t 2 [0; 1], t H (R) C t c H (R)  H tCt c  (R) : Strategy for EntSup(T)  Etop (T). Choose an " > 0. For L D 1; 2; 3; : : : , take a maximal (L; ")–separated–set FL X, then define 1 F D F" :D lim sup log(jFL j) : L!1 L 22 For any two sets B; B 0 X, the union @B [ @B 0 is a superset of the three boundaries @(B [ B0 ); @(B \ B0 ); @(B X B0 ).

Let 'L () be the equi-probable measure on FL ; each point has weight 1/jFL j. The desired invariant measure  will come from the Cesàro averages 1 X T ` 'L ; ˛L :D L `2[0:::L)

which get more and more invariant. Lemma 21 Let  be any weak- accumulation point of the above f˛L g1 1 . (Automatically,  is T-invariant.) Then E (T)  F. Indeed, if Q is any -nice partition with Diam(Q) < ", then E (T; Q)  F. Tactics As usual, Q[0:::N) means Q0 _Q1 _: : :_Q N1 . Our goal is 8N :

?

F

  1 H Q[0:::N) : N

(31)

Fix N and P :D Q[0:::N) , and a ı > 0. It suffices to verify: 8large L  N, ? 1 1 log(jFL j)  ı C H˛L (P) ; L N

(32)

since this and Corollary 17 will prove Eq. (31): Pushing L ! 1 along the sequence that produced  essentially sends LhS(32) to F, courtesy Eq. (24). And RhS(32) goes to ı C N1 H (P), by Corollary 17, since P is -nice. Descending ı & 0, hands us the needed Eq. (31). Remark 22 The idea in the following proof is to mostly fill interval [0::L) with N-blocks, starting with a offset K 2 [0::N). Averaging over the offset will create a Cesàro average over each N-block. Averaging over the N-blocks will allow us to compute distropy with respect to the averaged measure, ˛L . Proof (of Eq. (32)) Since L is frozen, agree to use ' for the ' L probability measure. Our dL -"-separated set F L has at most one point in any given atom of Q[0:::L) , thereupon   log(jFL j) D H' Q[0:::L) : Regardless of the “offset” K 2 [0 : : : N), we can always fit C :D b LN N c many N-blocks into [0 : : : L). Denote by G(K) :D [K : : : K C CN), this union of N-blocks, the good set of indices. Unsurprisingly, B(K) :D [0 : : : L) X G(K) is the bad index-set. Therefore, Bad(K)





‚  …„ _

H' Q[0:::L)  H'

j2B(K)

ƒ 

Good(K)

‚  …„ _

Q j C H'

ƒ  Qj

:

j2G(K)

(33)

Entropy in Ergodic Theory

this last inequality, when n is large. The upshot: E (T)  2 C Etop (T). Applied to a power T ` , this asserts that E (T ` )  2 C Etop (T ` ). Thus

Certainly Bad(K)  3Nlog(jQj). So 1 NL

X

Bad(K) 

K2[0:::N)

3N log(jQj) : L

This is less than ı, since L is large. Applying to Eq. (28) now produces 1 1 X Good(K) : log(jFL j)  ı C L NL

1 NL

P

K2[0:::N)

(34)

2 C Etop (T) ; ` using Lemma 9d and using Lemma 13g. Now coax ` ! 1.  E (T) 

K

Note _

j

T (Q) D

j2G(K)

Three Recent Results

_

:

c2[0:::C)T KCcN (P)

So Good(K) 

X

Having given an survey of older results in measure-theoretic entropy and in topological entropy, let us end this survey with a brief discussion of a few recent results, chosen from many.

H' (T KCc N P) :

c

P

This latter, by definition, equals c H T KCcN(') (P). We conclude that 1 X 1 XX Good(K)  H T KCcN ' (P) NL NL c K K 1 X H T ` ' (P) ;  NL `2[0:::L)

by adjoining a few translates of P; 1  H˛L (P) ; N by the Distropy-averaging lemma 18; P since ˛L is the average L1 ` T ` '. Thus Eq. (34) implies Eq. (32), our goal.  Proof (of EntSup(T)  Etop (T)) Fix a T-invariant . For partition Q D (B1 ; : : : ; B K ), choose a compact set A k B k with (B k X A k ) small. (This can be done, since F is automatically a regular measure [58].) Letting c and P :D (D; A1 ; : : : ; A K ), we can have D :D i Ai made H (PjQ) as small as desired. Courtesy of Lemma 8b, then, we only need consider partitions of the form that P has. Open-cover V D (U1 ; : : : ; U K ) has patches U k :D D[ A k . What atoms of, say, P[0:::3) , can the intersection U9 \ T 1 (U2 ) \ T 2 (U5 ) touch? Only the eight atoms (D or A9 ) \ T 1 (D or A2 ) \ T 2 (D or A5 ) : Thus ] P[0:::n)  2n  ] V[0:::n) . (Here, ] () counts the number of non-void atoms/patches.) So     1 1 H P[0:::n)  1 C log ] V [0:::n) n n  1 C 1 C Etop (T) ;

Ornstein–Weiss: Finitely-Observable Invariant In a landmark paper [10], Ornstein and Weiss show that all “finitely observable” properties of ergodic processes are secretly entropy; indeed, they are continuous functions of entropy. This was generalized by Gutman and Hochman [9]; some of the notation below is from their paper. Here is the setting. Consider an ergodic process, on a non-atomic space, taking on only finitely many values in N; let C be some family of such processes. An observation scheme is a metric space (˝; d) and a sequence of n functions S D (S n )1 1 , where Sn maps N  : : :  N into ˝. 1 On a point xE 2 N , the scheme converges if n 7! S n (x1 ; x2 ; : : : x n )

(35)

converges in ˝. And on a particular process X , say that S converges, if S converges on a.e. xE in X . A function J : C ! ˝ is isomorphism invariant if, whenever the underlying transformations of two processes X ; X 0 2 C are isomorphic, then J(X ) D J(X 0 ). Lastly, say that S “converges to J”, if for each X 2 C, scheme S converges to the value J(X ). The work of David Bailey [38], a student of Ornstein, produced an observation scheme for entropy. The Lempel-Ziv algorithm [43] was another entropy observer, with practical application. Ornstein and Weiss provided entropy schemes in [41] and [42]. Their recent paper “Entropy is the only finitelyobservable invariant” [10], gives a converse, a uniqueness result. Theorem 23 (Ornstein, Weiss) Suppose J is a finitely observable function, defined on all ergodic finite-valued processes. If J is an isomorphism invariant, then J is a continuous function of the entropy.

221

222

Entropy in Ergodic Theory

Gutman–Hochman: Finitely-Observable Extension Extending the Ornstein–Weiss result, Yonatan Gutman and Michael Hochman, in [9] proved that it holds even when the isomorphism invariant, J, is well-defined only on certain subclasses of the set of all ergodic processes. In particular they obtain the following result on three classes of zero-entropy transformations. Theorem 24 (Gutman, Hochman) Suppose J() is a finitely observable invariant on one of the following classes: (i)

The Kronecker systems; the class of systems with pure point spectrum. (ii) The zero-entropy mild mixing processes. (iii) The zero-entropy strong mixing processes. Then J() is constant.

Consider (G; G), a topological group and its Borel field (sigma-algebra). Let G  X be the field on G  X generated by the two coordinate-subfields. A map (36)

is measurable if 1

The paper introduces a new isomorphism invariant, the “f invariant”, and shows that, for Bernoulli actions, the f invariant agrees with entropy, that is, with the distropy of the independent generating partition. Exodos Ever since the pioneering work of Shannon, and of Kolmogorov and Sinai, entropy has been front and center as a major tool in Ergodic Theory. Simply mentioning all the substantial results in entropy theory would dwarf the length of this encyclopedia article many times over. And, as the above three results (cherry-picked out of many) show, Entropy shows no sign of fading away. . . Bibliography

Entropy of Actions of Free Groups

 : GxX ! X

Theorem 25 (Lewis Bowen) Let G be a finite-rank free group. Then two Bernoulli G-actions are isomorphic IFF they have the same entropy.

(X ) G  X :

Use g (x) for (g; x). This map in Eq. (36) is a (measure-preserving) group action if 8g; h 2 G: g ı h D g h , and each g : X ! X is measure preserving. This encyclopedia article has only discussed entropy for Z-actions, i. e., when G D Z. The ergodic theorem, our definition of entropy, and large parts of ergodic theory, involve taking averages (of some quantity of interest) over larger and larger “pieces of time”. In Z, we typically use the intervals I n :D [0 : : : n). When G is Z  Z, we might average over squares I n  I n . The amenable groups are those which possess, in a certain sense, larger and larger averaging sets. Parts of ergodic theory have been carried over to actions of amenable groups, e. g. [49] and [55]. Indeed, much of the Bernoulli theory was extended to certain amenable groups by Ornstein and Weiss, [50]. The stereotypical example of a non-amenable group, is a free group (on more than one generator). But recently, Lewis Bowen [8] succeeded in extending the definition of entropy to actions of finite-rank free groups.

Historical 1. Adler R, Weiss B (1967) Entropy, a complete metric invariant for automorphisms of the torus. Proc Natl Acad Sci USA 57:1573– 1576 2. Clausius R (1864) Abhandlungen ueber die mechanische Wärmetheorie, vol 1. Vieweg, Braunschweig 3. Clausius R (1867) Abhandlungen ueber die mechanische Wärmetheorie, vol 2. Vieweg, Braunschweig 4. Kolmogorov AN (1958) A new metric invariant of transitive automorphisms of Lebesgue spaces. Dokl Akad Nauk SSSR 119(5):861–864 5. McMillan B (1953) The basic theorems of information theory. Ann Math Stat 24:196–219 6. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423,623–656 7. Sinai Y (1959) On the Concept of Entropy of a Dynamical System, Dokl Akad Nauk SSSR 124:768–771

Recent Results 8. Bowen L (2008) A new measure-conjugacy invariant for actions of free groups. http://www.math.hawaii.edu/%7Elpbowen/ notes11.pdf 9. Gutman Y, Hochman M (2006) On processes which cannot be distinguished by finitary observation. http://arxiv.org/pdf/ math/0608310 10. Ornstein DS, Weiss B (2007) Entropy is the only finitely-observable invariant. J Mod Dyn 1:93–105; http://www.math.psu. edu/jmd

Ergodic Theory Books 11. Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge 12. Cornfeld I, Fomin S, Sinai Y (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New York

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13. Friedman NA (1970) Introduction to ergodic theory. Van Nostrand Reinhold, New York 14. Halmos PR (1956) Lectures on ergodic theory. The Mathematical Society of Japan, Tokyo 15. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. (With a supplementary chapter by Katok and Leonardo Mendoza). Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 16. Keller G, Greven A, Warnecke G (eds) (2003) Entropy. Princeton Series in Applied Mathematics. Princeton University Press, Princeton 17. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 18. Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, ser 3, vol 8. Springer, Berlin 19. Parry W (1969) Entropy and generators in ergodic theory. Benjamin, New York 20. Petersen K (1983) Ergodic theory. Cambridge University Press, Cambridge 21. Rudolph DJ (1990) Fundamentals of measurable dynamics. Clarendon Press, Oxford 22. Sinai Y (1994) Topics in ergodic theory. Princeton Mathematical Series, vol 44. Princeton University Press, Princeton 23. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, New York

Differentiable Entropy 24. Ledrappier F, Young L-S (1985) The metric entropy of diffeomorphisms. Ann Math 122:509–574 25. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32:55–114 26. Young L-S (1982) Dimension, entropy and Lyapunov exponents. Ergod Theory Dyn Syst 2(1):109–124

Finite Rank 27. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1):35– 65 28. King JLF (1988) Joining-rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227

Maximal-Entropy Measures 29. Buzzi J, Ruette S (2006) Large entropy implies existence of a maximal entropy measure for interval maps. Discret Contin Dyn Syst 14(4):673–688 30. Denker M (1976) Measures with maximal entropy. In: Conze J-P, Keane MS (eds) Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974. Lecture Notes in Mathematics, vol 532. Springer, Berlin, pp 70–112 31. Misiurewicz M (1973) Diffeomorphism without any measure with maximal entropy. Bull Acad Polon Sci Sér Sci Math Astron Phys 21:903–910

Topological Entropy 32. Adler R, Marcus B (1979) Topological entropy and equivalence of dynamical systems. Mem Amer Math Soc 20(219)

33. Adler RL, Konheim AG, McAndrew MH (1965) Topological entropy. Trans Am Math Soc 114(2):309–319 34. Bowen R (1971) Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc 153:401–414. Errata 181:509–510 (1973) 35. Bowen R (1973) Topological entropy for noncompact sets. Trans Am Math Soc 184:125–136 36. Dinaburg EI (1970) The relation between topological entropy and metric entropy. Sov Math Dokl 11:13–16 37. Hasselblatt B, Nitecki Z, Propp J (2005) Topological entropy for non-uniformly continuous maps. http://www.citebase.org/ abstract?id=oai:arXiv.org:math/0511495

Determinism and Zero-Entropy, and Entropy Observation 38. Bailey D (1976) Sequential schemes for classifying and predicting ergodic processes. Ph D Dissertation, Stanford University 39. Kalikow S, King JLF (1994) A countably-valued sleeping stockbroker process. J Theor Probab 7(4):703–708 40. King JLF (1992) Dilemma of the sleeping stockbroker. Am Math Monthly 99(4):335–338 41. Ornstein DS, Weiss B (1990) How sampling reveals a process. Ann Probab 18(3):905–930 42. Ornstein DS, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inf Theory 39(1):78–83 43. Ziv J, Lempel A (1977) A universal algorithm for sequential data compression. IEEE Trans Inf Theory 23(3):337–343

Bernoulli Transformations, K-Automorphisms, Amenable Groups 44. Berg KR (1975) Independence and additive entropy. Proc Am Math Soc 51(2):366–370; http://www.jstor.org/stable/2040323 45. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44 46. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 5:337–352 47. Ornstein DS (1974) Ergodic theory randomness and dynamical systems, Yale Math Monographs, vol 5. Yale University Press, New Haven 48. Ornstein DS, Shields P (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 49. Ornstein DS, Weiss B (1983) The Shannon–McMillan–Briman theorem for a class of amenable groups. Isr J Math 44(3):53– 60 50. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 51. Shields P (1973) The theory of Bernoulli shifts. University of Chicago Press, Chicago 52. Sinai YG (1962) A weak isomorphism of transformations having an invariant measure. Dokl Akad Nauk SSSR 147:797– 800 53. Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Isr J Math 21:177– 207 54. Thouvenot J-P (1977) On the stability of the weak Pinsker property. Isr J Math 27:150–162

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Abramov Formula 55. Ward T, Zhang Q (1992) The Abramov–Rohlin entropy addition formula for amenable group actions. Monatshefte Math 114:317–329

Miscellaneous 56. Newhouse SE (1989) Continuity properties of entropy. Ann Math 129:215–235 57. http://www.isib.cnr.it/control/entropy/ 58. Rudin W (1973) Functional analysis. McGraw-Hill, New York 59. Tribus M, McIrvine EC (1971) Energy and information. Sci Am 224:178–184 60. Wikipedia, http://en.wikipedia.org/wiki/. pages: http://en.

wikipedia.org/wiki/Spectral_radius, wiki/Information_entropy

http://en.wikipedia.org/

Books and Reviews Boyle M, Downarowicz T (2004) The entropy theory of symbolic extensions. Invent Math 156(1):119–161 Downarowicz T, Serafin J (2003) Possible entropy functions. Isr J Math 135:221–250 Hassner M (1980) A non-probabilistic source and channel coding theory. Ph D Dissertation, UCLA Katok A, Sinai YG, Stepin AM (1977) Theory of dynamical systems and general transformation groups with an invariant measure. J Sov Math 7(6):974–1065

Ergodicity and Mixing Properties

Ergodicity and Mixing Properties ANTHONY QUAS Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Article Outline Glossary Definition of the Subject Introduction Basics and Examples Ergodicity Ergodic Decomposition Mixing Hyperbolicity and Decay of Correlations Future Directions Bibliography Glossary Bernoulli shift Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments. Markov chain A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations. Measure-preserving transformation A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map. Measure-theoretic entropy A non-negative (possibly infinite) real number describing the complexity of a measure-preserving transformation. Product transformation Given a pair of measure-preserving transformations: T of X and S of Y, the product transformation is the map of X  Y given by (T  S)(x; y) D (T(x); S(y)). Definition of the Subject Many physical phenomena in equilibrium can be modeled as measure-preserving transformations. Ergodic theory is the abstract study of these transformations, dealing in particular with their long term average behavior. One of the basic steps in analyzing a measure-preserving transformation is to break it down into its simplest possible components. These simplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time

average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner in which a system can be split into ergodic components. A related (stronger) property of a measure-preserving transformation is mixing. Here one is investigating the correlation between the state of the system at different times. The system is mixing if the states are asymptotically independent: as the times between the measurements increase to infinity, the observed values of the measurements at those times become independent. Introduction The term ergodic was introduced by Boltzmann [8,9] in his work on statistical mechanics, where he was studying Hamiltonian systems with large numbers of particles. The system is described at any time by a point of phase space, a subset of R6N where N is the number of particles. The configuration describes the 3-dimensional position and velocity of each of the N particles. It has long been known that the Hamiltonian (i. e. the overall energy of the system) is invariant over time in these systems. Thus, given a starting configuration, all future configurations as the system evolves lie on the same energy surface as the initial one. Boltzmann’s ergodic hypothesis was that the trajectory of the configuration in phase space would fill out the entire energy surface. The term ergodic is thus an amalgamation of the Greek words for work and path. This hypothesis then allowed Boltzmann to conclude that the long-term average of a quantity as the system evolves would be equal to its average value over the phase space. Subsequently, it was realized that this hypothesis is rarely satisfied. The ergodic hypothesis was replaced in 1911 by the quasi-ergodic hypothesis of the Ehrenfests [17] which stated instead that each trajectory is dense in the energy surface, rather than filling out the entire energy surface. The modern notion of ergodicity (to be defined below) is due to Birkhoff and Smith [7]. Koopman [44] suggested studying a measure-preserving transformation by means of the associated isometry on Hilbert space, U T : L2 (X) ! L2 (X) defined by U T ( f ) D f ı T. This point of view was used by von Neumann [91] in his proof of the mean ergodic theorem. This was followed closely by Birkhoff [6] proving the pointwise ergodic theorem. An ergodic measure-preserving transformation enjoys the property that Boltzmann first intended to deduce from his hypothesis: that long-term averages of an observable quantity coincide with the integral of that quantity over the phase space.

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These theorems allow one to deduce a form of independence on the average: given two sets of configurations A and B, one can consider the volume of the phase space consisting of points that are in A at time 0 and in B at time t. In an ergodic measure-preserving transformation, if one computes the average of the volumes of these regions over time, the ergodic theorems mentioned above allow one to deduce that the limit is simply the product of the volume of A and the volume of B. This is the weakest mixing-type property. In this article, we will outline a rather full range of mixing properties with ergodicity at the weakest end and the Bernoulli property at the strongest end. We will set out in some detail the various mixing properties, basing our study on a number of concrete examples sitting at various points of this hierarchy. Many of the mixing properties may be characterized in terms of the Koopman operators operators mentioned above (i. e. they are spectral properties), but we will see that the strongest mixing properties are not spectral in nature. We shall also see that there are connections between the range of mixing properties that we discuss and measure-theoretic entropy. In measure-preserving transformations that arise in practice, there is a correlation between strong mixing properties and positive entropy, although many of these properties are logically independent. One important issue for which many questions remain open is that of higher-order mixing. Here, the issue is if instead of asking that the observations at two times separated by a large time T be approximately independent, one asks whether if one makes observations at more times, each pair suitably separated, the results can be expected to be approximately independent. This issue has an analogue in probability theory, where it is well-known that it is possible to have a collection of random variables that are pairwise independent, but not mutually independent. Basics and Examples In this article, except where otherwise stated, the measurepreserving transformations that we consider will be defined on probability spaces. More specifically, given a measurable space (X; B) and a probability measure  defined on B, a measure-preserving transformation of (X; B; ) is a B-measurable map T : X ! X such that (T 1 B) D (B) for all B 2 B. While this definition makes sense for arbitrary measures, not simply probability measures, most of the results and definitions below only make sense in the probability measure case. Sometimes it will be helpful to make the assumption that the underlying probability space is

a Lebesgue space (that is, the space together with its completed -algebra agrees up to a measure-preserving bijection with the unit interval with Lebesgue measure and the usual  -algebra of Lebesgue measurable sets). Although this sounds like a strong restriction, in practice it is barely a restriction at all, as almost all of the spaces that appear in the theory (and all of those that appear in this article) turn out to be Lebesgue spaces. For a detailed treatment of the theory of Lebesgue spaces, the reader is referred to Rudolph’s book [76]. The reader is referred also to the chapter on  Measure Preserving Systems. While many of the definitions that we shall present are valid for both invertible and non-invertible measure-preserving transformations, the strongest mixing conditions are most useful in the case of invertible transformations. It will be helpful to present a selection of simple examples, relative to which we will be able to explore ergodicity and the various notions of mixing. These examples and the lemmas necessary to show that they are measure-preserving transformations as claimed may be found in the books of Petersen [64], Rudolph [76] and Walters [92]. More details on these examples can also be found in the chapter on  Ergodic Theory: Basic Examples and Constructions. Example 1 (Rotation on the circle) Let ˛ 2 R. Let R˛ : [0; 1) ! [0; 1) be defined by R˛ (x) D x C ˛ mod 1. It is straightforward to verify that R˛ preserves the restriction of Lebesgue measure  to [0; 1) (it is sufficient to check that (R˛1 (J)) D (J) for an interval J). Example 2 (Doubling Map) Let M2 : [0; 1) ! [0; 1) be defined by M2 (x) D 2x mod 1. Again, Lebesgue measure is invariant under M 2 (to see this, one observes that for an interval J, M21 (J) consists of two intervals, each of half the length of J). This may be generalized in the obvious way to a map M k for any integer k  2. Example 3 (Interval Exchange Transformation) The class of interval exchange transformations was introduced by Sinai [85]. An interval exchange transformation is the map obtained by cutting the interval into a finite number of pieces and permuting them in such a way that the resulting map is invertible, and restricted to each interval is an order-preserving isometry. More formally, one takes a sequence of positive lengths `1 ; `2 ; : : : ; ` k summing to 1 and a permutation  P of f1; : : : ; kg and defines a i D j
Ergodicity and Mixing Properties

Example 4 (Bernoulli Shift) Let A be a finite set and fix a vector (p i ) i2A of positive numbers that sum to 1. Let AN denote the set of sequences of the form x0 x1 x2 : : : , where x n 2 A for each n 2 N and let AZ denote the set of biinfinite sequences of the form : : : x2 x1  x0 x1 x2 : : : (the  is a placeholder that allows us to distinguish (for example) between the sequences : : : 01010  10101 : : : and : : : 10101  01010 : : : ). We define a map (the shift map) S on AN by (S(x))n D x nC1 and define S on AZ by the same formula. Note that S is invertible as a transformation on AZ but non-invertible as a transformation on AN . We need to equip AN and AZ with measures. This is done by defining the measure of a preferred class of sets, checking certain consistency conditions and appealing to the Kolmogorov extension theorem. Here the preferred sets are the cylinder sets. Given m  n in the invertible case and a sequence a m : : : a n , we let [a m : : : a n ]nm denote fx 2 AZ : x m D a m ; : : : ; x n D a n g and define ([a m : : : a n ]nm ) D p a m p a mC1 : : : p a n . This is then shown to uniquely define a measure  on the -algebra of AZ generated by the cylinder sets. It is immediate to see that for any cylinder set C, (S 1 C) D (C), and it follows that S is a measure-preserving transformation of (AZ ; B; ). The construction is exactly analogous in the non-invertible case. See the chapter on  Measure Preserving Systems or the books of Walters [92] or Rudolph [76] for more details of defining measures in these systems.

of identical balls which move at constant velocity until two balls collide, whereupon they elastically swap momentum along the direction of contact. The phase space for this system is a region of R6N (with N 3-dimensional position vectors and N 3-dimensional velocity vectors). More abstractly, the system is equivalent to the motion of a single point particle in a region of R M  R M (with the first M-vector representing position and the second representing velocity). The system is constrained in that its position is required to lie in a bounded region S of R M with a piecewise smooth boundary. The system evolves by moving the position at a constant rate in the direction of the velocity vector until the point reaches @S, at which time the component of the velocity parallel to the normal to @S is reversed. This then defines a flow (i. e. a family of maps (Tt ) t2R satisfying TtCs D Tt ı Ts ) on the phase space. Since the magnitude of the velocity is conserved, it is convenient to restrict to flows with speed 1. This system is clearly the closest of the examples that we consider to the situation envisaged by Boltzmann. Perhaps not surprisingly, proofs of even the most basic properties for this system are much harder than those for the other examples that we consider.

The class of Bernoulli shifts will play a distinguished role in what follows.

1. g ı h and h ı g agree with the respective identity maps almost everywhere; 2. (g 1 F) D (F) and (h1 B) D (B) for all F 2 F and B 2 B; and 3. Sıg(x) D gıT(x) for -almost every x (or equivalently T ı h(y) D h ı S(y) for -almost every y).

Example 5 (Markov Shift) The spaces AN and AZ are exactly as above, as is the shift map. All that changes is the measure. To define a Markov shift, we need a stochastic matrix P (i. e. a matrix with non-negative entries whose rows sum to 1) with rows and columns indexed by A and a left eigenvector  for P with eigenvalue 1 with the property that the entries of  are non-negative and sum to 1. The existence of such an eigenvector is a consequence of the Perron–Frobenius theory of positive matrices. Provided that the matrix P is irreducible (for each a and a0 in A, there is n an n > 0 such that Pa;a 0 > 0), the eigenvector  is unique. Given the pair (P; ), one defines the measure of a cylinder set by ([a m : : : a n ]nm ) D  a m Pa m a mC1 : : : Pa n1 a n and extends  as before to a probability measure on AN or AZ . Example 6 (Hard Sphere Gases and Billiards) We wish to model the behavior of a gas in a bounded region. We make the assumption that the gas consists of a large number N

We will need to make use of the concept of measure-theoretic isomorphism. Two measure-preserving transformations T of (X; B; ) and S of (Y; F ; ) are measure-theoretically isomorphic (or just isomorphic) if there exist measurable maps g : X ! Y and h : Y ! X such that

Measure-theoretic isomorphism is the basic notion of ‘sameness’ in ergodic theory. It is in some sense quite weak, so that systems may be isomorphic that feel very different (for example, as we discuss later, the time one map of a geodesic flow is isomorphic to a Bernoulli shift). For comparison, the notion of sameness in topological dynamical systems (topological conjugacy) is far stronger. As an example of measure-theoretic isomorphism, it may be seen that the doubling map is isomorphic to the one-sided Bernoulli shift on f0; 1g with p0 D p1 D 1/2 (the map g takes an x 2 [0; 1) to the sequence of 0’s and 1’s in its binary expansion (choosing the sequence ending with 0’s, for example, if x is of the form p/2n ) and the inverse map h takes a sequence of 0’s and 1’s to the point in [0; 1) with that binary expansion.)

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Given a measure-preserving transformation T of a probability space (X; B; ), T is associated to an isometry of L2 (X; B; ) by U T ( f ) D f ı T. This operator is known as the Koopman Operator. In the case where T is invertible, the operator U T is unitary. Two measure-preserving transformations T and S of (X; B; ) and (Y; F ; ) are spectrally isomorphic if there is a Hilbert space isomorphism ‚ from L2 (X; B; ) to L2 (Y; F ; ) such that ‚ ı U T D U S ı ‚. As we shall see below, spectral isomorphism is a strictly weaker property than measure-theoretic isomorphism. Since in ergodic theory, measure-theoretic isomorphism is the basic notion of sameness, all properties that are used to describe measure-preserving systems are required to be invariant under measure-theoretic isomorphism (i. e. if two measure-preserving transformations are measure-theoretically isomorphic, the first has a given property if and only if the second does). On the other hand, we shall see that some mixing-type properties are invariant under spectral isomorphism, while others are not. If a property is invariant under spectral isomorphism, we say that it is a spectral property. There are a number of mixing type properties that occur in the probability literature (˛-mixing, ˇ-mixing, -mixing, -mixing etc.) (see Bradley’s survey [12] for a description of these conditions). Many of these are stronger than the Bernoulli property, and are therefore not preserved under measure-theoretic isomorphism. For this reason, these properties are not widely used in ergodic theory, although ˇ-mixing turns out to be equivalent to the so-called weak Bernoulli property (which turns out to be stronger than the Bernoulli property that we discuss in this article – see Smorodinsky’s paper [87]) and ˛-mixing is equivalent to strong-mixing. A basic construction (see the article on  Ergodic Theory: Basic Examples and Constructions) that we shall require in what follows is the product of a pair of measurepreserving transformations: given transformations T of (X; B; ) and S of (Y; F ; ), we define the product transformation T  S : (X  Y; B ˝ F ; ) by (T  S)(x; y) D (Tx; Sy). One issue that we face on occasion is that it is sometimes convenient to deal with invertible measure-preserving transformations. It turns out that given a non-invertible measure-preserving transformation, there is a natural way to uniquely associate an invertible measurepreserving transformation transformation sharing almost all of the ergodic properties of the original transformation. Specifically, given a non-invertible measure-preserving transformation T of (X; B; ), one lets X D f(x0 ; x1 ; : : : ) : x n 2 X and T(x n ) D x n1 for all ng, B

be the  -algebra generated by sets of the form A¯ n D ¯ A¯ n ) D (A) and T(x0 ; x1 ; : : : ) D fx¯ 2 X : x n 2 Ag, ( (T(x0 ); x0 ; x1 ; : : : ). The transformation T of (X; B; ) is called the natural extension of the transformation T of (X; B; ) (see the chapter on  Ergodic Theory: Basic Examples and Constructions for more details). In situations where one wants to use invertibility, it is often possible to pass to the natural extension, work there and then derive conclusions about the original non-invertible transformation. Ergodicity Given a measure-preserving transformation T : X ! X, if T 1 A D A, then T 1 Ac D Ac also. This allows us to decompose the transformation X into two pieces A and Ac and study the transformation T separately on each. In fact the same situation holds if T 1 A and A agree up to a set of measure 0. For this reason, we call a set A invariant if (T 1 AA) D 0. Returning to Boltzmann’s ergodic hypothesis, existence of an invariant set of measure between 0 and 1 would be a bad situation as his essential idea was that the orbit of a single point would ‘see’ all of X, whereas if X were decomposed in this way, the most that a point in A could see would be all of A, and similarly the most that a point in Ac could see would be all of Ac . A measure-preserving transformation will be called ergodic if it has no non-trivial decomposition of this form. More formally, let T be a measure-preserving transformation of a probability space (X; B; ). The transformation T is said to be ergodic if for all invariant sets, either the set or its complement has measure 0. Unlike the remaining concepts that we discuss in this article, this definition of ergodicity applies also to infinite measure-preserving transformations and even to certain non-measure-preserving transformations. See Aaronson’s book [1] for more information. The following lemma is often useful: Lemma 1 Let (X; B; ) be a probability space and let T : X ! X be a measure-preserving transformation. Then T is ergodic if and only if the only measurable functions f satisfying f ı T D f (up to sets of measure 0) are constant almost everywhere. For the straightforward proof, we notice that if the condition in the lemma holds and A is an invariant set, then 1A ı T D 1A almost everywhere, so that 1A is an a. e. constant function and so A or Ac is of measure 0. Conversely, if f is an invariant function, we see that for each ˛, fx : f (x) < ˛g is an invariant set and hence of measure 0

Ergodicity and Mixing Properties

or 1. It follows that f is constant almost everywhere. We remark for future use that it is sufficient to check that the bounded measurable invariant functions are constant. The following corollary of the lemma shows that ergodicity is a spectral property. Corollary 2 Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if 1 is a simple eigenvalue of U T . The ergodic theorems mentioned earlier due to von Neumann and Birkhoff are the following (see also the chapter on  Ergodic Theorems). Theorem 3 (von Neumann Mean Ergodic Theorem [91]) Let T be a measure-preserving transformation of the probability space (X; B; ). For f 2 L2 (X; B; ), let A N f D 1/N( f C f ı T C    C f ı T N1 ). Then for all f 2 L2 (X; B; ), A N f converges in L2 to an invariant function f . Theorem 4 (Birkhoff Pointwise Ergodic Theorem [6]) Let T be a measure-preserving transformation of the probability space (X; B; ). Let f 2 L1 (X; B; ). Let A N f be as above. Then for -almost every x 2 X, (A N f (x)) is a convergent sequence. Of these two theorems, the pointwise ergodic theorem is the deeper result, and it is straightforward to deduce the mean ergodic theorem from the pointwise ergodic theorem. The mean ergodic theorem was reproved very concisely by Riesz [71] and it is this proof that is widely known now. Riesz’s proof is reproduced in Parry’s book [63]. There have been many different proofs given of the pointwise ergodic theorem. Notable amongst these are the argument due to Garsia [23] and a proof due to Katznelson and Weiss [40] based on work of Kamae [35], which appears in a simplified form in work of Keane and Petersen [42]. If the measure-preserving transformation T is ergodic, then by virtue of Lemma 1, the limit functions appearing in the ergodic theorems are constant. One sees that the constant is simply the integral of f with respect to , so R that in this situation A N f (x) converges to f d in norm and pointwise almost everywhere, thereby providing a justification of Boltzmann’s original claim: for ergodic measure-preserving transformations, time averages agree with spatial averages. In the case where T is not ergodic, it is also possible to identify the limit in the ergodic theorems: we have f  D E( f jI ), where I is the -algebra of T-invariant sets. Note that the set on which the almost everywhere convergence in Birkhoff’s theorem takes place depends on the L1 function f that one is considering. Straightforward considerations show that there is no single full measure set

that works simultaneously for all L1 functions. In the case where X is a compact metric space, it is well known that C(X), the space of continuous functions on X with the uniform norm has a countable dense set, ( f n )n1 say. If the invariant measure  is ergodic, then for each n, there is aR set Bn of measure 1 such that for all x 2 B n , A N f n (x) ! T f n d. Letting B D n B n , one obtains a full Rmeasure set such that for all n and all x 2 B, A N f n (x) ! f n d. A simple approximation argument thenRshows that for all x 2 B and all f 2 C(X), A N f (x) ! f d. A point x with this property is said to be generic for . The observations above show that for an ergodic invariant measure , we have fx : x is generic for g D 1. If T is ergodic, but T n is not ergodic for some n, then one can show that the space X splits up as A1 ; : : : ; A d for some djn in such a way that T(A i ) D A iC1 for i < d and T(A d ) D A1 with T n acting ergodically on each Ai . The transformation T is totally ergodic if T n is ergodic for all n 2 N. One can check that a non-invertible transformation T is ergodic if and only if its natural extension is ergodic. The following lemma gives an alternative characterization of ergodicity, which in particular relates it to mixing. Lemma 5 (Ergodicity as a Mixing Property) Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if for all f and g in L2 , N 1 X h f ; g ı T n i ! h f ; 1ih1; gi: N nD0

P N1 In particular, if T is ergodic, then (1/N) nD0 (A \ T n B) ! (A)(B) for all measurable sets A and B. Proof Suppose that T is ergodic. Then the left-hand side P N1 of the equality is equal to h f ; (1/N) nD0 g ı T n i. The mean ergodic theorem shows that the second term conR verges in L2 to the constant function with value g d D hg; 1i, and the equality follows. Conversely, if the equation holds for all f and g in L2 , suppose that A is an invariant set. Let f D g D 1A . Then since g ı T n D 1A for all n, the left-hand side is h1A ; 1A i D (A). On the other hand, the right-hand side is (A)2 , so that the equation yields (A) D (A)2 , and (A) is either 0 or 1 as required. Taking f D 1A and g D 1B for measurable sets A and B gives the final statement.  We now examine the ergodicity of the examples presented above. Firstly, for the rotation of the circle, we claim that

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the transformation is ergodic if and only if the ‘angle’ ˛ is irrational. To see this, we argue as follows. If ˛ D p/q, then we see that f (x) D e 2 i qx is a non-constant R˛ invariant function, and hence R˛ is not ergodic. On the other hand, if ˛ is irrational, suppose f is a bounded measurable invariant function. Since f is bounded, it is an L2 function, and so f may be expressed in L2 as a Fourier seP 2 i nx . We then ries: f D n e n where e n (x) D e n2Z cP see that f ı R˛ D n2Z e 2 i n˛ c n e n . In order for f to be equal in L2 to f ı R˛ , they must have the same Fourier coefficients, so that c n D e 2 i n˛ c n for each n. Since ˛ is irrational, this forces c n D 0 for all n ¤ 0, so that f is constant as required. The doubling map and the Bernoulli shift are both ergodic, although we defer proof of this for the time being, since they in fact have the strong-mixing property. A Markov chain with matrix P and vector  is ergodic if and only if for all i and j in A with  i > 0 and  j > 0, there exists an n  0 with Pinj > 0. This follows from the ergodic theorem for Markov chains (which is derived from the Strong Law of Large Numbers) (see [18] for details). In particular, if the underlying Markov chain is irreducible, then the measure is ergodic. In the case of interval exchange transformations, there is a simple necessary condition on the permutation for irreducibility, namely for 1  j < k, we do not have f1; : : : ; jg D f1; : : : ; jg. Under this condition, Masur [49] and Veech [88] independently showed that for almost all values of the sequence of lengths (` i )1ik , the interval exchange transformation is ergodic. (In fact they showed the stronger condition of unique ergodicity: that the transformation has no other invariant measure than Lebesgue measure. This implies that Lebesgue measure is ergodic, because if there were a non-trivial invariant set, then the restriction of Lebesgue measure to that set would be another invariant measure). For the hard sphere systems, there are no results on ergodicity in full generality. Important special cases have been studied by Sinai [84], Sinai and Chernov [86], Krámli, Simányi and Szász [45], Simányi and Szász [81], Simnyi [79,80] and Young [95]. Ergodic Decomposition We already observed that if a transformation is not ergodic, then it may be decomposed into parts. Clearly if these parts are not ergodic, they may be further decomposed. It is natural to ask whether the transformation can be decomposed into ergodic parts, and if so what form does the decomposition take? In fact such a decomposition does exist, but rather than decompose the transforma-

tion, it is necessary to decompose the measure into ergodic pieces. This is known as ergodic decomposition. The set of invariant measures for a measurable map T of a measurable space (X; B) to itself forms a simplex. General functional analytic considerations (due to Choquet [14,15] – see also Phelps’ account [66] of this theory) mean that it is possible to write any member of the simplex as an integral-convex combination of the extreme points. Further, the extreme points of the simplex may be identified as precisely the ergodic invariant measures for T. It follows that any invariant probability measure  for T may be uniquely expressed in the form Z (A) D (A) dm() ; M erg (X;T)

where Merg (X; T) denotes the set of ergodic T-invariant measures on X and m is a measure on Merg (X; T). We will give a proof of this theorem in the special case of a continuous transformation of a compact space. Our proof is based on the Birkhoff ergodic theorem and the Riesz Representation Theorem identifying the dual space of the space of continuous functions on a compact space as the set of bounded signed measures on the space (see Rudin’s book [75] for details). We include it here because this special case covers many cases that arise in practice, and because few of the standard ergodic theory references include a proof of ergodic decomposition. An exception to this is Rudolph’s book [76] which gives a full proof in the case that X is a Lebesgue space. This is based on a detailed development of the theory of these spaces and builds measures using conditional exceptions. Kalikow’s notes [32] give a brief outline of a proof similar to that which follows. Oxtoby [62] also wrote a survey article containing much of the following (and much more besides). Theorem 6 Let X be a compact metric space, B be the Borel  -algebra,  be an invariant Borel probability measure and T be a continuous measure-preserving transformation of (X; B; ). Then for each x 2 X, there exists an invariant Borel measure x such that: R R R  1. For f 2 L1 (X; B; ), f d D f dx d(x); 2. Given f 2 L1 (X; R B; ), for -almost every x 2 X, one has A N f (x) ! f dx ; 3. The measure x is ergodic for -almost every x 2 X. Notice that conclusion (2) shows that x can be understood as the distribution on the phase space “seen” if one starts the system in an initial condition of x. This interpretation of the measures x corresponds closely with the ideas of Boltzmann and the Ehrenfests in the formulation

Ergodicity and Mixing Properties

of the ergodic and quasi-ergodic hypotheses, which can be seen as demanding that x is equal to  for (almost) all x. Proof The proof will be divided into 3 main steps: defining the measures x , proving measurability with respect to x and proving ergodicity of the measures. Step 1: Definition of x Given a function f 2 L1 (X; B; ), Birkhoff’s theorem states that for -almost every x 2 X, (A N f (x)) is convergent. It will be convenient to denote the limit by f˜(x). Let f1 ; f2 ; : : : be a sequence of continuous functions that is dense in C(X). For each k, there is a set Bk of x’s measure 1 for which (A n f k (x))1 nD1 is a convergent sequence. Intersecting these gives a set B of full measure such that for x 2 B, for each k  1, A n f k (x) is convergent. A simple approximation argument shows that for x 2 B and f an arbitrary continuous function, A n f (x) is convergent. Given x 2 B, define a map L x : C(X) ! R by L x ( f ) D f˜(x). This is a continuous linear functional on C(X), and hence by the Riesz Representation Theorem there exists a Borel R measure x such that f˜(x) D f dx for each f 2 C(X) and x 2 B. Since L x ( f )  0 when f is a non-negative function and L x (1) D 1, the measure x is a probability measure. Since L x ( f ı T) D L x ( f ) for f 2 C(X), one can check that x must be an invariant probability measure. For x 62 B, simply define x D . Since Bc is a set of measure 0, this will not affect any of the statements that we are trying to prove. Now for f continuous, we have A N f is a bounded seR to f d quence of functions withR A N f (x) converging x R almost everywhere and A N f d D f d since T is measure-preserving. It follows from the bounded convergence theorem that for f 2 C(X),  Z Z Z f d D f dx d(x) : (1) Step 2: Measurability of x 7! x (A) Lemma 7 Let C 2 B satisfy (C) D 0. Then x (C) D 0 for -almost every x 2 X. Proof Using regularity of Borel probability measures (see Rudin’s book [75] for details), there exist open sets U1  U2      C with (U k ) < 1/k. There exist continuous functions g k;m with (g k;m (x))1 mD1 increasing to 1U k everywhere R(e.Rg. g k;m (x) D min(1; m  d(x; U kc ))). By (1), we Rhave ( g k;m dx ) d(x) < 1/k for all k; m. Note that g k;m dx D lim n!1 A n g k;m (x) is a measurable function of x, so that using the monotone converR gence theorem (taking the limit in m), x ! 7 1 U k d x D R R x (U k ) is measurable and ( 1U k dx ) d(x)  1/k. T We now see that x 7! lim k!1 x (U k ) D x ( U k )

is measurable, and by monotone convergence we see R alsoT T x ( U k ) d(x) D 0. It follows that x ( U k ) D 0 T for -almost every x. Since U k  C, the lemma follows.  Given a set A 2 B, let f k be a sequence of continuous functions (uniformly bounded by 1) satisfying k f k  1A kL 1 () < 2n , so that in particular R f k (x) ! 1A (x) for -almost every x. For each k, x 7! f k dx D lim n!1 A n f k (x) is a measurable function. By Lemma 7, for -almost every x, f k ! 1A x -almost everywhere, so R that by the bounded convergence theorem limk!1 f k dx D x (A) for -almost every x. Since the limit of measurable functions is measurable, it follows that x 7! x (A) is measurable for any measurable set A 2 B. R This allows us to define a measure  by (A) D x (A) R d(x). RForR a bounded measurable function f , we have f d D ( f dx ) d(x). Since this agrees with R f d for continuous functions by (1), it follows that  D . Conclusion (1) of the theorem now follows easily. Given f 2 L1 (X), we let ( f k ) be a sequence of continuous functions such that k f k  f kL 1 () is summable. This implies that k f k  f kL 1 ( R for -almost evR x ) is summable ery x and in particular, f k dx ! f dx for almost every x. On the other hand, by the remark following the statement of Birkhoff ’s theorem, we have f˜k D E( f k jI ) so that k f˜  f˜k kL 1 () is summable and f˜k (x) ! f˜(x) for -almost every x. Combining these two statements, we see that for -almost every x, we have f˜(x) D lim f˜k (x) D lim k!1

k!1

Z

Z f k dx D

f dx :

This establishes conclusion (2) of the theorem. Step 3: Ergodicity of x We have shown how to disintegrate the invariant measure  as an integral combination of x ’s, and we have interpreted the x ’s as describing the average behavior starting from x. It remains to show that the x ’s are ergodic measures. Fix for now a continuous function f and a number 0 < < 1. Since A n f (x) ! f˜(x) -almost everywhere, there exists an N such that fx : jA N f (x) f˜(x)j > /2g < 3 /8. We now claim the following: R ˚  x : x fy : j f˜(y)  f dx j > g > < :

(2)

R To see this, note that fy : j f˜(y)  f dx j > g

fy : j f˜(y)A N f (y)j > /2g[fy : jA N f (y) f˜(x)j > /2g, R ˜ so that if x fy : j f (y)  f dx j > g > , then either

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x fy : j f˜(y) A N f (y)j > /2g > /2 or x fy : jA N f (y) f˜(x)j > /2g > /2. We show that the set of x’s satisfying each condition is small. 3 ˜ Firstly, R we have /8 > fy : j f (y)  A N f (y)j > /2g D x fy : j f˜(y)  A N f (y)j > /2g d(x), so that fx : x fy : j f˜(y)  A N f (y)j > /2g > /2g < 2 /4 < /2. For the second term, given c 2 R, let Fc (x) D jA N f (x)  cj and G(x) D F f˜(x) (x). Note that Z F f˜(x) (y) dx (y) D lim A n F f˜(x) (x) D lim A n G(x) n!1

n!1

(using the facts that y 7! F f˜(x) (y) is a continuous function and that since f˜(x) is R an invariant function, F f˜(x) (T k x) D G(T k x)). Since G(x) d(x) < 3 /8, it R follows that F f˜(x) (y) dx (y)  2 /4 except on a set of x’s of measure less than /2. Outside this bad set, we have x fy : jA N f (y)  f˜(x)j > /2g < /2 so that fx : x fy : jA N f (y)  f˜(x)j > /2g > /2g < /2 as required. This establishes our claim (2) above. Since > 0 is arbitrary, it follows that for each f 2 C(X), for R -almost every x, x -almost every y satisfies f˜(y) D f dx . As usual, taking a countable dense sequence ( f k ) in C(X), it ˜ is R the case that for all k and -almost every x, f k (y) D f k dx x -almost everywhere. Let the set of x’s with this property be D. We claim that for x 2 D, x is ergodic. Suppose not. Then let x 2 D and let J be an invariant set of x measure between ı and 1  ı for some ı > 0. Then by density of C(X) in L1 (x ), there exists an f k with k f k  1 J kL 1 (x ) < ı. Since 1 J is an invariant function, we have 1˜ J D 1 J . On the other hand, f˜k is a constant function. It follows that k f˜k  1˜ J kL 1 (x )  ı > k f k  1 J kL 1 (x ) . This contradicts the identification of the limit as a conditional expectation and concludes the proof of the theorem.  Mixing As mentioned above, ergodicity may be seen as an independence on average property. More specifically, one wants to know whether in some sense (A \ T n B) converges to (A)(B) as n ! 1. Ergodicity is the property that there is convergence in the Césaro sense. Weak-mixing is the property that there is convergence in the strong Césaro sense. That is, a measure-preserving transformation T is weak-mixing if N1 1 X j(A\ T n B)(A)(B)j ! 0 N nD0

as

N ! 1:

In order for T to be strong-mixing, we require simply (A \ T N B) ! (A)(B) as N ! 1. It is clear that strong-mixing implies weak-mixing and weak-mixing implies ergodicity. If T d is not ergodic (so that T d A D A for some A of measure strictly between 0 and 1), then j(T nd A \ A)  (A)2 j D (A)(1  (A)), so that T is not weak-mixing. An alternative characterization of weak-mixing is as follows: Lemma 8 The measure-preserving transformation T is weak-mixing if and only if for every pair of measurable sets A and B, there exists a subset J of N of density 1 (i. e. #(J \ f1; : : : ; Ng)/N ! 1) such that lim

n!1 n62 J

(A \ T n B) D (A)(B) :

(3)

By taking a countable family of measurable sets that are dense (with respect to the metric d(A; B) D (AB)) and taking a suitable intersection of the corresponding J sets, one shows that for a given weak-mixing measure-preserving transformation, there is a single set J N such that (3) holds for all measurable sets A and B (see Petersen [64] or Walters [92] for a proof). We show that an irrational rotation of the circle is not weak-mixing as follows: let ˛ 2 R n Q and let A be the interval [ 14 ; 34 ). There is a positive proportion of n’s in the natural numbers (in fact proportion 1/3) with the property that jT n ( 12 )  12 j < 16 . For these n’s (A \ T n A) > 13 , 1 . so that in particular j(A \ T n A)  (A)(A)j > 12 Clearly this precludes the required convergence to 0 in the definition of weak-mixing, so that an irrational rotation is ergodic but not weak-mixing. Since R˛n D R n˛ , the earlier argument shows that R˛n is ergodic, so that R˛ is totally ergodic. On the other hand, we show that any Bernoulli shift is strong-mixing. To see this, let A and B be arbitrary measurable sets. By standard measure-theoretic arguments, A and B may each be approximated arbitrarily closely by a finite union of cylinder sets. Since if A0 and B0 are finite unions of cylinder sets, we have that (A0 \ T n B0 ) is equal to (A0 )(B0 ) for large n, it is easy to deduce that (A \ T n B) ! (A)(B) as required. Since the doubling map is measure-theoretically isomorphic to a onesided Bernoulli shift, it follows that the doubling map is also strong-mixing. Similarly, if a Markov Chain is irreducible (i. e. for any states i and j, there exists an n  0 such that Pinj > 0) and aperiodic (there is a state i such that gcdfn : Pini > 0g D 1), then given any pair of cylinder sets A0 and B0 we have by standard theorems of Markov chains (A0 \ T n B0 ) !

Ergodicity and Mixing Properties

(A0 )(B0 ). The same argument as above then shows that an aperiodic irreducible Markov Chain is strong-mixing. On the other hand, if a Markov chain is periodic (d D gcdfn : Pini > 0g > 0), then letting A D B D fx : x0 D ig, we have that (A \ T n B) D 0 whenever d − n. It follows T d is not ergodic, so that T is not weak-mixing. Both weak- and strong-mixing have formulations in terms of functions: Lemma 9 Let T be a measure-preserving transformation of the probability space (X; B; ). 1. T is weak-mixing if and only if for every f ; g 2 L2 one has N1 1 X jh f ; g ı T n i  h f ; 1ih1; gij ! 0 as N ! 1 : N nD0

2. T is strong-mixing if and only if for every f ; g 2 L2 , one has h f ; g ı T N i ! h f ; 1ih1; gi as N ! 1 : Using this, one can see that both mixing conditions are spectral properties. Lemma 10 Weak- and strong-mixing are spectral properties. Proof Suppose S is a weak-mixing transformation of (Y; F ; ) and the transformation T of (X; B; ) is spectrally isomorphic to S by the Hilbert space isomorphism ‚. Then for f ; g 2 L2 (X; B; ), h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); ‚(1)iY h‚(1); ‚(g)iY . Since 1 is an eigenfunction of U T with eigenvalue 1, ‚(1) is an eigenfunction of U S with an eigenvalue 1, so since S is ergodic, ‚(1) must be a constant function. Since ‚ preserves norms, ‚(1) must have a constant value of absolute value 1 and hence h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); 1iY h1; ‚(g)iY . It follows from Lemma 9 that T is weak-mixing. A similar proof shows that strong-mixing is a spectral property.  Both weak- and strong-mixing properties are preserved by taking natural extensions. Recent work of Avila and Forni [4] shows that for interval exchange transformations of k  3 intervals with the underlying permutation satisfying the non-degeneracy condition above, almost all divisions of the interval (with respect to Lebesgue measure on the k1-dimensional simplex) lead to weak-mixing transformations. On the other

hand, work of Katok [36] shows that no interval exchange transformation is strong-mixing. It is of interest to understand the behavior of the ‘typical’ measure-preserving transformation. There are a number of Baire category results addressing this. In order to state them, one needs a set of measure-preserving transformations and a topology on them. As mentioned earlier, it is effectively no restriction to assume that a transformation is a Lebesgue-measurable map on the unit interval preserving Lebesgue measure. The classical category results are then on the collection of invertible Lebesgue-measure preserving transformations of the unit interval. One topology on these is the ‘weak’ topology, where a sub-base is given by sets of the form N(T; A; ) D fS : (S(A)T(A)) < g. With respect to this topology, Halmos [26] showed that a residual set (i. e. a dense Gı set) of invertible measurepreserving transformations is weak-mixing (see also work of Alpern [3]), while Rokhlin [72] showed that the set of strong-mixing transformations is meagre (i. e. a nowhere dense F  set), allowing one to conclude that with respect to this topology, the typical transformation is weak- but not strong-mixing. As often happens in these cases, even when a certain kind of behavior is typical, it may not be simple to exhibit concrete examples. In this case, a well-known example of a transformation that is weak-mixing but not strong-mixing was given by Chacon [13]. While on the face of it the formulation of weak-mixing is considerably less natural than that of strong-mixing, the notion of weak-mixing turns out to be extremely natural from a spectral point of view. Given a measure-preserving transformation T, let U T be the Koopman operator described above. Since this operator is an isometry, any eigenvalue must lie on the unit circle. The constant function 1 is always an eigenfunction with eigenvalue 1. If T is ergodic and g and h are eigenfunctions of U T with eigenvalue , then g h¯ is an eigenfunction with eigenvalue 1, hence invariant, so that g D Kh for some constant K. We see that for ergodic transformations, up to rescaling, there is at most one eigenfunction with any given eigenvalue. If U T has a non-constant eigenfunction f , then one has jhU Tn f ; f ij D k f k2 for each n, whereas by Cauchy– Schwartz, jh f ; 1ij2 < k f k2 . It follows that jhU Tn f ; f i  h f ; 1ih1; f ij  c for some positive constant c, so that using Lemma 9, T is not weak-mixing. Using the spectral theorem, the converse is shown to hold. Theorem 11 The measure-preserving transformation T is weak-mixing if and only U T has no non-constant eigenfunctions.

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Ergodicity and Mixing Properties

Of course this also shows that weak-mixing is a spectral property. Equivalently, this says that the transformation T is weak-mixing if and only if the apart from the constant eigenfunction, the operator U T has only continuous spectrum (that is, the operator has no other eigenfunctions). For a very nice and concise development of the part of spectral theory relevant to ergodic theory, the reader is referred to the Appendix in Parry’s book [63]. Using this theory, one can establish the following: Theorem 12 1. T is weak-mixing if and only if T  T is ergodic; 2. If T and S are ergodic, then T  S is ergodic if and only if U S and U T have no common eigenvalues other than 1. Proof The main factor in the proof is that the eigenvalues of U TS are precisely the set of ˛ˇ, where ˛ is an eigenvalue of U T and ˇ is an eigenvalue of U S . Further, the eigenfunctions of U TS with eigenvalue are spanned by eigenfunctions of the form f ˝ g, where f is an eigenfunction of U T , g is an eigenfunction of U S , and the product of the eigenvalues is . Suppose that T is weak-mixing. Then the only eigenfunction is the constant function, so that the only eigenfunction of U TT is the constant function, proving that T  T is ergodic. Conversely, if U T has an eigenvalue (so that f ı T D ˛T for some non-constant f ) then f ˝ f¯ is a non-constant invariant function of T  T so that T  T is not ergodic. For the second part, if U S and U T have a common eigenvalue other than 1 (say f ı T D ˛ f and g ı T D ˛g), then f ˝ g¯ is a non-constant invariant function. Conversely, if T  S has a non-constant invariant function h, then h can be decomposed into functions of the form f ˝g, where f and g are eigenfunctions of U T and U S respectively with eigenvalues ˛ and ˇ satisfying ˛ˇ D 1. Since the eigenvalues of S are closed under complex conjugation, we see that U T and U S have a common eigenvalue other than 1 as required.  For a measure-preserving transformation T, we let K be the subspace of L2 spanned by the eigenfunctions of U T . It is a remarkable fact that K may be identified as L2 (X; B0 ; ) where B0 is a sub--algebra of B. The space K is called the Kronecker factor of T. The terminology comes from the fact that any sub--algebra F of B gives rise to a factor mapping  : (X; B; ) ! (X; F ; ) with (x) D x. By construction L2 (X; B0 ; ) is the closed linear span of the eigenfunctions of T considered as a measure-preserving transformation of (X; B0 ; ). By the Discrete Spectrum Theorem of Halmos and von Neumann [27], T act-

ing on (X; B0 ; ) is measure-theoretically isomorphic to a rotation on a compact group. This allows one to split L2 (X; B; ) as L2 (X; B0 ; ) ˚ L2c (X; B; ), where, as mentioned above the first part is the discrete spectrum part, spanned by eigenfunctions, and the second part is the continuous spectrum part, consisting of functions whose spectral measure is continuous. Since we have split L2 into a discrete part and a continuous part, it is natural to ask whether the underlying transformation T can be split up in some way into a weak-mixing part and a discrete spectrum (compact group rotation) part, somewhat analogously to the ergodic decomposition. Unfortunately, there is nosuch decomposition available. However for some applications, for example to multiple recurrence (starting with the work of Furstenberg [20,21]), the decomposition of L2 (possibly into more complicated parts) plays a crucial role (see the chapters on  Ergodic Theory: Recurrence and  Ergodic Theory: Interactions with Combinatorics and Number Theory). For non-invertible measure-preserving transformations, the transformation is weak- or strong-mixing if and only if its natural extension has that property. The understanding of weak-mixing in terms of the discrete part of the spectrum of the operator also extends to total ergodicity. T n is ergodic if and only if T has no eigenvalues of the form e 2 i p/n other than 1. From this it follows that an ergodic measure-preserving transformation T is totally ergodic if and only if it has no rational spectrum (i. e. no eigenvalues of the form e 2 i p/q other than the simple eigenvalue 1). An intermediate mixing condition between strongand weak- mixing is that a measure-preserving transformation is mild-mixing if whenever f ı T n i ! f for an L2 function f and a sequence n i ! 1, then f is a.e. constant. Clearly mild-mixing is a spectral property. If a transformation has an eigenfunction f , then it is straightforward to find a sequence ni such that f ı T n i ! f , so we see that mild-mixing implies weak-mixing. To see that strong-mixing implies mild-mixing, suppose that T is and that f ı T n i ! f . Then we have R strong-mixing n ¯ i f ı T f ! k f k2 . On R the other hand, the strong mixing property implies that f ı T n i f¯ ! jh f ; 1ij2. The equality of these implies that f is a. e. constant. Mild-mixing has a useful reformulation in terms of ergodicity of general (not necessarily probability) measure-preserving transformations: A transformation T is mild-mixing if and only if for every conservative ergodic measure-preserving transformation S, T  S is ergodic. See Furstenberg and Weiss’ article [22] for further information on mild-mixing. The strongest spectral property that we consider is that of having countable Lebesgue spectrum. While we

Ergodicity and Mixing Properties

will avoid a detailed discussion of spectral theory in this article, this is a special case that can be described simply. Specifically, let T be an invertible measure-preserving transformation. Then T has countable Lebesgue spectrum if there is a sequence of functions f1 ; f2 ; : : : such that f1g [ fU Tn f j : n 2 Z; j 2 Ng forms an orthonormal basis for L2 (X). To see that this property is stronger than strongmixing, we simply observe that it implies that hU Tt U Tn f j ; U Tm f k i ! 0 as t ! 1. Then by approximating f and g by their expansions with respect to a finite part of the basis, we deduce that hU Tn f ; gi ! h f ; 1ih1; gi as required. Since already strong-mixing is atypical from the topological point of view, it follows that countable Lebesgue spectrum has to be atypical. In fact, Yuzvinskii [96] showed that the typical invertible measure-preserving transformation has simple singular spectrum. The property of countable Lebesgue spectrum is by definition a spectral property. Since it completely describes the transformation up to spectral isomorphism, there can be no stronger spectral properties. The remaining properties that we shall examine are invariant under measuretheoretic isomorphisms only. An invertible measure-preserving transformation T of (X; B; ) is said to be K (for Kolmogorov) if there is a sub-algebra F of B such that T n F is the trivial -algebra up to sets of mea1. 1 nD1 T sure 0 (i. e. the intersection consists only of null sets and sets of full measure). W n 2. 1 nD1 T F D B (i. e. the smallest -algebra containn ing T F for all n > 0 is B). The K property has a useful reformulation in terms of entropy as follows: T is K if and only if for every non-trivial partition P of X, the entropy of T with respect to the partition P is positive: T has completely positive entropy. See the chapter on  Entropy in Ergodic Theory for the relevant definitions. The equivalence of the K property and completely positive entropy was shown by Rokhlin and Sinai [74]. For a general transformation T, one can consider the collection of all subsets B of X such that with respect to the partition PB D fB; B c g, h(PB ) D 0. One can show that this is a -algebra. This -algebra is known as the Pinsker -algebra. The above reformulation allows us to say that a transformation is K if and only if it has a trivial Pinsker -algebra. The K property implies countable Lebesgue spectrum (see Parry’s book [63] for a proof). To see that K is not implied by countable Lebesgue spectrum, we point out that certain measure-preserving transformations derived from Gaussian systems (see for example the paper of Parry and

Newton [52]) have countable Lebesgue spectrum but zero entropy. The fact that (two-sided) Bernoulli shifts have the K property follows from Kolmogorov’s 0–1 law by taking W1 F D nD0 T n P , where P is the partition into cylinder sets (see Williams’s book [93] for details of the 0–1 law). Although the K property is explicitly an invertible property, it has a non-invertible counterpart, namely exactness. A transformation T of (X; B; ) is exact if T1 n B consists entirely of null sets and sets of meanD0 T sure 1. It is not hard to see that a non-invertible transformation is exact if and only if its natural extension is K. The final and strongest property in our list is that of being measure-theoretically isomorphic to a Bernoulli shift. If T is measure-theoretically isomorphic to a Bernoulli shift, we say that T has the Bernoulli property. While in principle this could apply to both invertible and non-invertible transformations, in practice the definition applies to a large class of invertible transformations, but occurs comparatively seldom for non-invertible transformations. For this reason, we will restrict ourselves to a discussion of the Bernoulli property for invertible transformations (see however work of Hoffman and Rudolph [29] and Heicklen and Hoffman [28] for work on the one-sided Bernoulli property). In the case of invertible Bernoulli shifts, Ornstein [53,58] developed in the early 1970s a powerful isomorphism theory, showing that two Bernoulli shifts are measure-theoretically isomorphic if and only if they have the same entropy. Entropy had already been identified as an invariant by Kolmogorov and Sinai [43,82], so this established that it was a complete invariant for Bernoulli shifts. Keane and Smorodinsky [41] gave a proof which showed that two Bernoulli shifts of the same entropy are isomorphic using a conjugating map that is continuous almost everywhere. With other authors, this theory was extended to show that the property of being isomorphic to a Bernoulli shift applied to a surprisingly large class of measure-preserving transformations (e. g. geodesic flows on manifolds of constant negative curvature (Ornstein and Weiss [60]), aperiodic irreducible Markov chains (Friedman and Ornstein [19]), toral automorphisms (Katznelson [39]) and more generally many Gibbs measures for hyperbolic dynamical systems (see the book of Bowen [11])). Initially, it was conjectured that the properties of being K and Bernoulli were the same, but since then a number of measure-preserving transformations that are K but not Bernoulli have been identified. The earliest was due to Ornstein [55]. Ornstein and Shields [59] then provided an uncountable family of non-isomorphic K automorphisms.

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Katok [37] gave an example of a smooth diffeomorphism that is K but not Bernoulli; and Kalikow [33] gave a very natural probabilistic example of a transformation that has this property (the T, T 1 process). While in systems that one regularly encounters there is a correlation between positive entropy and the stronger mixing properties that we have discussed, these properties are logically independent (for example taking the product of a Bernoulli shift and the identity transformation gives a positive entropy transformation that fails to be ergodic; also, the zero entropy Gaussian systems with countable Lebesgue spectrum mentioned above have relatively strong mixing properties but zero entropy). In many of the mixing criteria discussed above we have considered a pair of sets A and B and asked for asymptotic independence of A and B (so that for large n, A and T n B become independent). It is natural to ask, given a finite collection of sets A0 ; A1 ; : : : ; A k , under what conditions Q (A0 \T n 1 A1 \  \T n k A k ) converges to kjD0 (A j ). A measure-preserving transformation is said to be mixing of order k + 1 if for all measurable sets A0 ; : : : ; A k , lim

n 1 !1;n jC1n j !1

   A0 \ T n 1 A1 \    \ T n k A k D

k Y

(A j ) :

jD0

An outstanding open question asked by Rokhlin [73] appearing already in Halmos’ 1956 book [27] is to determine whether mixing (i. e. mixing of order 2) implies mixing of all orders. Kalikow [34] showed that mixing implies mixing of all orders for rank 1 transformations (existence of rank one mixing transformations having been previously established by Ornstein in [54]). Later Ryzhikov [77] used joining methods to establish the result for transformations with finite rank, and Host [30] also used joining methods to establish the result for measure-preserving transformations with singular spectrum, but the general question remains open. It is not hard to show using martingale arguments that K automorphisms and hence all Bernoulli measure-preserving transformations are mixing of all orders. For weak-mixing transformations, Furstenberg [21] has established the following weak-mixing of all orders statement: if a measure-preserving transformation T is weak-mixing, then given sets A0 ; : : : ; A k , there is a subsequence J of the integers of density 0 such that

lim

n!1 n62 J

k  Y   A0 \ T n A1 \    \ T kn A k D (A i ) : iD0

Bergelson [5] generalized this by showing that    A0 \ T p 1 (n) A1 \    \ T p k (n) A k n!1;n62 J k Y D (A i ) lim

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. The method of proof of both of these results was a Hilbert space version of the van der Corput inequality of analytic number theory. Furstenberg’s proof played a key role in his ergodic proof [20] of Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in a subset of the integers of positive density (see the chapter on  Ergodic Theory: Interactions with Combinatorics and Number Theory for more information about this direction of study). The conclusions that one draws here are much weaker than the requirement for mixing of all orders. For mixing of all orders, it was required that provided the gaps between 0; n1 ; : : : ; n k diverge to infinity, one achieves asymptotic independence, whereas for these weak-mixing results, the gaps are increasing along prescribed sequences with regular growth properties. It is interesting to note that the analogous question of whether mixing implies mixing of all orders is known to fail in higher-dimensional actions. Here, rather than a Z action, in which there is a single measure-preserving transformation (so that the integer n acts on a point x 2 X by mapping it to T n x), one takes a Zd action. For such an action, one has d commuting transformations T1 ; : : : ; Td and a vector (n1 ; : : : ; nd ) acts on a point x by sending it to n T1n 1    Td d x. Ledrappier [46] studied the following two2 dimensional action. Let X D fx 2 f0; 1gZ : xv C xvCe1 C xvCe2 D 0 (mod 2)g and let Ti (x)v D xvCe i . Since X is a compact Abelian group, it has a natural measure  invariant under the group operations (the Haar measure). It is not hard to show that this system is mixing (i. e. given any measurable sets A and B, (A \ T1n 1 T2n 2 B) ! (A)(B) as k(n1 ; n2 )k ! 1). Ledrappier showed that the system fails to be 3-mixing. Subsequently Masser [48] established necessary and sufficient conditions for similar higher-dimensional algebraic actions to be mixing of order k but not order k C 1 for any given k. Hyperbolicity and Decay of Correlations One class of systems in which the stronger mixing properties are often found is the class of smooth systems possessing uniform hyperbolicity (i. e. the tangent space to the manifold at each point splits into stable and unstable

Ergodicity and Mixing Properties

subspaces E s (x) and E u (x) such that the kDTj E s (x) k  a < 1 for all x and kDT 1 j E u (x) k  a and DT(Es (x)) D Es (T(x)) and DT(Eu (x)) D Eu (T(x))). In some cases similar conclusions are found in systems possessing non-uniform hyperbolicity. See Katok and Hasselblatt’s book [38] for an overview of hyperbolic dynamical systems, as well as the chapter in this volume on  Smooth Ergodic Theory. In the simple case of expanding piecewise continuous maps of the interval (that is, maps for which the absolute value of the derivative is uniformly bounded below by a constant greater than 1), it is known that if they are totally ergodic and topologically transitive (i. e. the forward images of any interval cover the entire interval), then provided that the map has sufficient smoothness (e. g. the map is C1 and the derivative satisfies a certain additional summability condition), the map has a unique absolutely continuous invariant measure which is exact and whose natural extension is Bernoulli (see the paper of Góra [25] for results of this type proved under some of the mildest hypotheses). These results were originally established for maps that were twice continuously differentiable, and the hypotheses were progressively weakened, approaching, but never meeting, C1 . Subsequent work of Quas [68,69] provided examples of C1 expanding maps of the interval for which Lebesgue measure was invariant, but respectively not ergodic and not weak-mixing. Some of the key tools in controlling mixing in one-dimensional expanding maps that are absent in the C1 case are bounded distortion estimates. Here, there is a constant 1  C < 1 such that given any interval I on which some power T n of T acts injectively and any sub-interval J of I, one has 1/C  (jT n Jj/jT n Ij)/(jJj/jIj)  C. An early place in which bounded distortion estimates appear is the work of Rényi [70]. One important class of results for expanding maps establishes an exponential decay of correlations. Here, one starts with f and g and one estiR a pair of smooth R functions R mates f  g ı T n d f d g d, where  is an absolutely continuous invariant measure. If  is mixing, we expect this to converge to 0. In fact though, in good cases this converges to 0 at an exponential rate for each pair of functions f and g belonging to a sufficiently smooth class. In this case, the measure-preserving transformation T is said to have exponential decay of correlations. See Liverani’s article [47] for an introduction to a method of establishing this based on cones. Exponential decay of correlations implies in particular that the natural extension is Bernoulli. Hu [31] has studied the situation of maps of the interval for which the derivative is bigger than 1 everywhere except at a fixed point, where the local behavior is of the form x 7! x C x 1C˛ for 0 < ˛ < 1. In this case, rather

than exhibiting exponential decay of correlations, the map has polynomial decay of correlations with a rate depending on ˛. In Young’s survey [94], a variety of techniques are outlined for understanding the strong ergodic properties of non-uniformly hyperbolic diffeomorphisms. In her article [95], methods are introduced for studying many classes of non-uniformly hyperbolic systems by looking at suitably high powers of the map, for which the power has strong hyperbolic behavior. The article shows how to understand the ergodic behavior of these systems. These methods are applied (for example) to billiards, one-dimensional quadratic maps and Hénon maps. Future Directions Problem 1 (Mixing of all orders) Does mixing imply mixing of all orders? Can the results of Kalikow, Ryzhikov and Host be extended to larger classes of measure-preserving transformations? Thouvenot observed that it is sufficient to establish the result for measure-preserving transformations of entropy 0. This observation (whose proof is based on the Pinsker  -algebra) was stated in Kalikow’s paper [34] and is reproduced as Proposition 3.2 in recent work of de la Rue [16] on the mixing of all orders problem. Problem 2 (Multiple weak-mixing) As mentioned above, Bergelson [5] showed that if T is a weak-mixing transformation, then there is a subset J of the integers of density 0 such that lim

n!1;n62 J

   A0 \ T p 1 (n) A1 \    \ T p k (n) A k D

k Y

(A i )

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. It is natural to ask what is the most general class of times that can replace the sequences (p1 (n)); : : : ; (p k (n)). In unpublished notes, Bergelson and Håland considered as times the values taken by a family of integer-valued generalized polynomials (those functions of an integer variable that can be obtained by the operations of addition, multiplication, addition of or multiplication by apreal constantp and taking integer parts (e. g. g(n) D b 2b nc C b 3nc2 c)). They conjectured necessary and sufficient conditions for the analogue of Bergelson’s weakmixing polynomial ergodic theorem to hold, and proved the conjecture in certain cases.

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In a recent paper of McCutcheon and Quas [50], the analogous question was addressed in the case where T is a mild-mixing transformation. Problem 3 (Pascal adic transformation) Vershik [89,90] introduced a family of transformations known as the adic transformations. The underlying spaces for these transformations are certain spaces of paths on infinite graphs, and the transformations act by taking a path to its lexicographic neighbor. Amongst the adic transformations, the so-called Pascal adic transformation (so-called because the underlying graph resembles Pascal’s triangle) has been singled out for attention in work of Petersen and others [2,10,51,65]. In particular, it is unresolved whether this transformation is weak-mixing with respect to any of its ergodic measures. Weak-mixing has been shown by Petersen and Schmidt to follow from a number-theoretic condition on the binomial coefficients [2,10]. Problem 4 (Weak Pinsker Conjecture) Pinsker [67] conjectured that in a measure-preserving transformation with positive entropy, one could express the transformation as a product of a Bernoulli shift with a system with zero entropy. This conjecture (now known as the Strong Pinsker Conjecture) was shown to be false by Ornstein [56,57]. Shields and Thouvenot [78] showed that the collection of transformations that can be written as a product of a zero entropy transformation with a Bernoulli shift ¯ is closed in the so-called d-metric that lies at the heart of Ornstein’s theory. It is, however, the case that if T : X ! X has entropy h > 0, then for all h0  h, T has a factor S with entropy h0 (this was originally proved by Sinai [83] and reproved using the Ornstein machinery by Ornstein and Weiss in [61]). The Weak Pinsker Conjecture states that if a measure-preserving transformation T has entropy h > 0, then for all > 0, T may be expressed as a product of a Bernoulli shift and a measure-preserving transformation with entropy less than . Bibliography 1. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence 2. Adams TM, Petersen K (1998) Binomial coefficient multiples of irrationals. Monatsh Math 125:269–278 3. Alpern S (1976) New proofs that weak mixing is generic. Invent Math 32:263–278 4. Avila A, Forni G (2007) Weak-mixing for interval exchange transformations and translation flows. Ann Math 165:637–664 5. Bergelson V (1987) Weakly mixing pet. Ergodic Theory Dynam Systems 7:337–349 6. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci 17:656–660

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57. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 58. Ornstein DS (1974) Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, Newhaven 59. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:89–102 60. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Israel J Math 14:184–198 61. Ornstein DS, Weiss B (1975) Unilateral codings of Bernoulli systems. Israel J Math 21:159–166 62. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116–136 63. Parry W (1981) Topics in Ergodic Theory. Cambridge, Cambridge 64. Petersen K (1983) Ergodic Theory. Cambridge, Cambridge 65. Petersen K, Schmidt K (1997) Symmetric Gibbs measures. Trans Amer Math Soc 349:2775–2811 66. Phelps R (1966) Lectures on Choquet’s Theorem. Van Nostrand, New York 67. Pinsker MS (1960) Dynamical systems with completely positive or zero entropy. Soviet Math Dokl 1:937–938 68. Quas A (1996) A C 1 expanding map of the circle which is not weak-mixing. Israel J Math 93:359–372 69. Quas A (1996) Non-ergodicity for C 1 expanding maps and g-measures. Ergodic Theory Dynam Systems 16:531–543 70. Rényi A (1957) Representations for real numbers and their ergodic properties. Acta Math Acad Sci Hungar 8:477–493 71. Riesz F (1938) Some mean ergodic theorems. J Lond Math Soc 13:274–278 72. Rokhlin VA (1948) A ‘general’ measure-preserving transformation is not mixing. Dokl Akad Nauk SSSR Ser Mat 60:349–351 73. Rokhlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 74. Rokhlin VA, Sinai Y (1961) Construction and properties of invariant measurable partitions. Dokl Akad Nauk SSSR 141:1038– 1041 75. Rudin W (1966) Real and Complex Analysis. McGraw Hill, New York 76. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford, Oxford University Press 77. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funct Anal Appl 27:128–140 78. Shields P, Thouvenot J-P (1975) Entropy zero × Bernoulli pro¯ cesses are closed in the d-metric. Ann Probab 3:732–736 79. Simányi N (2003) Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent Math 154:123– 178 80. Simányi N (2004) Proof of the ergodic hypothesis for typical hard ball systems. Ann Henri Poincaré 5:203–233 81. Simányi N, Szász D (1999) Hard ball systems are completely hyperbolic. Ann Math 149:35–96 82. Sinai YG (1959) On the notion of entropy of a dynamical system. Dokl Russ Acad Sci 124:768–771 83. Sinai YG (1964) On a weak isomorphism of transformations with invariant measure. Mat Sb (NS) 63:23–42 84. Sinai YG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25:141–192 85. Sinai Y (1976) Introduction to Ergodic Theory. Princeton, Princeton, translation of the 1973 Russian original

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Ergodic Theorems

Ergodic Theorems ANDRÉS DEL JUNCO Department of Mathematics, University of Toronto, Toronto, Canada

Article Outline Glossary Definition of the Subject Introduction Ergodic Theorems for Measure-Preserving Maps Generalizations to Continuous Time and Higher–Dimensional Time Pointwise Ergodic Theorems for Operators Subadditive and Multiplicative Ergodic Theorems Entropy and the Shannon–McMillan–Breiman Theorem Amenable Groups Subsequence and Weighted Theorems Ergodic Theorems and Multiple Recurrence Rates of Convergence Ergodic Theorems for Non-amenable Groups Future Directions Bibliography

Glossary Dynamical system in its broadest sense, any set X, with a map T : X ! X. The classical example is: X is a set whose points are the states of some physical system and the state x is succeeded by the state Tx after one unit of time. Iteration repeated applications of the map T above to arrive at the state of the system after n units of time. Orbit of x the forward images x; Tx; T 2 X : : : of x 2 X under iteration of T. When T is invertible one may consider the forward, backward or two-sided orbit of x. Automorphism a dynamical system T : X ! X, where X is a measure space and T is an invertible map preserving measure. Ergodic average if f is a function on X let A n f (x) D P i n1 n1 iD0 f (T x); the average of the values of f over the first n points in the orbit of x. Ergodic theorem an assertion that ergodic averages converge in some sense. Mean ergodic theorem an assertion that ergodic averages converge with respect to some norm on a space of functions.

Pointwise ergodic theorem an assertion that ergodic averages A n f (x) converge for some or all x 2 X, usually for a.e. x. Stationary process a sequence (X1 ; X2 ; : : :) of random variables (real or complex-valued measurable functions) on a probability space whose joint distributions are invariant under shifting (X1 ; X2 ; : : :) to (X2 ; X3 ; : : :). Uniform distribution a sequence fx n g in [0; 1] is uniformly distributed if for each interval I [0; 1], the time it spends in I is asymptotically proportional to the length of I. Maximal inequality an inequality which allows one to bound the pointwise oscillation of a sequence of functions. An essential tool for proving pointwise ergodic theorems. Operator any linear operator U on a vector space of functions on X, for example one arising from a dynamical system T by setting U f (x) D f (Tx). More generally any linear transformation on a real or complex vector space. Positive contraction an operator T on a space of functions endowed with a norm k  k such that T maps positive functions to positive functions and kT f k  j f j. Definition of the Subject Ergodic theorems are assertions about the long-term statistical behavior of a dynamical system. The subject arose out of Boltzmann’s ergodic hypothesis which sought to equate the spatial average of a function over the set of states in a physical system having a fixed energy with the time average of the function observed by starting with a particular state and following its evolution over a long time period. Introduction Suppose that (X; B; ) is a measure space and T : (X; B; ) ! (X; B; ) is a measurable and measurepreserving transformation, that is (T 1 E) D (E) for all E 2 B. One important motivation for studying such maps is that a Hamiltonian physical system (see the article by Petersen in this collection) gives rise to a one-parameter group fTt : t 2 Rg of maps in the phase space of the system which preserve Lebesgue measure. The ergodic theorem of Birkhoff asserts that for f 2 L1 () n1 1X f (T i x) n

(1)

iD0

converges a.e. and that if RT is ergodic (to be defined shortly) then the limit is f d. This may be viewed

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as a justification for Boltzmann’s ergodic hypothesis that “space averages equal time averages”. See Zund [214] for some history of the ergodic hypothesis. For physicists, then, the problem is reduced to showing that a given physical system is ergodic, which can be very difficult. However ergodic systems arise in many natural ways in mathematics. One example is rotation z 7! z of the unit circle if  is a complex number of modulus one which is not a root of unity. Another is the shift transformation on a sequence of i.i.d. random variables, for example a coin-tossing sequence. Another is an automorphism of a compact Abelian group. Often a transformation possesses an invariant measure which is not obvious at first sight. Knowledge of such a measure can be a very useful tool. See Petersen’s article for more examples. If (X; B; ) is a probability space and T is ergodic then Birkhoff’s ergodic theorem implies that if A is a measurable subset of X then for almost every x, the frequency with which x visits A is asymptotically equal to (A), a very satisfying justification of intuition. For example, applying this to the coin-tossing sequence one obtains the strong law of large numbers which asserts that almost every infinite sequence of coin tosses has tails occurring with asymptotic frequency 12 . One also obtains Borel’s theorem on normal numbers which asserts that for almost all x 2 [0; 1] each digit 0; 1; 2; : : : ; 9 occurs with limiting 1 frequency 10 . The so-called continued fraction transformation x 7! x 1 mod 1 on (0; 1) has a finite invariant dx . (Throughout this article x mod 1 denotes measure 1Cx the fractional part of x.) Applying Birkhoff’s theorem then gives precise information about the frequency of occurrence of any n 2 N in the continued fraction expansion of x, for a.e. x. See for example Billingsley [34]. These are the classical roots of the subject of ergodic theorems. The subject has evolved from these simple origins into a vast field in its own right, quite independent of physics or probability theory. Nonetheless it still has close ties to both these areas and has also forged new links with many other areas of mathematics. Our purpose here is to give a broad overview of the subject in a historical perspective. There are several excellent references, notably the books of Krengel [135] and Tempelman [194] which give a good picture of the state of the subject at the time they appeared. There has been tremendous progress since then. The time is ripe for a much more comprehensive survey of the field than is possible here. Many topics are necessarily absent and many are only glimpsed. For example this article will not touch on random ergodic theorems. See the articles [75,143] for some references on this topic.

I thank Mustafa Akcoglu, Ulrich Krengel, Michael Lin, Dan Rudolph and particularly Joe Rosenblatt and Vitaly Bergelson for many helpful comments and suggestions. I would like to dedicate this article to Mustafa Akcoglu who has been such an important contributor to the development of ergodic theorems over the past 40 years. He has also played a vital role in my mathematical development as well as of many other mathematicians. He remains a source of inspiration to me, and a valued friend. Ergodic Theorems for Measure-Preserving Maps Suppose (X; B; ) is a measure space. A (measure-preserving) endomorphism of (X; B; ) is a measurable mapping T : X ! X such that (T 1 E) D (E) for any measurable subset E X. If T has a measurable inverse then one says that it is an automorphism. In this article the unqualified terms “endomorphism” or “automorphism” will always mean measure-preserving. T is called ergodic if for all measurable E one has T 1 E D E ) (E) D 0

or (E c ) D 0 :

A very general class of examples comes from the notion of a stationary stochastic process in probability theory. A stochastic process is a sequence of measurable functions f1 ; f2 ; : : : on a probability space (X; B; ) taking values in a measurable space (Y; C ). The distribution of the process, a measure  on Y N , is defined as the image of  under the map ( f1 ; f2 ; : : :) : X ! Y N .  captures all the essential information about the process f f i g. In effect one may view any process f f i g as a probability measure on Y N . The process is said to be stationary if  is invariant under the left shift transformation S on Y N , S(y)(i) D y(i C 1), that is S is an endomorphism of the probability space (Y N ; ). From a probabilistic point of view the most natural examples of stationary stochastic processes are independent identically distributed processes, namely the case when  is a product measure N for some measure  on (Y; C ). More generally one can consider a stationary Markov process defined by transition probabilities on the state space Y and an invariant probability on Y. See for example Chap. 7 in [50], also Sect. “Pointwise Ergodic Theorems for Operators” below. The first and most fundamental result about endomorphisms is the celebrated recurrence theorem of Poincaré [173]). Theorem 1 Suppose  is finite, A 2 B and (A) > 0. Then for a.e. x 2 A there is an n > 0 such that T n x 2 A, in fact there are infinitely many such n. It may be viewed as an ergodic theorem, in that it is a qualitative statement about how x behaves under iteration of T.

Ergodic Theorems

For a proof, observe that if E A is the measurable set of points which never return to A then for each n > 0 the set T n E is disjoint from E. Applying T m one gets also T (nCm) E \ T m E D ;. Thus E; T 1 E; T 2 E; : : : is an infinite sequence of disjoint sets all having measure (E) so (E) D 0 since (X) < 1. Much later Kac [120] formulated the following quantitative version of Poincaré’s theorem. Theorem 2 Suppose that (X) D 1, T is ergodic and let r A x denote the time of first return to A, that is r A (x) is the least n > 0 such that T n x 2 A. Then Z 1 1 r A d D ; (2) (A) A (A) that is, the expected value of the return time to A is (A)1 . Koopman [131] made the observation that associated to an automorphism T there is a unitary operator U D U T defined on the Hilbert space L2 () by the formula U f D f ı T. This led von Neumann [151] to prove his mean ergodic theorem. Theorem 3 Suppose H is a Hilbert space, U is a unitary operator on H and let P denote the orthogonal projection on the subspace of U-invariant vectors. Then for any x 2 H one has  n1  1X    U i x  Px  ! 0 as n ! 1 : (3)  n  iD0

Von Neumann’s theorem is usually quoted as above but to be historically accurate he dealt with unitary operators indexed by a continuous time parameter. Inspired by von Neumann’s theorem Birkhoff very soon proved his pointwise ergodic theorem [35]. In spite of the later publication by von Neumann his result did come first. See [29,214] for an interesting discussion of the history of the two theorems and of the interaction between Birkhoff and von Neumann. Theorem 4 Suppose (X; B; ) is a measure space and T is an endomorphism of (X; B; ). Then for any f 2 L1 D L1 () there is a T-invariant function g 2 L1 such that A n f (x) D

n1 1X f (T i x) ! g(x) n

a.e.

(4)

iD0

Moreover if  is finite then the convergence alsoRholds with R respect to the L1 norm and one has E g d D E f d for all T-invariant subsets E.

Again this formulation of Birkhoff’s theorem is not historically accurate as he dealt with a smooth flow on a manifold. It was soon observed that the theorem, and its proof, remain valid for an abstract automorphism of a measure space although the realization that T need not be invertible seems to have taken a little longer. The notation A n f D A n (T) f as above will occur often in the sequel. Whenever T is an endomorphism one uses the notation T f D f ı T and with this notation P i A n (T) D n1 n1 iD0 T . When the scalars (R or C) for an L1 space are not specified the notation should be understood as referring to either possibility. In most of the theorems in this article the complex case follows easily from the real and any indications about proofs will refer to the real case. Although von Neumann originally used spectral theory to prove his result, there is a quick proof, attributed to Riesz by Hopf in his 1937 book [100], which uses only elementary properties of Hilbert space. Let I denote the (closed) subspace of U-invariant vectors and I 0 the (usually not closed) subspace of vectors of the form f  U f . It is easy to check that any vector orthogonal to I 0 must be in I, whence the subspace I C I 0 is dense in H . For any vector of the form x D y C y 0 ; y 2 I; y 0 2 I 0 it is clear P i that A n x D n1 n1 iD0 U x converges to y D Px, since 0 A n y D y and if y D z  Uz then the telescoping sum A n y 0 D n1 (z  U n z) converges to 0. This establishes the desired convergence for x 2 I C I 0 and it is easy to extend it to the closure of I C I 0 since the operators An are contractions of H (kA n k  1). Lorch [147] used a soft argument in a similar spirit to extend non Neumann’s theorem from the case of a unitary operator on a Hilbert space to that of an arbitrary linear contraction on any reflexive Banach space. Sine [188] gave a necessary and sufficient condition for the strong convergence of the ergodic averages of a contraction on an arbitrary Banach space. Birkhoff’s theorem has the distinction of being one of the most reproved theorems of twentieth century mathematics. One approach to the pointwise convergence, parallel to the argument to argument just seen, is to find a dense subspace E of L1 so that the convergence holds for all f 2 E and then try to extend the convergence to all f 2 L1 by an approximation argument. The first step is not too hard. For simplicity assume that  is a finite measure. As in the proof of von Neumann’s theorem the subspace E spanned by the T-invariant L1 functions together with functions of the form g  T g; g 2 L1 , is dense in L1 (). This can be seen by using the Hahn–Banach theorem and the duality of L1 and L1 . (Here one needs finiteness of  to know that L1 L1 .) The pointwise convergence of A n f for invariant f is trivial and for f D g  T g it follows from telescoping of the sum and the fact that n1 T n g ! 0 a.e. This

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last can be shown by using the Borel–Cantelli lemma. The second step, extending pointwise convergence, as opposed to norm convergence, for f in a dense subspace to all f in L1 is a delicate matter, requiring a maximal inequality. Roughly speaking a maximal inequality is an inequality which bounds the pointwise oscillation of A n f in terms of the norm of f . The now standard maximal inequality in the context of Birkhoff’s theorem is the following, due to Kakutani and Yosida [209]. Birkhoff’s proof of his theorem includes a weaker version of this result. Let S n f D nA n f , the nth partial sum of the iterates of f . S Theorem 5 Given any real f 2 L1 let A D n1 fS n f  0g. Then Z f d  0 : (5) A

Moreover if one sets M f D supn1 A n f then for any ˛ > 0 fM f > ˛g 

1 k f k1 ˛

(6)

A distributional inequality such as (6) will be referred to as a weak L1 inequality. Note that (6) follows easily from (5) by applying (5) to f ˛, at least in the case when  is finite. With the maximal inequality in hand it is straightforward to complete the proof of Birkhoff’s theorem. For a realvalued function f let Osc f D lim sup A n f  lim inf A n f :

(7)

Osc f D 0 a.e. if and only if A n f converges a.e.(to a possibly infinite limit). One has lim sup A n f  M f  Mj f j and by symmetry lim inf A n f  Mj f j, so Osc f  2Mj f j. To establish the convergence of A n f for a real-valued f 2 L1 let > 0 and write f D g C h with g 2 E (the subspace where convergence has already been established), h 2 L1 and khk < . Then since Osc g D 0 one has Osc f D Osc h. Thus for any fixed ˛ > 0, using (6) n ˛o fOsc f > ˛g D  fOsc h > ˛g   M h > 2 2khk1 2  < : (8) ˛ ˛ Since > 0 was arbitrary one concludes that fOsc f > ˛g D 0 and since ˛ > 0 is arbitrary it follows that fOsc f > 0g D 0, establishing the a.e. convergence. Moreover a simple application of Fatou’s lemma shows that the limiting function is in L1 , hence finite a.e. There are many proofs of (5). Two of particular interest are Garsia’s [89], perhaps the shortest and most mysterious, and the proof via the filling scheme of Chacón

and Ornstein [61], perhaps the most intuitive, which goes like this. Given a function g 2 L1 write g C D max(g; 0), g  D g C  g and let Ug D T g C  g  . Interpretation: the region between the graph of g C and the X-axis is a sheaf of vertical spaghetti sticks, the intervals [0; g C (x)], x in X, and g  is a hole. Now move the spaghetti (horizontally) by T and then let it drop (vertically) into the hole leaving a new hole and a new sheaf which are the negative and posS itive parts of Ug. Now let E 0 D n1 fU n f  0g, the set of points x at which the hole is eventually filled after finitely many iterations of U. The key point is that E D E 0 . Indeed if S n f (x)  0 for some n, then the total linear height of sticks over x; Tx; : : : T n1 x is greater than the total linear depth of holes at these points. The only way that spaghetti can escape from these points is by first filling the hole at x, which shows x 2 E 0 . Similar thinking shows that if x 2 E 0 and the hole at x is filled for the first time at time n then S n f (x)  0, so x 2 E, and that all the spaghetti that goes into the hole at x comes from points T i x which belong to E 0 . This shows that E D E 0 and that the part of the hole lying beneath E is eventually filled by spaghetti coming from E 0 D E. Thus the amount of spaghettiRover E is no less than the size of the hole under E, that is E f d  0. Most proofs of Birkhoff’s theorem use a maximal inequality in some form but a few avoid it altogether, for example [126,186]. It is also straightforward to deduce Birkhoff’s theorem directly from a maximal inequality, as Birkhoff does, without first establishing convergence on a dense subspace. However the technique of proving a pointwise convergence theorem by finding an appropriate dense subspace and a suitable maximal inequality has proved extremely useful, not only in ergodic theory. Indeed, in some sense maximal inequalities are unavoidable: this is the content of the following principle proved already in 1926 by Banach. The principle has many formulations; the following one is a slight simplification of the one to be found in [135]. Suppose B is a Banach space, (X; B; ) is a finite measure space and let E denote the space of -equivalence classes of measurable real-valued functions on X. A linear map T : B ! E is said to be continuous in measure if for each > 0 kx n  xk ! 0 ) fjTx n  Txj > g ! 0 : Suppose that T n is a sequence of linear maps from B to E which are continuous in measure and let Mx D sup n jTn xj. Of course if Tn x converges a.e. to a finite limit then Mx < 1 a.e. Theorem 6 (Banach principle) Suppose Mx < 1 a.e. for each x 2 X. Then there is a function C() such that

Ergodic Theorems

C() ! 0 as  ! 1 such that for all x 2 B and  > 0 one has fMx  kxkg  C() :

(9)

Moreover the set of x for which Tn x converges a.e. is closed in B. The first chapter of Garsia’s book [89] contains a nice introduction to the Banach principle. It should be noted that, for general integrable f , M f need not be in L1 . However if f 2 L p for p > 1 then M f does belong to Lp and one has the estimate kM f k p 

p k f kp : p1

(10)

This can be derived from (6) using also the (obvious) fact that kM f k1  k f k1 . Any such estimate on the norm of a maximal function will be called a strong Lp inequality. It also follows from (6) that if (X) is finite and f 2 L log L, R that is j f j logC j f jd < 1, then M f 2 L1 . In fact Ornstein [160] has shown that the converse of this last statement holds provided T is ergodic. There is a special setting where one has uniform convergence in the ergodic theorem. Suppose T is a homeomorphism of a compact metric space X. By a theorem of Krylov and Bogoliouboff [138] there is at least one probability measure on the Borel -algebra of X which is invariant under T. T is said to be uniquely ergodic if there is only one Borel probability measure, say , invariant under T. It is easy to see that when this is the case then T is an ergodic automorphism of (X; B; ). As an example, if ˛ is an irrational number then the rotation z 7! e2 i˛ z is a uniquely ergodic transformation of the circle fjzj D 1g. Equivalently x 7! x C ˛ mod 1 is a uniquely ergodic map on [0; 1]. A quick way to see this is to show that the ˆ Fourier co-efficients (n) of any invariant probability  are zero for n ¤ 0. The Jewett–Krieger theorem (see Jewett [109] and Krieger [137]) guarantees that unique ergodicity is ubiquitous in the sense that any automorphism of a probability space is measure-theoretically isomorphic to a uniquely ergodic homeomorphism. The following important result is due to Oxtoby [165]). Theorem 7 If T is uniquely ergodic,  is its unique invariant probability measure and f 2 C(X) R then the ergodic averages A n ( f ) converge uniformly to f d. This result can be proved along the same lines as the proof given above of von Neumann’s theorem. In a nutshell, one uses the fact that the dual of CR (X) is the space of finite signed measures on X and the unique ergodicity to show

that functions of the form f  f ı T together with the invariant functions (which are just constant functions) span a dense subspace of C(X). A sequence fx n g in the interval [0; 1]R is said to be uniP 1 formly distributed if n1 niD1 f (x n ) ! 0 f (x)dx for any f 2 C[0; 1] (equivalently for any Riemann integrable f or for any f D 1I where I is any subinterval of [0; 1]). As a simple application of Theorem 7 one obtains the linear case of the following result of Weyl [202]. Theorem 8 If ˛ is irrational and p(x) is any non-constant polynomial with integer coefficients then the sequence fp(n)˛mod1g is uniformly distributed. Furstenberg [84], see also [86], has shown that Weyl’s result, in full generality, can be deduced from the unique ergodicity of certain affine transformations of higher dimensional tori. For polynomials of degree k > 1 Weyl’s result is usually proved by inductively reducing to the case k D 1, using the following important lemma of van der Corput (see for example [139]). Theorem 9 Suppose that for each fixed h > 0 the sequence fx nCh  x n

mod 1gn

is uniformly distributed. Then fx n g is uniformly distributed. When  is infinite and T is ergodic the limiting function in Birkhoff’s theorem is 0 a.e. In 1937 Hopf [100] proved a generalization of Birkhoff’s theorem which is more meaningful in the case of an infinite invariant measure. It is a special case of a later theorem of Hurewicz [105], which we will discuss first. Suppose that (X; B; ) is a  -finite measure space. If  is another  -finite measure on B write    ( is absolutely continuous relative to ) if (E) D 0 implies (E) D 0 and one writes   if    and   . Consider a non-singular automorphism  : X ! X, meaning that  is measurable with a measurable inverse and that (E) D 0 if and only if ( E) D 0. In other words  D  ı  . By the Radon–Nikodym theorem there Ris a function 2 L1 () such that > 0 a.e. and (E) D E d for all measurable E. In order to obtain an associated operator T on L1 which is an (invertible) isometry one defines T f (x) D (x) f ( x) :

(11)

The dual operator on L1 is then given by T  g D g ı  1 . If  is a  -finite measure equivalent to  which is invariant under  then the ergodic theory of  can be reduced to the measure-preserving case using  . The interesting case is when there is no such . It was an open

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Ergodic Theorems

problem for some time whether there is always an equivalent invariant measure. In 1960 Ornstein [163] gave an example of a  which does not have an equivalent invariant measure. It is curious that, with hindsight, such examples were already known in the fifties to people studying von Neumann algebras. Pn i For f 2 L1 let S n f D iD0 T f .  is said to be conservative if there is no set E with (E) > 0 such that  i E; i D 0; 1; : : : are pairwise disjoint, that is if the Poincaré recurrence theorem remains valid. For example, the shift on Z is not conservative. Hurewicz [105] proved the following ratio ergodic theorem. Theorem 10 Suppose  is conservative, f ; g 2 L1 and a.e. to a -invarig(x) > 0 a.e. Then S n f /S n g converges R f d ant limit h. If  is ergodic then h D R gd . In the case when  is -invariant one has T f D f ı . If  is invariant and finite, taking g D 1 one recovers Birkhoff’s theorem. If  is invariant and -finite then Hurewicz’s theorem becomes the theorem of Hopf alluded to earlier. Wiener and Wintner [204] proved the following variant of Birkhoff’s theorem. Theorem 11 (Wiener–Wintner) Suppose T is an automorphism of a probability space (X; B; ). Then for any f 2 L1 there is a subset X 0 X of measure one such that for each x 2 X 0 and  2 C of modulus 1 the sequence 1 P n1 i i iD0  T f (x) converges. n It is an easy consequence of the ergodic theorem that one has a.e. convergence for a given f and  but the point here is that the set on which the convergence occurs is independent of . Generalizations to Continuous Time and Higher-Dimensional Time A (measure-preserving) flow is a one-parameter group fTt ; t 2 Rg of automorphisms of (X; B; ), that is TtCs D Tt Ts , such that Tt x is measurable as a function of (t; x). It will always be implicitly assumed that the map (t; x) 7! Tt x from R  X to X is measurable. Theorem 4 generalizes to flows by replacing sums with integrals and this generalization follows without difficulty from Theorem 4. (As already observed this observation reverses the historical record.) Theorem 4 may be viewed as a theorem about the “discrete flow” fTn D T n : n 2 Zg. Wiener was the first to generalize Birkhoff’s theorem to families of automorphisms fTg g indexed by groups more general than R or Z. A measure-preserving flow is an action of R while a single automorphism corresponds to an action of Z.

A (measure-preserving) action of a group G is a homomorphism T : g 7! Tg from G into the group of automorphisms of a measure space (X; B; ) (satisfying the appropriate joint measurability condition in case G is not discrete). Suppose now that G D R k or Z k and T is an action of G on (X; B; ). In the case of Z k an action amounts to an arbitrary choice of commuting maps T1 ; : : : Tk , Ti D T(e i ) where ei is the standard basis of Z k . In the case of R k one must specify k commuting flows. Let m denote counting measure on G in case G D Z k and Lebesgue measure in case G D R k . For any subset E of G with m(E) < 1 let Z 1 f (Tg x)dm(g) : (12) A E f (x) D m(E) E One may then ask whether A E f converges to a limit, either in the mean or pointwise, as E varies through some sequence of sets which “grow large” or, in case G D R k , “shrink to 0”. The second case is referred to as a local ergodic theorem. In the case of ergodic theorems at infinity the continuous and discrete theories are rather similar and often the continuous analogue of a discrete result can be deduced from the discrete result. In Wiener [203] proved the following result for actions of G D R k and ergodic averages over Euclidean balls Br D fx 2 R k : kxk2  rg. Theorem 12 Suppose T is an action of R k on (X; B; ) and f 2 L1 (). (a) For f 2 L1 limr!1 A B r f D g exists a.e. If  is finite the convergence also holds with respect to the L1 -norm, R R g is T-invariant and I gd D I f d for every T-invariant set I. (b) limr!0 A B r f D f a.e. The local aspect of Wiener’s theorem is closely related to the Lebesgue differentiation theorem, see for example Proposition 3.5.4 of [132], which, in its simplest form, states Rthat for f 2 L1 (R k ; m) one has a.e. convergence of 1 m(B r ) B r f (x C t)dt to f (x) as r ! 0. The local ergodic theorem implies Lebesgue’s theorem, simply by considering the action of R k on itself by translation. In fact the local ergodic theorem can also be deduced from Lebesgue’s theorem by a simple application of Fubini’s theorem (see for example [135], Chap. 1, Theorem 2.4 in the case k D 1). The key point in Wiener’s proof is the following weak L1 maximal inequality, similar to (6). Theorem 13 Let M f D supr>0 jA B r f j. Then one has fM f > ˛g 

C k f k1 ; ˛

(13)

Ergodic Theorems

where C is a constant depending only on the dimension d. (In fact one may take C D 3d .) In the case when T is the action of Rk on itself by translation (13) is the well-known maximal inequality for the Hardy–Littlewood maximal function ([132], Lemma 3.5.3). Wiener proves (13) by way of the following covering lemma. If B is a ball in R k let B0 denote the concentric ball with three times the radius. Rk

Theorem 14 Suppose a compact subset K of is covered by a (finite) collection U of open balls. Then there exist pairS wise disjoint B i 2 U; i D 1; : : : k such that i B0i covers K. To find the Bi it suffices to let B1 be the largest ball in U, then B2 the largest ball which does not intersect B1 and in general Bn the largest ball which does not intersect Sn1 0 iD1 B i . Then it is not hard to argue that the B i cover K. In general, a covering lemma is, roughly speaking, an assertion that, given a cover U of a set K in some space (often a group), one may find a subcollection U0 U which covers a substantial part of K and is in some sense efficient P in that U 2U0 1U  C, C some absolute constant. Covering lemmas play an important role in the proofs of many maximal inequalities. See Sect. 3.5 of [132] for a discussion of several of the best-known classical covering lemmas. As Wiener was likely aware, the same kind of covering argument easily leads to maximal inequalities and ergodic theorems for averages over “sufficiently regular” sets. For example, in the case of G D Z k use the standard total order  on G and for n 2 N k let S n D fm 2 Z k : 0  m  ng. Let e(n) denote the maximum value of n i /n j . Then one has the following result.

vergence if f 2 L p for some p > 1, provided  is finite. Let T1 ; : : : ; Tk be any k (possibly non-commuting!) automorphisms of a finite measure space (X; B; ). For n n D (n1 ; n2 ; : : : ; n k ) let Tn D T1n 1 : : : Tk k . (This is not an action of Z k unless the T i commute with each other.) As before when F Z k is a finite subset write A F f D 1 P n2F Tn f . Finally let P j f D lim n A n (T j ) f . jFj Theorem 16 For f 2 L p lim A S n f D P1 : : : Pk f

n!1

both a.e. and in Lp . The proof of Theorem 16 uses repeated applications of Birkhoff’s theorem and hinges on (10). Somewhat surprisingly Hurewicz’s theorem was not generalized to higher dimensions until the very recent work of Feldman [79]. In fact the theorem fails if one considers averages over the cube [0; n  1]d in Zd . However Feldman was able to prove a suitably formulated generalization of Hurewicz’s theorem for averages over symmetric cubes. For f 2 L1 (R) the classical Hilbert transform is defined for a.e. t by Z f (t  s) 1 lim ds (16) H f (t) D  !0C jsj> s R f (ts) Let H  f (t) D 1 sup >0 j jsj> s dsj, the corresponding maximal function. The proof that the limit (16) exists a.e. is based on the maximal inequality mfH f > g 

Theorem 15 For any e > 0 and any integrable f lim

n!1;e(n)<e

AS n f

exists a.e. and the limit is characterized by the same properties as in Theorem 12. Moreover there is a maximal inequality ( ) C  sup jA S n f j > ˛  k f k1 ; (14) ˛ e(n)<e

(15)

C k f k1 ; 

(17)

where C is an absolute constant. See Loomis [146] (1946) for a proof of (17) using real-variable methods. Cotlar [65] proved the existence of the following ergodic analogue of the Hilbert transform (actually for the n-dimensional Hilbert transform). Theorem 17 Suppose fTt g is a measure-preserving flow on a probability space (X; B; ) and f 2 L1 . Then Z f (Tt x) dt (18) lim !0
where C is a constant depending only on e and the dimension k.

exists for a.a. x.

If one does not impose a bound on e(n) the a.e. convergence of A S n f may fail for f 2 L1 , (although it does hold for any increasing sequence of n’s by Theorem 30 below). Nonetheless Dunford [72] and independently Zygmund [215] showed that one does have unrestricted con-

Motivated by Cotlar’s result Calderón [54]proved a general transfer principle which allows one to transfer maximal inequalities and convergence theorems for functions of a real or integer variable to the ergodic setting. Although stated for functions on R it applies equally well to R k

247

248

Ergodic Theorems

or Z k . This principle subsumes Birkhoff’s theorem, Wiener’s theorem and Cotlar’s result. For simplicity only a special case of Calderón’s result, in the discrete case, will be stated here. Let T be an automorphism of a probability space (X; B; ), let m denote counting measure on Z and suppose  is a probability measure on Z. For g 2 L1 (m) define P  (g)(n) D i2Z (i)g(nC i). For f 2 L1 () let (T) f D P i i2Z (i)T f , which is easily seen to converge a.e. and in L1 -norm. Given a fixed sequence  n of probabilities define M g D supn n g and M(T) f D supn n (T) f . Theorem 18 (Calderón) Suppose there is a constant C such that mfM g > g <

C kgk1 

for all g 2 L1 (m) :

(19)

Then one also has fM(T) f > g <

C k f k1 

for all T

and g 2 L1 () : (20)

In other words, in order to prove a maximal inequality for general T it suffices to prove it in case T is the shift map on the integers. The idea of transference could already be seen in Wiener’s proof of the Rn ergodic theorem. Transfer principles in various forms of have become an important tool in the study of ergodic theorems. See Bellow [18] for a very readable overview.

If  is a R -finite measure on X one may define the measure P D Px d(x). P is also meaningful if  is a finite signed measure. The case when P D  is the stochastic analogue of measure-preserving dynamics and the case when P   is the analogue of non-singular dynamics. It is easy to see that given any -finite measure  there is always a  such that    and P  . Let L˜ 1 denote the space of finite signed measures  such that   , which is identified with L1 D L1 (; R) via the Radon– Nikodym theorem. If P   then P maps L˜ 1 () into itself so the restriction of P is an operator T on L1 (). T is a positive contraction, that is kTk  1 and T maps nonnegative functions to non-negative functions. As proved in, for example, [153], every positive contraction arises in this way from a substochastic kernel under the assumption that (X; B) is standard. This simply means that there is some complete metric on X for which B is the -algebra of Borel sets. Virtually all measurable spaces encountered in analysis are standard so this should be viewed as a technicality only. See [153], [80] and [135] for more about the relation between kernels and positive contractions. The case when X is finite, P is stochastic and  is a a probability measure is classical in probability theory. P and  determine a probability measure  on X N characterized by its values on cylinder sets, namely for all x1 ; : : : ; x n 2 X fx 2 X N : x(i) D x i ; 1  i  n  1g D (x1 )

n1 Y

Px i (x iC1 ) :

(21)

iD1

Pointwise Ergodic Theorems for Operators Early in the history of ergodic theorems there were attempts to generalize the ergodic theorem to more general linear operators on Lp spaces, that is, operators which do not arise by composition with a mapping of X . In the case p D 1 the main motivation for this comes from the theory of Markov processes. If (X; B) is a measurable space a sub-stochastic kernel on X is a non-negative function P on X  B such that (a) for each x 2 XP(x; ) D Px is a measure on B such that Px (X)  1 and (b) P(; A) is a measurable function for each A 2 B. It is most intuitive to think about the stochastic case, namely when each Px is a probability measure. One then views P(x; A) as the probability that the point x moves into the set A in one unit of time, so one has stochastic dynamics as opposed to deterministic dynamics, namely the case when Px D ıT x for a map T. In this case the measures Px are called transition probabilities.

The co-ordinate functions X i (x) D x(i) on the space X N endowed with the probability  form a Markov process, which will be stationary if and only if  is P-invariant. For a general X the analogous construction is possible provided X is standard. Hopf [101] initiated the systematic study of positive L1 contractions and proved the following ergodic theorem. Theorem 19 Suppose (X; B; ) is a probability space and T is a positive contraction on L1 () satisfying T1 D 1 and P k T  1 D 1. Then limn!1 n1 n1 kD0 T f exists a.e. The importance of Hopf’s article lies less in this convergence result than in the methods he developed. He proved that the maximal inequality (5) generalizes to all positive L1 contractions T and used this to obtain the decomposition of X into its conservative and dissipative parts C and D, characterized by the fact that for any p 2 L1 such that P k p > 0 a.e. 1 kD0 T p is infinite a.e. on C and finite a.e. on D. These results are the cornerstone of much of the subsequent work on L1 contractions.

Ergodic Theorems

Theorem 19 contains Birkhoff’s theorem for an automorphism  of (X; B; ), simply by defining T f D f ı . In fact one can also deduce Theorem 19 from Birkhoff’s theorem (if one assumes only that X is standard). Indeed the hypotheses of Hopf’s theorem imply that the kernel P associated to T is stochastic (T  1 D 1) and and that  is P-invariant (T1 D 1). Hopf’s theorem then follows by applying Birkhoff’s theorem to the shift on the stationary Markov process associated to P and . In fact Kakutani [123] (see also Doob [71]) had already made essentially the same observation, except that his result assumes the stationary Markov process to be already given. In 1955 Dunford and Schwartz [73] made essential use of Hopf’s work to prove the following result. Theorem 20 Suppose (X) D 1 and T is a (not necessarily positive) contraction with respect to both the L1 and L1 norms. Then the conclusion of Hopf’s theorem remains valid. Note that the assumption that T contracts the L1 -norm is meaningful, as L1 L1 . The proof of the result is reduced to the positive case by defining a positive contraction jTj analogous to the total variation of a complex measure. Then in 1960 Chacón and Ornstein [61] proved a definitive ratio ergodic theorem for all positive contractions of L1 which generalizes both Hopf’s theorem and the Hurewicz theorem. Theorem 21 Suppose T is a positive contraction of L1 (), where  is -finite, f ; g 2 L1 and g  0. Then Pn Ti f PiD0 n i iD0 T g converges a.e. on the set f

(22) P1

iD0

T i g > 0g.

In 1963 Chacón [58] proved a very general theorem for non-positive operators which includes the Chacón–Ornstein theorem as well as the Dunford–Schwartz theorem. Theorem 22 Suppose T is a contraction of L1 and p n  0 is a sequence of measurable functions with the property that g 2 L1 ; jgj  p n ) jT gj  p nC1 :

(23)

Then Pn Ti f PiD0 n iD0 p i converges a.e. to a finite limit on the set f

(24) P1

iD0

p i > 0g.

If T is an L1 ; L1 -contraction and p n D 1 for all n then the hypotheses of this theorem are satisfied, so Theorem 22

reduces to the result of Dunford and Schwartz. See [59] for a concise overview of all of the above theorems in this section and the relations between them. The identification of the limit in the Chacón–Ornstein theorem on the conservative part C of X is a difficult problem. It was solved by Neveu [152] in case C D X, and in general by Chacón [57]. Chacón [60] has shown that there is a non-singular automorphism  of (X; B; ) such that for the associated invertible isometry T of L1 () given by (11) there is an f 2 L1 () such that lim sup A n f D 1 and lim inf A n f D 1. In 1975 Akcoglu [3] solved a major open problem when he proved the following celebrated theorem. Theorem 23 Suppose T : L p 7! L p is a positive contracP i tion. Then A n f D n1 n1 iD0 T f converges a.e. Moreover one has the strong Lp inequality     sup jA n f j     n

p

p k f kp : p1

(25)

As usual, the maximal inequality (25) is the key to the convergence. Note that it is identical in form to (10) the classical strong Lp inequality for automorphisms. (25) was proved by A. Ionesco–Tulcea (now Bellow) [106] in the case of positive invertible isometries of Lp . It is a result of Banach [16], see also [140], that in this case T arises from a non-singular automorphism  of (X; B; ) in the form T f D 1/p f ı . By a series of reductions Bellow was able to show that in this case (25) can be deduced from (10). Akcoglu’s brilliant idea was to consider a dilation S of T which is a positive invertible isometry on a larger Lp space L˜ p D L p (Y; C ; ). What this means is that there is a positive isometric injection D : L p ! L˜ p and a positive projection P on L˜ p whose range is D(L p ) such that DT n D PS n D for all n  0. Given the existence of such an S it is not hard to deduce (25) for T from (25) for S. In fact Akcoglu constructs a dilation only in the case when Lp is finite dimensional and shows how to reduce the proof of (25) to this case. In the finite dimensional case the construction is very concrete and P is a conditional expectation operator. Later Akcoglu and Kopp [7] gave a construction in the general case. It is noteworthy that the proof of Akcoglu’s theorem consists ultimately of a long string of reductions to the classical strong Lp inequality (10), which in turn is a consequence of (5). Subadditive and Multiplicative Ergodic Theorems Consider a family fX n;m g of real-valued random variables on a probability space indexed by the set of pairs (n; m) 2

249

250

Ergodic Theorems

Z2 such that 0  n < m. fX(n;m) g is called a (stationary) subadditive process if (a) the joint distribution of fX n;m g is the same as that of fX nC1;mC1 g (b) X n;m  X n;l C X l ;m whenever n < l < m. Denoting the index set by fn < mg Z2 the distribution of the process is a measure  on Rfn<mg which is invariant under the shift Tx(n; m) D x(n C 1; m C 1). Thus there is no loss of generality in assuming that there is an underlying endomorphism T such that X n;m ı T D X nC1;mC1 . In 1968 Kingman [129] proved the following generalizaR tion of Birkhoff’s theorem. D inf n1 X 0;n d is called the time constant of the process. Theorem 24 If the X n;m are integrable and > 1 then 1 n X 0;n converges a.e. and R in L1 -norm to a T -invariant limit ¯ D . X¯ 2 L1 () satisfying Xd It is easy to deduce from the above that if one assumes only C that X 0;1 is integrable then n1 X0;n still converges a.e. to a T-invariant limit X¯ taking values in [1; 1). Subadditive processes first arose in the work of Hammersley and Welsh [99] on percolation theory. Here is an example. Let G be the graph with vertex set Z2 and with edges joining every pair of nearest neighbors. Let E denote the edge set and let fTe : e 2 Eg be non-negative integrable i.i.d. random variables. To each finite path P in G P associate the “travel time” T(P) D e2E Te . For integers m > n  0 let X n;m be the infimum of T(P) over all paths P joining (0; R n) to (0; m). This is a subadditive process with 0  < Te d and it is not hard to see that the underlying endomorphism is ergodic. Thus Kingman’s theorem yields the result that n1 X0;n ! a.e. Suppose now that T is an ergodic automorphism of a probability space (X; B; ) and P is a function on X taking values in the space of d  d real matrices. Define Pn;m D P(T m1 x)P(T m2 x) : : : P(T n x) and let Pn;m (i; j) denote the i; j entry of Pn;m . Then X n;m D log(kPn;m k) (use any matrix norm) is a subadditive process so one obtains the first part of the following result of Furstenberg and Kesten [87] (1960) originally proved by more elaborate methods. The second part can also be deduced from the subadditive theorem with a little more work. See Kingman [130] for details and for some other applications of subadditive processes. Theorem 25 R (a) Suppose logC (jPj)d < 1. Then kP0;n k1/n converges a.e. to a finite limit. (b) Suppose that for each i; j P(i; j) is a strictly positive function such that log P(i; j) is integrable. Then the

1

limit p D lim(P0;n (i; j)) n exists a.e. and is independent of i and j. Partial results generalizing Kingman’s theorem to the multiparameter case were obtained by Smythe [189] and Nguyen [157]. In 1981 Akcoglu and Krengel [8] obtained a definitive multi-parameter subadditive theorem. They d consider an action fTm g of the semigroup G D Z0 by endomorphisms of a measure space (X; B; ). Using the standard total ordering  of G an interval in G is any set of the form fk 2 G : m  k  ng for any m  n 2 G. Let I denote the set of non-empty intervals. Reversing the direction of the inequality, they define a superadditive process as a collection of integrable functions FI ; I 2 I , such that (a) FI ı Tm D FICm , (b) FI  FI 1 C : : : C FI k whenever I is the disjoint union of I1 ; : : : ; I k and R (c) D supI2I jIj1 FI d < 1. A sequence fI n g of sets in I is called regular if there is an increasing sequence I 0n such that I n I 0n and jI 0n j  CjI n j for some constant C. Theorem 26 (Akcoglu–Krengel) Suppose F I is a superadditive process and fI n g is regular. Then jI1n j FI n converges a.e. fFI g is additive if the inequality in (b) is replaced by equalP ity. In this case FI D n2I f ı Tn where f is an integrable function. Thus in the additive case the Akcoglu–Krengel result is a theorem about ordinary multi-dimensional ergodic averages, which is in fact a special case of an earlier result of Tempelman [196] (see Sect. “Amenable Groups” below). Kingman’s proof of Theorem 24 hinged on the existence of a certain (typically non-unique) decomposition for subadditive processes. Akcoglu and Krengel’s proof of the multi-parameter result does not depend on a Kingman-type decomposition, in fact they show that there is no such decomposition in general. They prove a weak maximal inequality fsup jI n j1 FI n > g <

C

; 

(26)

where C is a constant depending only on the dimension, and show that this is sufficient to prove their result. In the case d D 1 the Akcoglu–Krengel argument provides a new and more natural proof of Kingman’s theorem, similar in spirit to Wiener’s arguments. Akcoglu and Sucheston [9] have proved a ratio ergodic theorem for subadditive processes with respect to

Ergodic Theorems

a positive L1 contraction, generalizing both the Chacón– Ornstein theorem and Kingman’s theorem. In 1968 Oseledec [164] proved his celebrated multiplicative ergodic theorem, which gives very precise information about the random matrix products studied by Furstenberg and Kesten. His theorem is an important tool for the study of Lyapunov exponents in differentiable dynamics, see notably Pesin [169]. If A is a d  d matrix let kAk D supfkAxk : kxk D 1g where kxk is the Euclidean norm on Rn . Theorem 27 Suppose T is an endomorphism of the probability space (X; B; ). Suppose P is a measurable function on XR whose values are d  d real matrices such that logC kPkd < 1 and let Pn (x) D P(T n1 x)P(T n2 x) : : : P(x). Then there is a T-invariant subset X 0 of X with measure 1 such that for x 2 X 0 the following hold. 1

(a) lim n!1 (Pn (x)Pn (x)) 2n D A(x) exists. (b) Let 0  exp 1 (x)  exp 2 (x)  : : :  exp r (x) be the distinct eigenvalues of A(x) (r D r(x) may depend on x and 1 may be 1) with multiplicities m1 (x); : : : mr (x). Let E i (x) be the eigenspace corresponding to exp( i (x)) and set F i (x) D E1 (x) C : : : C E i (x) : Then for each u 2 F i (x)nF i1 (x) lim

1 log kPn (x)uk D  i (x) : n

Entropy and the Shannon–McMillan–Breiman Theorem The notion of entropy was introduced by Shannon in his landmark work [185] which laid the foundations for a mathematical theory of information. Suppose (X; B; ) is a probability space, P is a finite measurable partition of X and T is an automorphism of (X; B; ). P(x) denotes the atom of P containing x. The entropy of P is h(P) D 

X Z

D

log((P(x)) d(x)  0 :

(d) If T is ergodic, det P(x) D 1 a.e. and lim sup n

1 n

Z log kPn kd > 0

then the i are constants, 1 < 0 and r > 0. Raghunathan [174] gave a much shorter proof of Oseledec’s theorem, valid for matrices with entries in a locally compact normed field. He showed that it could be reduced to the Furstenberg–Kesten theorem by considering the exterior powers of P. Ruelle [180] extended Oseledec’s theorem to the case where P takes values in the set of bounded operators on a Hilbert space. Walters [199] has given a proof (under slightly stronger hypotheses) which avoids the matrix calculations and tools from multilinear algebra used in other proofs.

(27)

 log((A)) may be viewed as a quantitative measure of the amount of information contained in the statement that a randomly chosen x 2 X happens to belong to A. So h(P) is the expected information if one is about to observe which atom of P a randomly chosen point falls in. See Billingsley [34] for more motivation of this concept. See also the article in this collection on entropy by J. King or any introductory book on ergodic theory, e. g. Petersen [171]. W If P and Q are partitions P Q denotes the common refinement which consists of all sets p\q, p 2 P; q 2 Q. It W is intuitive and not hard to show that h(P Q)  h(P) C W n i h(Q). Now let P0n D n1 iD0 T P and h n D h(P0 ). The subadditivity of entropy implies that h nCm  h n C h m , so by a well-known elementary lemma the limit h(P; T) D lim n

(c) The functions m i and i are T-invariant.

(p) log((p))

p2P

hn hn D inf 0 n n n

(28)

exists. If one thinks of P(x) as a measurement performed on the space X and Tx as the state succeeding x after one second has elapsed then h(P; T) is the expected information per second obtained by repeating the experiment every second for a very long time. See [177] for an alternative and very useful approach to h(P; T) via name-counting. The following result, which is known as the Shannon– McMillan–Breiman theorem, has proved to be of fundamental importance in ergodic theory, notably, for example, in the proof of Ornstein’s celebrated isomorphism theorem for Bernoulli shifts [159]. Theorem 28 If T is ergodic then 1 lim  log((P0n1 (x)) D h(P; T) n

n!1

a.e. and in L1 -norm.

(29)

251

252

Ergodic Theorems

In other words, the actual information obtained per second by observing x over time converges to the constant h(P; T), namely the limiting expected information per second. Shannon [185] formulated Theorem 28 and proved convergence in probability. McMillan [149] proved L1 convergence and Breiman [49] obtained the a.e. convergence. The original proofs of a.e. convergence used the martingale convergence theorem, were not very intuitive and did not generalize to Z n -actions, where the martingale theorem is not available. Ornstein and Weiss [161] (1983) found a beautiful and more natural argument which bypasses the martingale theorem and allows generalization to a class of groups which includes Z n . Amenable Groups Let G be any countable group and T D fTg g an action of G by automorphisms of a probability space. Suppose  is a complex measure on G, that is, f(g)g g2G is an absolutely summable sequence. Let T g act on functions via Tg f D f ı Tg , so T g is an isometry of Lp for every P 1  p  1. Let (T) D g2G (g)Tg . A very general framework for formulating ergodic theorems is to consider a sequence fn g and ask whether n (T) f converges, a.e. or in mean, for f in some Lp space. When n (T) f converges for all actions T and all f in Lp in p-norm or a.e. then one says that  n is mean or pointwise good in Lp . When the  n are probability measures it is natural to call such results weighted ergodic theorems and this terminology is retained for complex  as well. Birkhoff’s theorem says that if G D Z and  n is the normalized counting measure on f0; 1; : : : n  1g then fn g is pointwise good in L1 . This section will be concerned only with sequences fn g such that  n is normalized counting measure on a finite subset Fn G so one speaks of mean or pointwise good sequences fFn g. A natural condition to require of fFn g, which will ensure that the limiting function is invariant, is that it be asymptotically (left) invariant in the sense that 

jgFn Fn j !0 jFn j

8g 2 G :

(30)

Such a sequence is called a Følner sequence and a group G is amenable if it has a Følner sequence. As in most of this article G is restricted to be a discrete countable group for simplicity but most of the results to be seen actually hold for a general locally compact group. Amenability of G is equivalent to the existence of a finitely additive left invariant probability measure on G. It is not hard to see that any Abelian, and more generally

any solvable, group is amenable. On the other hand the free group F 2 on two generators is not amenable. See Paterson [167] for more information on amenable groups. The Følner property by itself is enough to give a mean ergodic theorem. Theorem 29 Any Følner sequence is mean good in Lp for 1  p < 1. The proof of this result is rather similar to the proof of Theorem 3. In fact Theorem 29 is only a special case of quite general results concerning amenable semi-groups acting on abstract Banach spaces. See the book of Paterson [167] for more on this. Turning to pointwise theorems, the Følner condition alone does not yield a pointwise theorem, even when G D Z and the F n are intervals. For example Akcoglu and del Junco [6] have shown that when G D Z and p Fn D [n; n C n] \ Z the pointwise ergodic theorem fails for any aperiodic T and for some characteristic function f . See also del Junco and Rosenblatt [119]. The following pointwise result of Tempelman [196] is often quoted. A Følner sequence fFn g is called regular if there is a constant C such that jFn1 Fn j  CjFn j and there is an increasing sequence Fn0 such that Fn Fn0 and jFn0 j  CjFn j. Theorem 30 Any regular Følner sequence is pointwise good in L1 . In case the F n are intervals in Z n this result can be proved by a variant of Wiener’s covering argument and in the general case by an abstraction thereof. The condition jFn1 Fn j  CjFn j captures the property of rectangles which is needed for the covering argument. Emerson [78] independently proved a very similar result. The work on ergodic theorems for abstract locally compact groups was pioneered by Calderón [53] who built on Wiener’s methods. The main result in this paper is somewhat technical but it already contains the germ of Tempelman’s theorem. Other ergodic theorems for amenable groups, whose main interest lies in the case of continuous groups, include Tempelman [195], Renaud [175], Greenleaf [93] and Greenleaf and Emerson[94]. The discrete versions of these results are all rather close to Tempelman’s theorem. Among pointwise theorems for discrete groups Tempelman’s result was essentially the best available for a long time. It was not known whether every amenable group had a Følner sequence which is pointwise good for some Lp . In 1988 Shulman [187] introduced the notion of a tempered

Ergodic Theorems

Følner sequence fFn g, namely one for which ˇ ˇ ˇ[ ˇ ˇ 1 ˇ F i Fn ˇ < CjFn j ; ˇ ˇ ˇ

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i
for some constant C. The advantage of the tempered condition is that any Følner sequence has a tempered subsequence, and in particular any amenable group has a tempered Følner sequence. Shulman proved a maximal inequality in L2 for such F n which implies that fFn g is pointwise good in L2 . An account of this work may be found in Section 5.6 of Tempelman’s book [194]. Lindenstrauss [145] was able to extend the result to L1 . Theorem 31 Any tempered Følner sequence is pointwise good in L1 . The key new idea in his proof is to use a probabilistic argument to establish a covering lemma. In the discrete case Ornstein and Weiss [201] have given a non-probabilistic proof of Lindenstrauss’s covering lemma. Lindenstrauss also generalizes the a.e. convergence in the Shannon–McMillan–Breiman theorem to this setting. L1 convergence was already established by Kieffer [128] in 1975. Subsequence and Weighted Theorems In this section G and T are as in the previous section and  n is a sequence of complex measures on G. This section will be concerned with conditions on  n that ensure that it is mean or pointwise good. For the most part G will be Z. Hopf’s ergodic theorem gives a class of examples for free. Choose any probability measure  on G, define the operator T D (T) and observe that (T )n D Tn , where n denotes the convolution power. Since T  satisfies the hypotheses of Hopf’s theorem it follows that the P i sequence n D n1 n1 iD0  is pointwise good in L1 . Another sort of example is given by choosing a seP quence g n in G and letting n D n1 n1 iD0 ı g n . If one has convergence for such a sequence one speaks of a subsequence ergodic theorem. This section will mainly focus on the case G D Z and sequences which are the increasing enumeration of a subset S N. Given S N write S;n for the corresponding probabilities  n and say S is good if S;n is good. Perhaps the first subsequence ergodic theorem is due to Blum and Hanson [38] who proved that an automorphism T of a probability space is strongly mixing if and P m i f converges in L norm for every only if n1 n1 2 iD0 T f 2 L2 and every increasing fm i g. Strong mixing means

that (T n A \ B) ! (A)(B). In 1969 Brunel and Keane [51] proved the first pointwise subsequence ergodic theorem Theorem 32 Suppose that T is a translation on a compact Abelian group G with Haar measure , g 2 G, and E G is any Borel set with (E) > 0 and (@E) D 0. Let S D fi > 0 : T i g 2 Eg. Then S is pointwise good in L1 . Krengel [133] constructed the first example of a sequence S N which is pointwise universally bad, in the strong sense that for any aperiodic T the a.e. convergence of S;n (T) f fails for some characteristic function f . Bellow [17] proved that any lacunary sequence (meaning a nC1 > ca n for some c > 1) is pointwise universally bad in L1 . Later Akcoglu at al. [1] were able to show that for lacunary sequences fS;n g is even strongly sweeping out. A sequence fn g of probability measures on Z is said to be strongly sweeping out if for any ergodic TRand for all ı > 0 there is a characteristic function f with f d < ı such that lim sup n (T) f D 1 a.e. It is not difficult to show that if fn g is strongly sweeping out then there are characteristic functions f such that lim inf n (T) f D 0 and lim sup n (T) f D 1. Thus for lacunary sequences the ergodic theorem fails in the worst possible way. Bellow and Losert [21] gave the first example of a sequence S Z of density 0 which is universally good for pointwise convergence, answering a question posed by Furstenberg. They construct an S which is pointwise good in L1 . This paper also contains a good overview of the progress on weighted and subsequence ergodic theorems at that time. Weyl’s theorem on uniform distribution (Theorem 9) suggests the possibility of an ergodic theorem for the sequence fn2 g. It is not hard to see that fn2 g is mean good in L2 . In fact the spectral theorem and the dominated convergence theorem show that it is enough to prove that theL1 P n 2 on the unit bounded sequence of functions n1 n1 iD0 z circle converges at each point z of the unit circle. When z is not a root of unity the sequence converges to 0 by Weyl’s result and when z is a root of unity the convergence is 2 trivial because fz n g is periodic. In 1987 Bourgain [39,43] proved his celebrated pointwise ergodic theorem for polynomial subsequences. Theorem 33 If p is any polynomial with rational coefficients taking integer values on the integers then S D fp(n)g is pointwise good in L2 . The first step in Bourgain’s argument is to reduce the problem of proving a maximal inequality to the case of the shift map on the integers, via Calderón’s transfer princi-

253

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Ergodic Theorems

ple. Then the problem is transferred to the circle by using Fourier transforms. At this point the problem becomes a very delicate question about exponential sums and a whole arsenal of tools is brought to bear. See Rosenblatt and Wierdl [176] and Quas and Wierdl [28](Appendix B) for nice expositions of Bourgain’s methods. Bourgain subsequently improved this to all Lp , p > 1 and also extended it to sequences f[q(n)]g where now q is an arbitrary real polynomial and [] denotes the greatest integer function. He also announced that his methods can be used to show that the sequence of primes is pointwise p 1C 3 good in Lp for any p > 2 . Wierdl [205] (1988) soon extended the result for primes to all p > 1. Theorem 34 The primes are pointwise good in Lp for p >1. It has remained a major open question for quite some time whether any of these results hold for p D 1. In 2005 there appeared a preprint of Mauldin and Buczolich [148], which remains unpublished, showing that polynomial sequences are L1 -universally bad. Another major result of Bourgain’s is the so-called return times theorem [44]. A simplification of Bourgain’s original proof was published jointly with Furstenberg, Katznelson and Ornstein as an appendix to an article [47] of Bourgain. To state it let us agree to say that a a sequence of complex numbers fa(n)gn0 has property P if the seP quence of complex measures n D n1 n1 iD0 a(i)ı i has property P, where ı i denotes the point mass at i. Theorem 35 (Bourgain) Suppose T is an automorphism of a probability space (X; B; ), 1  p; q  1 are conjugate exponents and f 2 L p (). Then for almost all x the sequence f f (T n x)g is pointwise good in Lq . Applying this to characteristic functions f D 1E one sees that the return time sequence fi > 0 : S i x 2 Eg is good for pointwise convergence in L1 . Theorem 32 is a very special case. It is also easy to see that Theorem 35 contains the Wiener–Wintner theorem. In 1998 Rudolph [179] proved a far-reaching generalization of the return times theorem using the technique of joinings. For an introduction to joinings see the article by de la Rue in this collection and also Thouvenot [198], Glasner [90] (2003) and Rudolph’s book [177]. Rudolph’s result concerns the convergence of multiple averages N1 k 1 XY f j (T jn x) N nD0

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jD1

where each T j is an automorphism of a probability space (X j ; B j ;  j ) and the f j are L1 functions. The point is that the convergence occurs whenever each x j 2 X 0j , sets of

measure one which may be chosen sequentially for j D 1; : : : ; k without knowing what T i or f i are for any i > j. He actually proves something stronger, namely he identifies an intrinsic property of a sequence fa i g, which he calls fully generic, such that the following hold. (a) The constant sequence {1} is fully generic. (b) If fa i g is fully generic then for any T and f 2 L1 the sequence a i f (T i x) is fully generic for almost all x. (c) Fully generic implies pointwise good in L1 . The definition of fully generic will not be quoted here as it is somewhat technical. For a proof of the basic return times theorem using joinings see Rudolph [178]. Assani, Lesigne and Rudolph [13] took a first step towards the multiple theorem, a Wiener–Wintner version of the return times theorem. Also Assani [11] independently gave a proof of Rudolph’s result in the case when all the T j are weakly mixing. Ornstein and Weiss [162] have proved the following version of the return times theorem for abstract discrete groups. As with Z, let us say that a sequence fa g g g2G of complex numbers has property P for fFn g if the sequence P n D jF1 j g2Fn a(g)ı g of complex measures has propn erty P. Theorem 36 Suppose that the increasing Følner sequence fFn g satisfies the Tempelman condition supn jFn1 Fn j/ S jFn j < 1 and Fn D G. If b 2 L1 then for a.a. x the sequence fb(Tg x)g is pointwise good in L1 for fFn g. Recently Demeter, Lacey, Tao and Thiele [67] have proved that the return times theorem remains valid for any 1 < p  1 and q  2. On the other hand Assani, Buczolich and Mauldin [14] (2005) showed that it fails for p D q D 1. Bellow, Jones and Rosenblatt have a series of papers [22,23,24,25] studying general weighted averages associated to a sequence  n of probability measures on Z, and, in some cases, more general groups. The following are a few of their results. [23] is concerned with Z-actions and moving block averages given by n D m I n , where the I n are finite intervals and mI denotes normalized counting measure on I. They resolve the problem completely, obtaining a checkable necessary and sufficient condition for such a sequence to be pointwise good in L1 . [24] gives sufficient conditions on a sequence  n for it to be pointwise good in Lp , p > 1, via properties of the Fourier transforms ˆ n . A particular consequence is P that if lim n!1 k2Z jn (k)  n (k  1)j D 0 then fn g has a subsequence which is pointwise good in Lp ,

Ergodic Theorems

p > 1. In [25] they obtain convergence results for sequences n D  n , the convolution powers of a probability measure . A consequence of one of their main results P is that if the expectation k2Z k(k) is zero, the second P moment k2Z k 2 (k) is finite and  is aperiodic (its support is not contained in any proper coset in Z) then  n is pointwise good in Lp for p > 1. Bellow and Calderón [19] later showed that this last result is valid also for p D 1. This is a consequence of the following sufficient condition for a sequence T to satisfy a weak L1 inequality. Given an automorphism of a probability space (X; B; ) let M f D sup jn (T) f (x)j be the maximal operator associated to fn g. Theorem 37 (Bellow and Calderón) Suppose there is an ˛ 2 (0; 1] and C > 0 such that for each n > 1 one has jn (x C y)  n (x)j  C

jyj˛ jxj1C˛

for all x; y 2 Z

Theorem 39 Suppose T is an automorphism of a probability space (X; B; ), (B) > 0 and k  1. Then there is T an n > 0 such that ( kiD1 T in B) > 0. In 1977, Furstenberg [85] proved the following ergodic theorem which implies the multiple recurrence theorem. He also established a general correspondence principle which puts the shaky analogy between the multiple recurrence theorem and Szemerédi’s theorem on a firm footing and allows each to be deduced from the other. Thus he obtained an ergodic theoretic proof of Szemerédi’s combinatorial result. Theorem 40 Suppose T is an automorphism of a probabilR ity space (X; B; ), f 2 L1 , f  0, f d > 0 and k  1. Then lim inf N

N1 Z k 1 X Y in T f d > 0 : N nD0

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iD1

such that 0 < 2jyj  jxj Then there is a constant D such that fM f > g 

D k f k1 

f 2 L1 ()

for all T;

and  > 0 :

Ergodic Theorems and Multiple Recurrence Suppose S N. The upper density of S is ¯ D lim sup jS \ [1; n]j : d(S) n n

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and the density d(S) is the limit of the same quantity, if it exists. In 1975 Szemerédi [190] proved the following celebrated theorem, answering an old question of Erd˝os and Turán. Theorem 38 Any subset of N with positive upper density contains an arithmetic progression of length k for each k  1. This result has a distinctly ergodic-theoretic flavor. Letting T denote the shift map on Z, it says that for each k there T is an n such that S 0 D kiD1 T i n S is non-empty. In fact ¯ 0 ) > 0. the result gives more: there is an n for which d(S In this light Szemerédi’s theorem becomes a multiple recurrence theorem for the shift map on N, equipped with ¯ Of course d¯ is not the invariant “measure-like” quantity d. even finitely additive so it is not a measure. d, however, is at least finitely additive, when defined, and d(N) D 1. This point of view suggests the following multiple recurrence theorem.

Furstenberg’s result opened the door to the study of socalled ergodic Ramsey theory which has yielded a vast array of deep results in combinatorics, many of which have no non-ergodic proof as yet. The focus of this article is not on this direction but the reader is referred to Furstenberg’s book [86] for an excellent introduction and to Bergelson [27,28] for surveys of later developments. There is also the article by Frantzikinakis and McCutcheon in this collection. Furstenberg’s proof relies on a deep structure theorem for a general automorphism which was also developed independently by Zimmer [213], [212] in a more general context. A factor of T is any sub- -algebra F B such that T(F ) D F . (It is more accurate to think of the factor as the action of T on the measure space (X; F ; jF ).) The structure theorem asserts that there is a transfinite increasing sequence of factors fF˛ g of T such that the following conditions hold. (a) F˛C1 is compact relative to F˛ . W (b) F˛ D ˇ <˛ Fˇ whenever ˛ is a limit ordinal. W (c) B is weakly mixing relative to F˛ . W ( denotes the  -algebra generated by a collection of  -alW gebras.) ˛ F˛ is called the maximal distal factor of T and W if F˛ D B then T is called distal. The definitions of the relative properties in (a) and (c) above are somewhat technical so only their absolute versions (i. e. relative to the trivial -algebra f;; Xg) will be described here. T is said to be compact if for each f 2 L2 the orbit fT i f : i 2 Zg is pre-compact in the norm topology of L2 . This turns out to be equivalent to the statement that T is

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Ergodic Theorems

a translation on a compact Abelian group endowed with its Haar measure. The property of weak mixing is a fundamental notion in ergodic theory which has many equivalent definitions. The most appropriate for our purposes is that T is weakly mixing if it has no compact factors. This turns out to be equivalent to the ergodicity of T  T acting on the product measure space (X; B; )  (X; B; ). The verification of 37 in the case of compact T is rather easy. In this case it is not hard to prove that for any f 2 L1 and > 0 the set fn 2 Z : kT n f  f k2 < g has bounded gaps and (34) follows easily. In the case when T is weakly mixing (34) is a consequence of the following theorem which Furstenberg proves in [85] (as a warm-up for it’s much harder relative version). Theorem 41 If T is weakly mixing and f1 ; f2 ; : : : ; f k are L1 functions then lim N

N1 Z k k Z Y 1 X Y in fi T f i d D N nD0 iD1

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iD1

Later Bergelson [26] showed that the result can be obtained easily by an induction argument using the following Hilbert space generalization of van der Corput’s lemma. Theorem 42 (Bergelson) Suppose fx n g is a bounded sequence of vectors in Hilbert space such that for each h > 0 P N1 one has N1 nD0 hx nCh ; x n i ! 0 as N ! 1. Then P N1 1 k N nD0 x n k ! 0. Ryzhikov has also given a beautiful short proof of Theorem 41 using joinings ([182]). Bergelson’s van der Corput lemma and variants of it have been a key tool in subsequent developments in ergodic Ramsey theory and in the convergence results to be be discussed in this section. Bergelson [26] used it to prove the following mean ergodic theorem for weakly mixing automorphisms. Theorem 43 Suppose T is weakly mixing, f1 ; : : : ; f k are L1 functions and p1 ; : : : ; p k are polynomials with rational coefficients taking integer values on the integers such that no p i  p j is constant for i ¤ j. Then   N1 k k Z  1 XY Y   p i (n) f i d D 0 : (36) T fi  lim   N!1  N nD0 iD1

iD1

Theorems 40 and 41 immediately raise the question of P N1 Q k in convergence of the multiple averages N1 nD0 iD1 T f i for a general T. Several authors obtained partial results on the question of mean convergence. It was finally resolved only recently by Host and Kra [104], who proved the following landmark theorem.

Theorem 44 Suppose f1 ; f2 ; : : : ; f k 2 L1 . Then there is a g 2 L1 such that   N1 k  1 XY   T in f i  g  D 0 : lim   N nD0 iD1

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2

Independently and somewhat later Ziegler [211]obtained the same result by somewhat different methods. Furstenberg had already established Theorem 44 for k D 2 in [85]. It was proved for k D 3 in the case of a totally ergodic T by Conze and Lesigne [63]and in general by Host and Kra [102]. It can also be obtained using the methods developed by Furstenberg and Weiss [88]. In this paper Furstenberg and Weiss proved a result for polynomial powers of T in the case k D 2. They also formalized the key notion of a characteristic factor. A factor C of T is said to be characteristic for the averages (37) if, roughly speaking, the L2 limiting behavior of the averages is unchanged when any one of the f i ’s is replaced by its conditional expectation on C . This means that the question of convergence of these averages may be reduced to the case when f i are all C -measurable. So the problem is to find the right (smallest) characteristic factor and prove convergence for that factor. The importance of characteristic factors was already apparent in Furstenberg’s original paper [85], where he showed that the maximal distal factor is characteristic for the averages (37). In fact he showed that for a given k a k-step distal factor is characteristic. (An automorphism is k-step distal if it is the top rung in a k-step ladder of factors as in the Furstenberg–Zimmer structure-theorem.) It turns out, though, that the right characteristic factor for (37) is considerably smaller. In their seminal paper [63] Conze and Lesigne identified the characteristic factor for k D 3, now called the Conze–Lesigne factor. As shown in [104], and [211], the characteristic factor for a general k is (isomorphic to) an inverse limit of k-step nilflows. A k-step nilflow is a compact homogeneous space N/ of a k-step nilpotent Lie group N, endowed with its unique left-invariant probability measure, on which T acts via left translation by an element of N. Ergodic properties of nilflows have been studied for some time in ergodic theory, for example in Parry [166]. In this way the problem of L2 -convergence of (37) is reduced to the case when T is a nilflow. In this case one has more: the averages converge pointwise by a result of Leibman [141] (See also Ziegler [210]). There have already been a good many generalizations of (37). Host and Kra [103], Frantzikinakis and Kra [82, 83], and Leibman [141] have proved results which replace

Ergodic Theorems

linear powers of T by polynomial powers. In increasing degrees of generality Conze and Lesigne [63], Frankzikinakis and Kra [81] and Tao [192] have obtained results which replace the maps T; T 2 ; : : : ; T k in (37) by commuting maps T1 ; : : : ; Tk . Bergelson and Leibman [30,31] have obtained results, both positive and negative, in the case of two noncommuting maps. In the direction of pointwise convergence the only general result is the following theorem of Bourgain [48] which asserts pointwise convergence in the case k D 2. Theorem 45 Suppose S and T are powers of a single autoPN f (T n x)g(S n x) morphism R and f ; g 2 L1 . Then N1 nD1 converges a.e. When T is a K-automorphism Derrien and Lesigne [68] have proved that the averages (35) converge pointwise to the product of the integrals, even with polynomial powers of T replacing the linear powers. Gowers [91] has given a new proof of Szemerédi’s theorem by purely finite methods using only harmonic analysis on Z n . His results give better quantitative estimates in the statement of finite versions of Szemerédi’s theorem. Although his proof contains no ergodic theory it is to some extent guided by Furstenberg’s approach. This section would be incomplete without mentioning the spectacular recent result of Green and Tao [92] on primes in arithmetic progression and the subsequent extensions of the Green-Tao theorem due to Tao [191] and Tao and Ziegler [193]. Rates of Convergence There are many results which say in various ways that, in general, there are no estimates for the rate of convergence of the averages A n f in Birkhoff’s theorem. For example there is the following result of Krengel [134]. Theorem 46 Suppose limn!1 c n D 1 and T is any ergodic automorphism of a probability space. Then there R is a bounded measurable f with f d D 0 such that lim sup c n A n f D 1 a.e. See Part 1 of Derriennic [69] for a selection of other results in this direction. In spite of these negative results one can obtain quantitative estimates by reformulating the ergodic theorem in various ways. Bishop [37] proved the following result which is purely finite and constructive in nature and evidently implies the a.e. convergence in Birkhoff’s theorem. If y D (y1 ; : : : ; y n ) is a finite sequence of real numbers and a < b, an upcrossing of y over [a; b] is a minimal integer interval [k; l] [1; n] satisfying y k < a and y l > b.

Theorem 47 Suppose T is an automorphism of the probability space (X; B; ). Let U(n; a; b; f ; x) be the number of upcrossings of the sequence A0 f (x); : : : ; A n f (x) over [a; b]. Then for every n fx : U(n; a; b; f ; x) > kg 

k f k1 : (b  a)k

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Ivanov [107] has obtained the following stronger upcrossing inequality for an arbitrary positive measurable f , which also implies Birkhoff’s theorem. Theorem 48 For any positive measurable f and 0 < a kg 

 a k b

:

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Note the exponential decay and the remarkable fact that the estimate does not depend on f . Ivanov has also obtained the following result (Theorem 23 in [121]) about fluctuations of A n f . An -fluctuation of a real sequence y D (y1 : : : y n ) is a minimal integer interval [k; l] satisfying jy k  y l j  . If f 2 L1 (R) let F( ; f ; x) be the number of -fluctuations of the sequence fA n f (x)g1 nD0 . Theorem 49 1

fx : F( ; f ; x)  kg  C

(log k) 2 k

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where C is a constant depending only on k f k1 / . If f 2 L1 then fx : F( ; f ; x)  kg  AeBk ;

(41)

where A and B are constants depending only on k f k1 / . See Kachurovskii’s survey [121] of results on rates of convergence in the ergodic theorem and also the article of Jones et al. [115] for more results on oscillation type inequalities. Under special assumptions on f and T it is possible to give more precise results on speed of convergence. If the sequence T i f is independent then there is a vast literature in probability theory giving very precise results, for example the central limit theorem and the law of the iterated logarithm (see, for example, [50]). See the surveys of Derrienic [69] (2006) and of Merlevède, Peligrad and Utev [150] for results on the central limit theorem for dynamical systems.

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Ergodic Theorems

Ergodic Theorems for Non-amenable Groups Guivarch [95] was the first to prove an ergodic theorem for a general pair of non-commuting unitary operators. Much work has been done in the last 15 years on mean and pointwise theorems for non-amenable groups. See the extensive survey by Nevo [155]. Here is just one result of Nevo [154] as a sample. Suppose G is a discrete group and fn g is a sequence of probability measures on G. Say that fn g is mean ergodic if for any unitary representation  of G on a Hilbert space H and any x 2 H one P has g2G n (g)(g)x converges in norm to the projection of x on the subspace of -invariant vectors. Say that fn g is pointwise ergodic if for any measure preserving action T of G on a probability space and f 2 L2 one has P g2G  n (g)Tg f converges a.e. to the projection of f on the subspace of T-invariant functions. Theorem 50 Let G be the free group on k generators, k  1. Let  n be the normalized counting measure on the set of elements whose word length (in terms of the generators and their inverses) is n. Let n D (n C nC1 )/2 and P n0 D n1 niD1  i . Then  n and n0 are mean and pointwise ergodic but  n is not mean ergodic. Future Directions This section will be devoted to a few open problems and some questions. Many of these have been suggested by my colleagues acknowledged in the introduction. The topic of convergence of Furstenberg’s multiple averages has seen some remarkable achievements in the last few years and will likely continue to be vital for some time to come. The question of pointwise convergence of multiple averages is completely open, beyond Bourgain’s result (Theorem 45). Even extending Bourgain’s result to three different powers of R or to any two commuting automorphisms S and T would be a very significant achievement. Another natural question is whether one has pointwise convergence of the averages of the sequence 2 f (T n (x))g(T n (x)), which are mean convergent by the result of Furstenberg and Weiss [88] (which is now subsumed in much more general results). A long-standing open problem relating to Akcoglu’s ergodic theorem (Theorem 23) for positive contractions of Lp is whether it can be extended to power-bounded operators T (this means that kTkn is bounded). It is also an open question whether it extends to non-positive contractions, excepting the case p D 2 where Burkholder [52] has shown that it fails. There are many natural questions in the area of subsequence and weighted ergodic theorems. For example,

which of Bourgain’s several pointwise theorems can be extended to L1 ? Are there other natural subsequences of an arithmetic character which have density 0 and for which an ergodic theorem is valid, either mean or pointwise, and in what Lp spaces might such theorems be valid? Are there higher dimensional analogues? Since lacunary sequences are bad, to have any sort of an pointwise theorem l n D log a nC1 must get close to 0 and for simplicity let us assume that lim l n D 0. How fast must the convergence be in order to get an ergodic theorem? Jones and Wierdl [113] have shown that if l n > 1 (log n) 2 C then the pointwise ergodic theorem fails in L2 while Jones, Lacey and Wierdl [116] have shown that an only slightly faster rate permits a sequence which is pointwise good in L2 . How well or badly does the ergodic theorem succeed or fail depending on the rate of convergence of ln ? In particular is there a (slow) rate which still guarantees strong sweeping out? [116] contains some interesting conjectures in this direction. There are also interesting questions concerning the mean and pointwise ergodic theorems for subsequences which are chosen randomly in some sense. See Bourgain [42] and [116] for some results in this direction. Again [116] contains some interesting conjectures along these lines. In a recent paper [32] Bergelson and Leibman prove some very interesting and surprising results about the distribution of generalized polynomials. A generalized polynomial is any function which can be built starting with polynomials in R[x] using the operations of addition, multiplication and taking the greatest integer. As a consequence they derive a generalization of von Neumann’s mean ergodic theorem to averages along generalized polynomial sequences. The following is a special case. Theorem 51 Suppose p is a generalized polynomial taking integer values on the integers and U is a unitary operator P p(i) x is norm convergent for all x 2 on H . Then n1 n1 iD0 U H. This begs the question: does one have pointwise convergence? If so, this would be a far-reaching generalization of Bourgain’s polynomial ergodic theorem. There are also lots of questions concerning the nature of Følner sequences fFn g in an amenable group which give a pointwise theorem. For example Lindenstrauss [145] has shown that in the lamplighter group, a semi-direct product L of Z with i2Z Z/2Z on which Z acts by the shift, there is no sequence satisfying Tempelman’s condition and that any fFn g satisfying the Shulman condition must grow super-exponentially. So, it is natural to ask for slower rates of growth. In particular, in any amenable group is there al-

Ergodic Theorems

ways a sequence fFn g which is pointwise good and grows at most exponentially? Can one do better either in general or in particular groups? Lindenstrauss’s theorem at least guarantees the existence of Følner sequences which are pointwise good in L1 but in particular groups there are often natural sequences L which one hopes might be good. For example in 1 iD1 Z one may take F n to be a cube based at 0 of sidelength ln and dimension dn (that is, all but the first dn co-ordinates are zero), where both sequences increase to 1. What conditions on ln and dn will give a good sequence? Note that no such sequence is regular in Tempelman’s sense. If d n D n then fl n g must be superexponential to ensure Shulman’s condition. Can one do better? What about l n D d n D n? Bibliography 1. Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynam Systems 16(2):207–253 2. Akcoglu M, Jones RL, Rosenblatt JM (2000) The worst sums in ergodic theory. Michigan Math J 47(2):265–285 3. Akcoglu MA (1975) A pointwise ergodic theorem in Lp-spaces. Canad J Math 27(5):1075–1082 4. Akcoglu MA, Chacon RV (1965) A convexity theorem for positive operators. Z Wahrsch Verw Gebiete 3:328–332 (1965) 5. Akcoglu MA, Chacon RV (1970) A local ratio theorem. Canad J Math 22:545–552 6. Akcoglu MA, del Junco A (1975) Convergence of averages of point transformations. Proc Amer Math Soc 49:265–266 7. Akcoglu MA, Kopp PE (1977) Construction of dilations of positive Lp -contractions. Math Z 155(2):119–127 8. Akcoglu MA, Krengel U (1981) Ergodic theorems for superadditive processes. J Reine Angew Math 323:53–67 9. Akcoglu MA, Sucheston L (1978) A ratio ergodic theorem for superadditive processes. Z Wahrsch Verw Gebiete 44(4):269– 278 10. Alaoglu L, Birkhoff G (1939) General ergodic theorems. Proc Nat Acad Sci USA 25:628–630 11. Assani I (2000) Multiple return times theorems for weakly mixing systems. Ann Inst H Poincaré Probab Statist 36(2):153–165 12. Assani I (2003) Wiener–Wintner ergodic theorems. World Scientific Publishing Co. Inc., River Edge, NJ 13. Assani I, Lesigne E, Rudolph D (1995) Wiener–Wintner returntimes ergodic theorem. Israel J Math 92(1–3):375–395 14. Assani I, Buczolich Z, Mauldin RD (2005) An L1 counting problem in ergodic theory. J Anal Math 95:221–241 15. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. With the assistance of Markus L, Massey W and an appendix by Greenberg L Annals of Mathematics Studies, No 53. Princeton University Press, Princeton, NJ 16. Banach S (1993) Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, reprint of the 1932 original 17. Bellow A (1983) On “bad universal” sequences in ergodic theory II. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lecture Notes in Math, vol 1033. Springer, Berlin, pp 74–78

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Ergodic Theory: Basic Examples and Constructions MATTHEW N ICOL1 , KARL PETERSEN2 Department of Mathematics, University of Houston, Houston, USA 2 Department of Mathematics, University of North Carolina, Chapel Hill, USA

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Article Outline Glossary Definition of the Subject Introduction Examples Constructions Future Directions Bibliography Glossary  A transformation T of a measure space (X; B; ) is measure-preserving if (T 1 A) D (A) for all measurable A 2 B.  A measure-preserving transformation (X; B; ; T) is ergodic if T 1 (A) D A (mod ) implies (A) D 0 or (Ac ) D 0 for each measurable set A 2 B.  A measure-preserving transformation (X; B; ; T) of a probability space is weak-mixing if lim n!1 n1 P n1 i iD0 j(T A \ B)  (A)(B)j D 0 for all measurable sets A; B 2 B. A measure-preserving transformation (X; B; ; T) of a probability space is strong-mixing if limn!1  (T n A\B) D (A)(B) for all measurable sets A; B 2 B.  A continuous transformation T of a compact metric space X is uniquely ergodic if there is only one T-invariant Borel probability measure on X. A continous transformation of a topological space X is topologically mixing for any two open sets U; V X there exists N > 0 such that T n (U) \ V ¤ ;, for each n  N.  Suppose (X; B; ) is a probability space. A finite partition P of X is a finite collection of disjoint (mod , i. e., up to sets of measure 0) measurable sets fP1 ; : : : ; Pn g such that X D [Pi (mod ). The entropy of P with reP spect to  is H(P ) D  i (Pi ) ln (Pi ) (other bases are sometimes used for the logarithm).  The metric (or measure-theoretic) entropy of T with respect to P is h (T; P ) D lim n!1 n1 H(P _    _ T nC1 (P )), where P _    _ T nC1 (P ) is the partition of X into sets of points with the same coding with







 









respect to P under T i , i D 0; : : : ; n  1. That is x, y are in the same set of the partition P _    _ T nC1 (P ) if and only if T i (x) and T i (y) lie in the same set of the partition P for i D 0; : : : ; n  1. The metric entropy h (T) of (X; B; ; T) is the supremum of h (T; P ) over all finite measurable partitions P . If T is a continuous transformation of a compact metric space X, then the topological entropy of T is the supremum of the metric entropies h (T), where the supremum is taken over all T-invariant Borel probability measures. A system (X; B; ; T) is loosely Bernoulli if it is isomorphic to the first-return system to a subset of positive measure of an irrational rotation or a (positive or infinite entropy) Bernoulli system. Two systems are spectrally isomorphic if the unitary operators that they induce on their L2 spaces are unitarily equivalent. A smooth dynamical system consists of a differentiable manifold M and a differentiable map f : M ! M. The degree of differentiability may be specified. Two submanifolds S1 , S2 of a manifold M intersect transversely at p 2 M if Tp (S1 ) C Tp (S2 ) D Tp (M). An ( -) small Cr perturbation of a Cr map f of a manifold M is a map g such that dC r ( f ; g) < i. e. the distance between f and g is less than in the Cr topology. A map T of an interval I D [a; b] is piecewise smooth (Ck for k  1) if there is a finite set of points a D x1 < x2 <    < x n D b such that Tj(x i ; x iC1 ) is Ck for each i. The degree of differentiability may be specified. A measure  on a measure space (X; B) is absolutely continuous with respect to a measure  on (X; B) if (A) D 0 implies (A) D 0 for all measurable A 2 B. A Borel measure  on a Riemannian manifold M is absolutely continuous if it is absolutely continuous with respect to the Riemannian volume on M. A measure  on a measure space (X; B) is equivalent to a measure  on (X; B) if  is absolutely continuous with respect to  and  is absolutely continuous with respect to .

Definition of the Subject Measure-preserving systems are a common model of processes which evolve in time and for which the rules governing the time evolution don’t change. For example, in Newtonian mechanics the planets in a solar system undergo motion according to Newton’s laws of motion: the planets

Ergodic Theory: Basic Examples and Constructions

move but the underlying rule governing the planets’ motion remains constant. The model adopted here is to consider the time-evolution as a transformation (either a map in discrete time or a flow in continuous time) on a probability space or more generally a measure space. This is the setting of the subject called ergodic theory. Applications of this point of view include the areas of statistical physics, classical mechanics, number theory, population dynamics, statistics, information theory and economics. The purpose of this chapter is to present a flavor of the diverse range of examples of measure-preserving transformations which have played a role in the development and application of ergodic theory and smooth dynamical systems theory. We also present common constructions involving measurepreserving systems. Such constructions may be considered a way of putting ‘building-block’ dynamical systems together to construct examples or decomposing a complicated system into simple ‘building-blocks’ to understand it better. Introduction In this chapter we collect a brief list of some important examples of measure-preserving dynamical systems, which we denote typically by (X; B; ; T) or (T; X; B; ) or slight variations. These examples have played a formative role in the development of dynamical systems theory, either because they occur often in applications in one guise or another or because they have been useful simple models to understand certain features of dynamical systems. There is a fundamental difference in the dynamical properties of those systems which display hyperbolicity: roughly speaking there is some exponential divergence of nearby orbits under iteration of the transformation. In differentiable systems this is associated with the derivative of the transformation possessing eigenvalues of modulus greater than one on a ‘dynamically significant’ subset of phase space. Hyperbolicity leads to complex dynamical behavior such as positive topological entropy, exponential divergence of nearby orbits (“sensitivity to initial conditions”) often coexisting with a dense set of periodic orbits. If ; are sufficiently regular functions on the phase space X of a hyperbolic measure-preserving transformation (T; X; ), then typically we have fast decay of correlations in the sense that ˇZ ˇ Z Z ˇ ˇ ˇ (T n x) (x) d   d ˇ  Ca(n) d ˇ ˇ X

where a(n) ! 0. If a(n) ! 0 at an exponential rate we say that the system has exponential decay of correlations. A theme in dynamical systems is that the time se-

ries formed by sufficiently regular observations on systems with some degree of hyperbolicity often behave statistically like independent identically distributed random variables. At this point it is appropriate to point out two pervasive differences between the usual probabilistic setting of a stationary stochastic process fX n g and the (smooth) dynamical systems setting of a time series of observations on a measure-preserving system f ı T n g. The most crucial is that for deterministic dynamical systems the time series is usually not an independent process, which is a common assumption in the strictly probabilistic setting. Even if some weak-mixing is assumed in the probabilistic setting it is usually a mixing condition on the  -algebras Fn D  (X1 ; : : : ; X n ) generated by successive random variables, a condition which is not natural (and usually very difficult to check) for dynamical systems. Mixing conditions on dynamical systems are given more naturally in terms of the mixing of the sets of the  -algebra B of the probability space (X; B; ) under the action of T and not by mixing properties of the  -algebras generated by the random variables f ı T n g. The other difference is that in the probabilistic setting, although fX n g satisfy moment conditions, usually no regularity properties, such as the Hölder property or smoothness, are assumed. In contrast in dynamical systems theory the transformation T is often a smooth or piecewise smooth transformation of a Riemannian manifold X and the observation  : X ! R is often assumed continuous or Hölder. The regularity of the observation  turns out to play a crucial role in proving properties such as rates of decay of correlation, central limit theorems and so on. An example of a hyperbolic transformation is an expanding map of the unit interval T(x) D (2x) (where (x) is x modulo the integers). Here the derivative has modulus 2 at all points in phase space. This map preserves Lebesgue measure, has positive topological entropy, Lebesgue almost every point x has a dense orbit and periodic points for the map are dense in [0; 1). Non-hyperbolic systems are of course also an important class of examples, and in contrast to hyperbolic systems they tend to model systems of ‘low complexity’, for example systems displaying quasiperiodic behavior. The simplest non-trivial example is perhaps an irrational rotation of the unit interval [0; 1) given by a map T(x) D (x C ˛), ˛ 2 R n Q. T preserves Lebesgue measure, every point has a dense orbit (there are no periodic orbits), yet the topological entropy is zero and nearby points stay the same distance from each other under iteration under T. There is a natural notion of equivalence for measurepreserving systems. We say that measure-preserving systems (T; X; B; ) and (S; Y; C ; ) are isomorphic if (possi-

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bly after deleting sets of measure 0 from X and Y) there is a one-to-one onto measurable map  : X ! Y with measurable inverse  1 such that  ı T D S ı   a.e. and ( 1 (A)) D (A) for all A 2 C . If X, Y are compact topological spaces we say that T is topologically conjugate to S if there exists a homeomorphism  : X ! Y such that  ı T D S ı . In this case we call  a conjugacy. If  is Cr for some r  1 we will call  a Cr -conjugacy and similarly for other degrees of regularity. We will consider X D [0; 1)(mod 1) as a representation of the unit circle S 1 D fz 2 C : jzj D 1g (under the map x ! e2 i x ) and similarly represent the k-dimensional torus T k D S 1      S 1 (k-times). If the -algebra is clear from the context we will write (T; X; ) instead of (T; X; B; ) when denoting a measure-preserving system. Examples Rigid Rotation of a Compact Group If G is a compact group equipped with Haar measure and a 2 G, then the transformation T(x) D ax preserves Haar measure and is called a rigid rotation of G. If G is abelian and the transformation is ergodic (in this setting transitivity implies ergodicity), then the transformation is uniquely ergodic. Such systems always have zero topological entropy. The simplest example of such a system is a circle rotation. Take X D [0; 1)(mod 1), with T(x) D (x C ˛) where ˛ 2 R : Then T preserves Lebesgue (Haar) measure and is ergodic (in fact uniquely ergodic) if and only if ˛ is irrational. Similarly, the map T(x1 ; : : : ; x k ) D (x1 C ˛1 ; : : : ; x k C ˛ k );

defined by  (k1  1; k2  1; : : : ) D (0; 0; : : : ) if each entry in the Z k i component is k i  1 , while  (k1  1; k2  1; : : : ; k n  1; x1 ; x2 ; : : : ) n times

‚ …„ ƒ D (0; 0; : : : ; 0; x1 C 1; x2 ; x3 ; : : : ) when x1 6D k nC1  1. The map  may be thought of as “add one and carry” and also as mapping each point to its successor in a certain order. See Sect. “Adic Transformations” for generalizations. If each k i D 2 then the system is called the dyadic (or von Neumann-Kakutani) adding machine or 2-odometer. Adding machines give examples of naturally occurring minimal systems of low orbit complexity in the sense that the topological entropy of an adding machine is zero. In fact if f is a continuous map of an interval with zero topological entropy and S is a closed, topologically transitive invariant set without periodic orbits, then the restriction of f to S is topologically conjugate to the dyadic adding machine (Theorem 11.3.13 in [46]). We say a non-empty set  is an attractor for a map T if there is an open set U containing  such that  D \n0 T n (U) (other definitions are found in the literature). The dyadic adding machine is topologically conjugate to the Feigenbaum attractor at the limit point of period doubling bifurcations (see Sect. “Unimodal Maps”). Furthermore, attractors for continuous unimodal maps of the interval are either periodic orbits, transitive cycles of intervals, or Cantor sets on which the dynamics is topologically conjugate to an adding machine [32].

where ˛1 ; : : : ; ˛ k 2 R ; preserves k-dimensional Lebesgue (Haar) measure and is ergodic (uniquely ergodic) if and only if there are no integers m1 ; : : : ; m k , not all 0, which satisfy m1 ˛1 C    C m k ˛ k 2 Z. Adding Machines Let fk i g i2N be a sequence of integers with k i  2. Equip each cyclic group Z k i with the discrete topology and form Q the product space ˙ D 1 iD1 Z k i equipped with the product topology. An adding machine corresponding to the seQ quence fk i g i2N is the topological space ˙ D 1 iD1 Z k i together with the map : ˙ !˙

Interval Exchange Maps A map T : [0; 1] ! [0; 1] is an interval exchange transformation if it is defined in the following way. Suppose that  is a permutation of f1; : : : ; ng and l i > 0, i D 1; : : : ; n, is a sequence of subintervals of I (open or closed) with P i l i D 1. Define ti by l i D t i  t i1 with t0 D 0. Suppose also that  is an n-vector with entries ˙1. T is defined by sending the interval t i1  x < t i1 of length li to the interval X X l( j)  x < l( j) ( j)<(i)

( j)>(i)

with orientation preserved if the ith entry of  is C1 and orientation reversed if the ith entry of  is -1. Thus on each

Ergodic Theory: Basic Examples and Constructions

interval li , T has the form T(x) D  i x Ca i , where  i is ˙1. If  i D 1 for each i, the transformation is called orientation preserving. The transformation T has finitely many discontinuities (at the endpoints of each li ), and modulo this set of discontinuities is smooth. T is also invertible (neglecting the finite set of discontinuities) and preserves Lebesgue measure. These maps have zero topological entropy and arise naturally in studies of polygonal billiards and more generally area-preserving flows. There are examples of minimal but non-ergodic interval exchange maps [56,69]. Full Shifts and Shifts of Finite Type Given a finite set (or alphabet) A D f0; : : : ; d  1g,