Perspectives of nonlinear dynamics Volume 2
VOLUME 2
Perspectives of nonlinear dynamics E. ATLEE JACKSON Professor of Physics, University of Illinois at Urbana-Champaign
AMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521354585
© Cambridge University Press 1990
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1990 First paperback edition (with corrections) 1991 Reprinted 1993, 1994 Re-issued in this digitally printed version 2008
A catalogue record for this publication is available from the British Library ISBN 978-0-521-35458-5 hardback ISBN 978-0-521-42633-6 paperback
-7;
1
Contents of Volume 2
Preface Acknowledgements
6 Models based on second order difference equations 6.1 Some origins of maps in R2: Delayed and coupled logistic maps; Poincare surface of section in extended phase space (R2 x I); area-preserving maps; nonconserva-
tive vs conservative maps; Levinson-Smith relaxation oscillator; Henon and Heiles Hamiltonian 6.2 Rotation and winding numbers: Maps and flows; knots, algebraic constants of the motion
xv xviii 1
1
9
6.3 The Cartwright-Littlewood, Levinson and Levi analyses: The extraordinary family of solutions, KO, of the forced self-exciting oscillator; equivalence to Bernoulli sequences; Levi's extensions
16
6.4 Some abstract nonconservative maps in R2: Henon's map; strange attractor; contracting map; geometrically wild vs dynamically-wild sets; Lyapunov exponents; period-three does not imply chaos in R2; a fractal boundary between basins
of attraction; Julia and Mandelbrot sets; Newton map
22
6.5 The standard map; twist maps: Ball on a vibrating plate; the microtron accelerator;
harmonic lattice in a periodic potential; toroidal magnetic fields; twist maps; characteristic multipliers 6.6 'Near-integrable' systems: Poincare's last geometric theorem; hyperbolic and
33
elliptic fixed point pairs; stable and unstable manifolds; homoclinic and heteroclinic points; KAM surfaces; Poincare's chaotic tangle; Bernoulli sequence
41
6.7 The breakup of KAM curves: Small divisors, irrational rotation numbers; computer study; continued fraction representation; the standard map and the golden mean
57
6.8 Physical regularity in mathematical chaos: Chaotic magnetic field lines and differentiable magnetic fields 6.9 Chirikov's resonance-overlap criterion: A periodically kicked oscillator example 6.10 The numerical Poincare map and discontinuous dynamics: Henon's integration method
65 67 72
6.11 The Henon-Heiles and Toda-Hamiltonian systems: Poincare maps for a nonintegrable and integrable system
74
Contents of Volume 2
viii
6.12 Abstract area-preserving maps on R2 and T2: Henon's map; involutive maps; Arnold's cat map; mixing 6.13 Maps of sets: The folds and kinks of a periodically forced conservative oscillator 6.14 Maps on a lattice: Rannou's study of the standard map and generalization on a N x N lattice; the random-map ensemble; Arnold's cat map on a lattice; short Poincare recurrence; ghosts
80 90
93
6.15 Dynamic entropies and information production: The shuffling of partitions in phase
space; Kolmogorov-Sinai, and topological dynamic entropies; informational interpretation 6.16 Epilogue: order-order, order-chaos, chaos-chaos in the house! 7 Models based on third order differential systems 7.1 Linear third order equations: Characterization of fixed points; stable, unstable, and center manifolds; various representations of flows 7.2 Nonlinear flows: Center cycles, saddle cycles; Poincare's first return map; local
Mobius band manifolds; some `interacting' flows from neighboring saddlenodes, or spiral nodes 7.3 The Lorenz model: Historical origin; bifurcation of the fixed points as a function of r; contracting and global attracting character 7.4 Lorenz chaotic dynamics: The Lorenz `map'; the Lorenz fractal `mask', a strange
attractor 7.5 A 'Lorenz-dynamic'.fuid system: A fluid flow in circular tubes which obeys the Lorenz equations 7.6 Dynamo dynamics: The amazing dynamics of the Earth's magnetic field; a physically unrelated simple disk dynamo with similar chaotic dynamics (a Lorenz system)
101
118
125 126
132 138 145
151
155
7.7 The Lorenz homoclinic and heteroclinic bifurcations: The global topological properties of the flow, for r above the first homoclinic bifurcation, and below the first heteroclinic bifurcation - a highly convoluted picture; `preturbulence'. 7.8 The Lorenz-Hopf bifurcation: The subcritical bifurcation
162 177
7.9 Lorenz dynamics for various parameter values: Stable limit cycles subharmonic (saddle-node) bifurcations; Mobius strips and knotted limit cycles; intermittencies; bistabilities; and much chaos! 7.10 The Lyapunov exponents: Definition and methods of obtaining them; application to the Lorenz system, and bistability
178
190
7.11 Rossler's models: The `lifting' approach to modeling; model R1, Lyapunov exponents, Poincare maps, Lorenz `maps' and Cantor set; the `walking stick' folding and second Cantor set; subharmonic bifurcations, largest Lyapunov exponent; `funnel' attractor; phase coherence; the `Dali' limit cycle; model R2, bistability chaos; the taffey-machine-on-a-lazy-susan flow
98
7.12 Lyapunov exponents and the dimension of a strange attractor: A heuristic discussion of possible relations; the Kaplan-Yorke conjecture 7.13 Open systems - chemical oscillations: The Belousov-Zhabotinskii discoveries; the Field-Noyes `Oregonator' equations; relaxation oscillations
217 222
Contents of Volume 2 8 'Moderate-order' systems
8.1 Linear systems: Poincare's variational equations; near fixed points, asymptotic properties (Lyapunov exponents), with periodic and quasi-periodic coefficients; lattice normal modes and Schrodinger's solution 8.2 Turing's linear chemical morphogenesis system: Instabilities in cellular chains which are coupled diffusively 8.3 'Integrable' Hamiltonian systems: Motion on an n-dimensional torus; separation of adjacent states 8.4 The Kolmogrov-Arnold-Moser theorem: `Near-integrable' systems 8.5 Poincare's and Fermi's theorems; Arnold diffusion. 8.6 The Fermi-Pasta-Ulam phenomenon and equipartitioning: The nonequipartitioning of energy; early example of'synergetics'; equipartitioning vis-a-vis ergodicity; influence on irreversibility (lattice heat conduction) 8.7 Molecular models: Polynomial potentials; Toda's exponential potential 8.8 Toda's solitary waves in a lattice: Analytic solutions for one and two `conserved' compression pulses; cnoidal waves 8.9 The dynamics of various Toda lattices: The Ford-Stoddard-Turner numerical prediction of the integrability of equal-mass Toda lattices; the Flaschka and
ix 231
232
240 246 253 256
259 268 274
Henon integrals; `physical' significance? diatomic Toda lattices (1 >1 ml/m2 1> 0);
influence on chaotic behavior and lattice heat conduction
81
8.10 The Painleve property and integrability conjecture: Kovalevskaya's use of complex time to search for integrability 8.11 Chemical oscillations and dissipative open-system structures: The 'Brusselator' and diffusively coupled Brusselators (Turing structures) 8.12 Smale's analysis of Turing's morphogenic system: Generating `life' from `dead' components, using diffusion; global aspects 8.13 Embedding the dynamics of high-order dissipative systems in lower dimensional R": The method of Takens and Crutchfield, Farmer, Packard, and Shaw; torus-knot
example; application to chemical oscillations and Taylor vortices 8.14 Some dynamics of living systems: Definition problems; Eigen and Schuster's hypercycle quasi-species, error catastrophe; replicator equations; mean fitness;
294 304 311
317
general Lotka-Volterra equation; Hofbauer's theorem, topological orbital equivalence; Smale's observation 8.15 Epilogue: open systems; open sesame! 9 Solitaires: solitons and nonsolitons
9.1 The continuum limit of lattices and 'solitaire' solutions 9.2 Riemann invariants and the Korteweg-de Vries (KdV) equation 9.3 A comparison of the Burgers and KdV equations: Dissipation vs dispersion 9.4 The exact solution of Burgers equation - The Hopf-Cole transformation 9.5 A brief history leading to the inverse scattering transform (1ST): Zabusky and Kruskal's discovery of'solitons'; conservation laws; Miura's transformation; the 'Schrodinger equation'
30 343 348
350 358
361 365
367
x
Contents of Volume 2
9.6 The general solution of the KdV equation: Gardner-Green-Kruskal-Miura analysis; the Gel'fand-Levitan theorem; the inverse scattering transform 9.7 Pure soliton solutions: KdV equation; the Landau-Lifshitz subset 9.8 The Lax formulation: The KdV example 9.9 The sine-Gordon equation: Kinks; topological solitons; breather modes 9.10 Hirota's 'direct method' in soliton theory: Hirota's binary operators; nonlinear transformations (types A, B, and C); bilinear differential equation and special exact solution 9.11 The AKNS formulation of the 1ST: Ablowitz-Kaup-Newell-Segur extension of Zaharov and Shabat's method; connection of IST general solutions with the sine-Gordon, sinh-Gordon, nonlinear Schrodinger, modified KdV, and Dym's equations
373 381 387 390
398
406
9.12 Some Backlund transformations between difference equations: Backlund transform-
ation; integrability condition; examples 9.13 Invariant Backlund transformations: Free parameter; nonlinear superposition; Riccati equation
410 413
9.14 Infinite number of conservation laws: Relationship to invariant Backlund transformation 9.15 Onward: Higher dimensions; resonant interactions 10 Coupled maps (CM) and cellular automata (CA) 10.1 An overview: Lagrangian and Eulerian models, with continuous and discrete variables and functions 10.2 Some coupled maps (CM): Diffusive, and other coupling of cells obeying logisticmap dynamics; multiple-periodic regions; coexistence of many periods; 'semiperiodic turbulence'; spatial-temporal intermittency; inhomogeneous CM 10.3 Coupled lattice maps (CLM = CA): General family; chemical turbulence model of Oono and Kohmoto; `turbulence', `solitons' and periodic space-time- patterns; nonperiodic, excitable cells, diffusively coupled; bistable cells 10.4 General cellular automata (CA): Von Neumann's question; Ulam's suggestion; dynamics defined 10.5 'Legal' cellular automata: Quiescent and symmetry conditions 10.6 A general association for legal CA: Some possible physical relations to CA rules, states and configurations 10.7 Simple examples: 'Self-reproduction' 10.8 Neighborhood configurations and dynamic rules: Rule number; `totalistic' CA 10.9 Several classifications of CA dynamic properties: Four qualitative categories of dynamics 10.10 Entropies of one-dimensional CA: Some possibilities and shortcomings of entropy measurements of `chaos' or `turbulence' 10.11 Particle-like dynamics from partial CA rules: Colliding `particles' with and without delays; oscillating `molecule' 10.12 Two-dimensional CA: Von Neumann neighborhoods; Fredkin's 'selfreproducing' rule
418 420
427
427
431
445
454 457 459 460 462
464 473 478
482
Contents of Volume 2 10.13 Garden-of-Eden configurations: Configurations which cannot dynamical arise; Moore's theorem 10.14 J.H. Conway's 'Life': Moore neighborhood; snakes, ponds, blinkers, beehives, barges, barberpoles, eaters, gliders and glider guns 10.15 Excitable medium: Quiescent, excited, and refractory states 10.16 Invertible CA and physical dynamics: Invertible vis-d-vis reversible dynamics; the description of nature, how can it best be achieved? Appendix: von Neumann's questions Epilogue: `Understanding' complex systems: Order; organization; Endnote-models, causality, irreversibility
xi
484 487 491
493 499 505
Appendixes
J On the Cartwright-Littlewood and Levinson studies of the forced relaxation oscillator K Smale's horseshoe map L Notes on the Kolmogorov-Arnold-Moser theorem
529 539 543
Bibliography
553
References added at 1991 reprinting Cumulative index (Volumes 1 and 2)
62.1
623
Contents of Volume 1
Concepts related to nonlinear dynamics: a brief historical outline
I In the beginning... ... there was Poincare 1.2 What are 'nonlinear phenomena'?: Projections, models, and some relationships between linear and nonlinear differential equations 1.3 Two myths: A linear and analytic myth 1.4 Remarks on modeling: Pure mathematics vis a vis `empirical' mathematics 1.5 The ordering and organization of ideas: Dynamic dimensions, continuous and discrete variables; the analytic, qualitative, computational, and experimental approaches to nonlinear dynamics; sneaking up on the complexity of dynamics 1.6 Some thoughts: Albert Einstein, Victor Hugo, A.B. Pippard, Richard Feynman, Henri Poincare 1.1
2 A potpourri of basic concepts 2.1 Dynamic equations; topological orbital equivalence:
z = F(x, t; c)
(xER", ceRk)
Autonomous and nonautonomous systems; phase space (x); control parameter space (c); Hamiltonian systems; gradient systems; phase portraits; topological orbital equivalence; manifolds
2.2 Existence, uniqueness, and constants of the motion: Lipschitz condition; universal differential equations; Winter's condition; general solution; singular (fixed) points; dynamics viewed as diffeomorphism; constants (integrals) of the motion, numbers and types; the implicit function theorem; obtaining time-independent integrals; isolating integrals 2.3 Types ofstabilities: Lyapunov, Poincare, Laplace; Lyapunov exponents; global stability; the Lyapunov function 2.4 Integral invariants: The Poincare integral invariants; generalized Liouville theorem; unbounded solutions, Liouville theorem on integral manifolds 2.5 More abstract dynamic systems: Classic dynamic systems; flows and orbits in phase
space; Poincare's recurrence theorems; the Poincare map; first-return map; areapreserving maps; maps and difference equations
xk+1 =F(xk;c) (k= 1,2,...;xeRm,ceR");
Contents of Volume 1
xiii
2.6 Dimensions and measures of sets: Capacity and information dimensions; self-similar sets; Cantor sets; fractal structures; thin and fat Cantor sets; measures vs dimension. Some physical 'fractals'
3 First order differential systems (n = 1) 3.1 Selected aspects: Classic examples; Riccati equations and linear second order differential
equations; the logistic and Landau equations; nonlinear superpositions; integrating factors; singular solutions and caustics 3.2 Control space effects: Simple bifurcations: definition of a bifurcation; `dynamic phase transitions'; fixed point bifurcations and the implicit function theorem, singular points, double points, the exchange of stability, Euler strut, imperfect bifurcation; hysteresis, a discharge tube; simple laser model 3.3 Structural stability, gradient systems and elementary catastrophe sets 3.4 Thom's `universal unfolding' and general theorem (for k 5 5); Brief summary 3.5 Catastrophe machines: Poston's (k = 1); Benjamin's (k = 1); Zeeman's (k = 2) 3.6 Some of Rene Thom's perspectives 4 Models based on first order difference equations 4.1 General considerations: x1 = F(x"; c) ('mappings') 4.2 Two-to-one maps: the logistic map Possible connections with differential equations, and differences. The logistic map, x"+1
= cx"(1 - x"); Schwarzian maps; tent maps; fixed points, periodic and eventually periodic points; criterion for stable periodic points; sequence of period 2" bifurcations; an attracting Cantor set; superstable cycles 4.3 Universal sequences and scalings: The U-sequence of Metropolis, Stein, and Stein; qualitative `universality'; Feigenbaum's quantitative `universality' and scaling; aperiodic solutions, reverse bifurcation; Sharkovsky's theorem 4.4 Tangent bifurcations, intermittencies: Windows, microcosms, crisis 4.5 Characterizing 'deterministic chaos': Partitioning phase space; correspondence with Bernoulli sequences; Li-Yorke characterization of chaos; other characterizations 4.6 Lyapunov exponent: sensitivity to initial conditions vs attractors; a strange attractor concept 4.7 The dimensions of 'near self-similar' Cantor sets 4.8 Invariant measures, mixing, and ergodicity: The mixed drinks of Arnold, Avez, and Halmos 4.9 The circle map: Model of coupled oscillators; rotation number, entrainment, Arnold `Tongues'; chaotic region 4.10 The `suspension' of a tent map 4.11 Mathematics, computations, and empirical sciences: THE FINITE vs THE INFINITE; pseudo-orbits, #-shadowing; discrete logistic map; where is the chaos? 5 Second order differential systems (n = 2) 5.1 The phase plane: fixed (singular) points: center, nodes, focus, saddle point, classification of
(linear) flows near fixed points; hyperbolic point, Lyapunov theorem, nonlinear modifications, global analysis, limit cycle, separatrix
Contents of Volume 1
xiv
5.2 Integrating factors: A few examples 5.3 Poincare index of a curve in a vector field: Brouwer's fixed point theorem 5.4 The pendulum and polynomial oscillators: Elliptic functions, frequency shift, heteroclinic and homoclinic orbits 5.5 The averaging method of Krylov-Bogoliubov-Mitropolsky (KBM): autonomous systems; eliminating secular terms, the Duffing equation (passive oscillator)
5.6 The Lotka-Volterra equations: Predator-prey equations, structurally unstable; one generalization; Lyapunov function 5.7 The Rayleigh and van der Pol equations: Andronov-Hopf bifurcation; self-exciting oscillator; limit cycles; the Poincare-Bendixson theorem 5.8 Singular perturbation problems: Levinson's model; relaxation oscillations 5.9 Global bifurcations: Saddle connection; homoclinic orbit 5.10 Periodically forced passive oscillators: A cusp catastrophe resonance and hysteresis effect 5.11 Harmonic excitations: Ultraharmonic, subharmonic, and ultrasubharmonic excitations 5.12 Averaging method for nonautonomous system (KBM) 5.13 Forced van der Pol equation -frequency entrainment: Van der Pol variables, heterodyning. Entrainments of the heat, piano strings, and physiological circadean pacemakers 5.14 Nonperturbative forced oscillators: Extended phase space; Poincare's first return (stroboscopic) map; inverted and non-harmonic Duffing equations - chaotic motion; Kneading action; the Cartwright-Littlewood, Levinson, and Levi studies 5.15 Experimental Poincare (stroboscopic) maps of forced passive oscillators 5.16 Epilogue Appendixes
A brief glossary of mathematical terms A Notations B Notes on topology, dimensions, measures, embeddings, and homotopy C Integral invariants D The Schwarzian derivative E The digraph method F Elliptic integrals and elliptic functions G The Poincare-Bendixson theorem and Birkhoff's limit sets H The fourth-order Runge-Kutta integration method I The Stoker-Haag model of relaxation oscillations Bibliography Index to Volume I
Preface Ah, but a man's reach should exceed his grasp, or what's a heaven for? Robert Browning
This book represents an attempt to give an introductory presentation of a variety of complementary methods and viewpoints that can be used in the study of a fairly broad spectrum of nonlinear dynamic systems. The skeleton of this organization consists of the three perspectives afforded by classic and some modern analytic methods, together with topological and other global viewpoints introduced by the genius of Poincare around the turn of the century and, finally, the computational and heuristic opportunities arising from modern computers, as partially foreseen by von Neumann in 1946. On a more profound level, the interplay between computational
concepts and physical theories, and what they may teach each other, has become a subject of growing interest since von Neumann's and Ulam's introduction of cellular automata. Filling out this skeleton are different viewpoints which stimulate other perspectives, such as: bifurcation concepts; the beautiful, if often ethereal, visions of catastrophe theory as conceived by Thom and practiced, with varying degrees of abandon, by his disciples; a variety of mapping concepts, dating again back to Poincare, ultimately giving rise to such abstract perspectives as symbolic dynamics; the mind-stretching world of chaotic dynamics, uncovered first by Poincare's imagination, and abstractly studied by Birkhoff; `curious' attractors, first discovered in the solutions of `physical' differential equations by Cartwright and Littlewood and by Levinson in the 1940s; the subsequent discovery of numerous physical strange attractors; the complementary situation of persisting oscillatory, nonchaotic dynamics (Fermi-Pasta-Ulam and Kolmogorov, in the 1950s) and its relationship to chaos in conservative systems (the Kolmogorov-Arnold-Moser theorem); the coherent dynamics of integrable systems and solitary waves (Russell, 1844; Korteweg and deVries, 1895); the remarkable joint discovery of the inverse scattering transforms, and its yet mystical connection with other methods of solution, such as the Painleve property; the development (morphogenesis) and dynamics of spatial structures in reactive chemical and biological systems, initiated by Turing (1952); the rudimentary dynamic modeling of `living' and `cognitive'
xvi
Preface
systems. A more complete outline of the development of dynamic concepts can be found in the Historical Outline. It is clear that any book of the present length can only present such a variety of concepts at a rather superficial level. Moreover, as the opening quotation suggests, my grasp frequently falls short of my reach. Nonetheless I think there is a need to introduce students and researchers to this broad spectrum of concepts, if only to make them aware that such ideas exist and can be useful, hopefully to stimulate their imaginations to more profound studies and applications. Indeed the raison d'etre for
this book is to afford an introductory access to concepts which will stimulate imaginations in the future. Since one of the significant impediments to the introduction of these concepts to nonmathematicians (the writer, and the intended readers of this book) is the technical jargon, which tends to obscure many presentations in the mathematical literature, an attempt has been made to lower this barrier without losing too much precision. The necessary technical terms needed for clarity, and to make it possible to read the general literature with some degree of ease, have been collected in a simplified glossary (Appendix A), or can be found in the index. However, it should be made abundantly clear that this book is in no sense a pure mathematical text, despite the mathematical
terms which are retained for clarity. The topics which are discussed have been selected with the hope that they will prove useful (perhaps in the distant future!) in analyzing some dynamic effect arising in models which, hopefully, bears some relationship to observed phenomena. Indeed the interrelation between models and observed phenomena, and the degree to which models engender an `understanding' of the phenomena, is becoming a more serious question with our growing understanding of the richness and variety of the dynamics described by even `simple' models. This is illustrated by the surprising fact that any
complicated (prescribed) dynamics, x(t), can be `modeled' by a solution of one `universal' low order differential equation, to any prescribed accuracy. Clearly such a `model' says nothing about the underlying physical causes in the real world, and is an extreme example of the precaution which we need to take in ascribing physical or observational significance to mathematical concepts, regardless of how `beautiful' they appear to be. This ever-increasing encroachment of nonempirical mathematical concepts into empirical sciences is a phenomenon which should be recognized and at least viewed with caution. Therefore the bias of this book is strongly on the `pragmatic' side of exploring mathematical concepts which are likely to be useful in describing observable phenomena, rather than `pure' mathematics. As examples, such beautiful concepts as
Cantor sets, fractals, and asymptotic features need to be examined in physical (empirical) contexts, which is often a challenging process.
Preface
xvii
In keeping with the introductory level of this book, the exercises are generally intended to be simple, frequently requiring only a few minutes' thought. They are discussed in the comments at the end of each chapter. Those who wish to read more about the mathematical refinements and proofs will find discussions in the cited literature. While this list is not encyclopedic, it is at least representative of the rapidly growing literature in this area. Urbana, 1988
Acknowledgements
I am indeed indebted to many people for their contribution to my knowledge and awareness in this area, not the least being my students over the past fifteen years who have endured my groping presentations of new ideas. Many colleagues have suggested physical and mathematical models which I have found fruitful, or have attempted to explain concepts and to correct numerous misconceptions on my part. Unfortunately they were not always successful, but their patience and generosity is warmly appreciated. I have been particularly fortunate to have had the opportunity to interact with a number of such knowledgeable and generous people. In particular I would like to acknowledge my indebtedness to: F. Albrecht, A. Bondeson, S.J. Chang, J. Dawson, J. Ford, G. Francis, J. Greene, Y. Ichikawa, M. Kruskal, D. Noid, Y. Oono, N. Packard, J. Palmore, J. Pasta, R.E. Peierls, M. Raether, L. Rubel, R. Shaw, R. Schult, M. Toda, N. Wax, M.P.H. Weenink, H. Wilhelmsson, J. Wright, and N. Zabusky as well as a number of industrious students, among them K. Miura, R. Martin, A. Mistriotis, P. Nakroshis, F. Nori, S. Puri, and M. Zimmer. Finally, the typing of this manuscript benefited significantly from the dedication, fast fingers, and keen eyes of Mary Ostendorf, to whom I am sincerely indebted. Special thanks are due to R. Schult for detecting a number of typographical errors in the first printing.
Models based on second order difference equations 6.1 Some origins of maps in R2 Second order difference equations, or maps in R2, xn+ 1 = F(xn, yn);
yn+ 1 = G(xn, yn)
(6.1.1)
can arise in a variety of ways. Historically such maps arose in two very different contexts, both due to Poincare. We will discuss these in some detail shortly. More recent sources of the map (6.1.1) occur in the area of biology and ecology. An example of this is the delayed logistic equation
Xn+1 =cxn(1 -xn-1)
(6.1.2)
(Smith 1971; Aronson et al. 1982). If x,, is again interpreted as the population of the nth generation of some species, (6.1.2) represents the case when the negative influence of competition or toxicity is due to the past second generation rather than the last generation, as in the logistic equation (4.2.1). (6.1.2) can of course easily be put into the form (6.1.1) xn+1
=cXn(1 -y,,);
yn+1 =Xn.
Another form of (6.1.1) comes from the interaction between two `cells', each of
which has identical, or similar, forms of dynamics. A case of some interest is represented by 'diffusively' coupled logistic systems, such as xn+ 1 = cxn(1 - xn) + D(yn - xn) (6.1.3) yn+1 =cyn(1 -yn)+D(xn-yn). In this case two identical systems (x, y), both have the growth parameter c, and a `diffusion' constant 1 > D > 0 (see Fig. 6.1). The `in phase' dynamics xn = yn is the
same as the logistic map, and not of present interest. However, if xn y,,, the larger population/concentration tends to migrate to the cell with lower population/concentration, thereby introducing a spatial aspect into the dynamics. The idea of considering diffusive coupling between dynamic spatial cells, was due to Turing (1952), and will be discussed in some detail in Chapter 8.
Models based on second order difference equations
2
D-membrane
Fig. 6.1
Exercise 6.1 Consider the special case c = 2D + 1 in (6.1.3), so that 1 < c < 3. For these values of c, the uncoupled system would simply be period-one. (a) Obtain the transformation (xn, y,,) -+ (un, vn) such that Un+1
-Un+D(Un+vn-Un);
vn+1 =vn+D(U,+lln-Un)
and determine two fixed points in the (u, v)-plane. Call them X 1 and X2. (b) Determine the stability of X 1 and X2 as a function of D. (c) What happens at the first bifurcation above D = 0?
(d) Write a simple computer program and determine numerically the next bifurcation point, D2. What happens at D2 in the (u, v)-plane? Does this look like a discrete version of a bifurcation discussed in Chapter 5? (e) Explore higher bifurcations, Dk, using graphics in the (u, v)-plane and plotting each (un, vn) for many values of n.
Maps in R2 can also arise from flows in higher dimensional spaces. Poincare introduced several such maps, for very different systems. If a system has a two-dimensional phase space, and is acted upon by a periodic `force', with period T, x = F(x, t);
F(x, t + T) - F(x, t),
(xeR2)
(6.1.4)
then Poincare introduced the concept of an extended phase space, discussed in the last chapter. If the dynamics in R 2 is limited to some disk D = {x: I x I < M }, then the extended phase space is simply the solid torus (D x S 1) (see Fig. 6.2). The time axis has the period T of F(x, t) in (6.1.4). The important feature of this phase Fig. 6.2 Time around equals period, T.
Some origins of maps in R2
3
space is that two distinct orbits do not intersect, just as in the case of the phase space (x, z) for autonomous systems (see Exercise 5.40). The map (6.1.1) is generated from (6.1.4), by introducing an arbitrary disk
(Poincare surface) transverse to the t-axis, associated with some sequence of times t" = to + nT. If we set
x"=x1(t"),
Y"x2(t"),
we obtain the map (6.1.1). As mentioned before, this Poincare map is equivalent to turning on a stroboscopic light every T seconds and recording the phase point (x1(tn), x2(tn))
We can now usefully divide the general maps (6.1.1) into two important categories,
which we will call conservative maps and nonconservative maps. This distinction depends on whether the map does or does not conserve the area of regions in the plane R2. Specifically, we make the following distinction: a(Xn+1,yn+1 )= aF/ax"
aF/ay"
± 1, conservative
aG/ax"
aG/ay"
otherwise, nonconservative.
(6.1.5)
a(x", Y")
If this determinant is + 1, the map is also orientation preserving. The significance of this distinction, for the case of the Poincare map, is made clear by recalling the generalized Liouville theorem (Chapter 2). If the system (6.1.4) satisfies the generalized Liouville theorem
x=F(x,t)
and
V (pF)=0
(xeR2)
for some p(x) > 0, then the motion in the extended phase space
xi=Fi(x1,x2,x3),
x3= 1
also satisfies V (p F) = 0 (xER3) a
a
ax, (pF) + ax 2 (pF) + ax3
(p(x1x2).1) = 0-
It is obvious from the stroboscopic point of view that this Poincare map in the extended phase space preserves areas. That is, 10,p dx 1 dx2 = faZ p dx 1 dx2, if 1Z2 is the region mapped from C 1(522 = P(1) as shown in Fig. 6.3. In most cases, and specifically in the Poincare map, (6.1.5), the function p is simply taken to be unity. However, see the magnetic example, Section 6.5. Nonconservative systems may have a map which has an attractive set of points, or attractor (Fig. 6.4(a)). The simplest attractor is a single point, so that P"x -+ z (n -+ co), as in the case of a Poincare map of a simple limit cycle (in the extended phase space). More generally there may be some finite set of such points, again representing a limit
cycle, but with a period differing from the period, T, used in Poincare's extended
4
Models based on second order difference equations
Fig. 6.3
x2
Fig. 6.4 (a) Nonconservative map. (b) Area-preserving conservative maps. (a)
Attracting set (inside)
(b)
phase space map. We will discuss these periodic solutions in the next section. Moreover, as we will see in Sections 3 and 4, there are much more interesting types
of attractors, which are now called strange attractors, that have an attracting set containing an infinite number of points with very strange properties. On the other hand, the conservative systems such as area-preserving Poincare maps, cannot have attracting sets of points (attractors) (Fig. 6.4(b)). This does not mean, however, that they cannot have their own very strange sets of points ('chaotic' sets), as Poincare appreciated many years ago. We will consider briefly in the remainder of this section how specific flows in phase spaces can be associated with other nonconservative or conservative maps. We begin with some nonconservative systems. Nonconservative maps typically represent the dynamics of systems which have either damping, or self-excitation effects. One of the basic questions of interest is
whether such a system has a periodic solution. The Poincare map, together with Brouwer's fixed point theorem can sometimes be used to give an affirmative answer to this question.
Some origins of maps in R2
5
Fig. 6.5
The idea is that, if we can show that the Poincare map P:R2 --> R2 maps some region 0 into itself, then Brouwer's fixed point theorem shows that there is at least one point in f which is invariant under this map, P:(x°,x2)-*(x°,x2) (Fig. 6.5). This means that there is a solution of F(x, t) such that x(O) = x(T). Moreover, at the time (t + T) the equation x = F(x, t + T) is the same as x = F(x, t), so that the initial state x(O) = (x°, x2) must repeat at the time 2T, 3T, etc. In other words, this proves that a periodic solution does exist, and it has the same period as F(x, t). This approach therefore rests on establishing conditions under which F maps some region 1 into itself. An important example of this is a theorem due to Levinson and Smith (1942; see also Cesari, 1963 p. 171). This concerns the generalized Lienard equation
x + f (x, x)x + g(x) = h(t)
(L)
where h(t) is a bounded continuous function. The objective is to find conditions on the functions f (x, x) and g(x) such that this equation represents a relaxation oscillator, thereby generalizing the van der Pol oscillator. Assume there are constants a, m, M > 0 in (L) such that:
(a) f(x,x)> m if both Ixl %a, Ixl > a, (b) f (x, x) > - M for all (x, x), (c) xg(x) > 0 for all I x I >, a,
(L* )
(d) l g(x) I - oo as I x I - oo, (e) If G(x) = f og(u) du, then g(x)/G(x) =- 0 as I x I --+ oo.
Levinson and Smith then proved that: There is a closed curve C in the (x, x)-plane, enclosing the square a >, I x 1, 1x 1, such
that the solutions of (L) crossing C at any time t pass from the exterior to the interior of C. Moreover, through every point (x, z) sufficiently remote from the origin there passes such a curve C (Fig. 6.6). Therefore, if (L) satisfies the conditions (L*), there is a region in phase space, inside the curve C which maps into itself under a Poincare map. Hence, by Brouwer's fixed
Models based on second order difference equations
6 Fig. 6.6
point theorem, there is a point (xo,.z0) in this region which maps into itself P:(xo, zo) -> (xo, zo). If, in addition, h(t + T) = h(t) this fixed point will also be a fixed point for every subsequent map. This proves that:
If (L) satisfies (L*) and h(t + T) = h(t), then (L) has a periodic solution with period T.
Exercise 6.2 Does the forced Duffing equation, x + 2µz + coox + ex3 = A cos (S2t) or
the forced van der Pol equation x - A(1 - x2)z + coax =A cos (i2t) have a periodic solution for all (µ, coo, e, A, and S2)?
We next consider some conservative systems, namely the Poincare map of Hamiltonian systems of two degrees of freedom
Pi = - aHlagi,
4, = aH/api
(i = 1, 2).
(6.1.6)
Since H(p, q) is a time-independent integral of the motion, the solutions are on a three-dimensional manifold, defined by H(pi, p2, q1, q2) = Ko =_ E
(6.1.7)
where E is a constant. Furthermore, on this manifold, it may be possible to find a two-dimensional surface, S, through which essentially all trajectories repeatedly pass. In this case a Poincare map, P: S - S, can be defined and will be of the form (6.1.1). As discussed in Chapter 2, S is called a Poincare surface of section, whose general properties were analyzed in detail by Birkhoff (1917, 1927). In general we can similarly define a surface of section of dimension (n - 1) in any n-dimensional phase space, but our interest presently concerns (6.1.7), where n = 3. An explicit example of such a map was treated in Exercises 2.26, 2.27, and 2.28. To illustrate this map, consider the Hamiltonian
H=i(pi+p2+q2+q2)+q g2-3g3= 1(p2+p2)+V(g1,g2)
(6.1.8)
which is a system studied by Henon and Heiles (1964), that will be discussed in greater
Some origins of maps in R2
7
detail in Section 6.11. If E is not too large, the manifold (6.1.7) has a finite (bounded) portion. We can `represent' this portion in the space (q1, q2, P2), where it lies on, and interior to, the surface
1 2+V(g1,g2)=E, shown in Fig. 6.7. This figure should be approached with caution, because through Fig. 6.7 A 'near'-immersion of the energy manifold. ql
P2
any point interior to the surface can pass two trajectories (for a given E), namely pl and -pl. Hence the figure is not a proper embedding of the energy manifold in R3 (see the discussion concerning embeddings and immersions in Chapter 8, Section 13). Nevertheless, such a figure is helpful. Exercise 6.3 What is the maximum value of E for which (6.1.7) has a bounded region,
as illustrated, when H is given by (6.1.8)? Is there an unbounded portion to this surface, even in that case?
The bounded plane section S = {q1 = 0, Zp2 + V(0, q2) < E} is bordered by the closed trajectory ip2 + iq2 - 3qi = E, which is also a solution of (6.1.6). Aside from this trajectory, all solutions of (6.1.6) pass through S in the positive q1 direction (41 > 0) at least once every T seconds, where T is finite. A schematic example is illustrated in Fig. 6.8. When the trajectory passes through S in the sense 41 > 0, the series of points s1, s2, ... , sk, ... defines the Poincare map sk + 1 = P(sk). An example of
a sequence of such points is illustrated in Fig. 6.9 (Henon and Heiles, 1964), for the case E = It appears, in this case, that the points progress in a regular fashion and i2.
8
Models based on second order difference equations
Fig. 6.8
sl
I
q2-
Fig. 6.9
P2
0.2
0
-0.2 0
0.4
0.2
q2
look like they lie on a smooth closed curve (essentially a circle). We will see later that looks can sometimes be deceiving. Note that in the space (P2, q1, q2) the volume of some propagating region, ((t), is
not invariant, d/dt f ,,,,) dp2 dq 1 dq2
0. However, using the results of Chapter 2,
Section 4, if we eliminate p,, and take A(ko, g 1,g2,P2)=
R
8H OP, P1] OH
,
where
Pt - ± [2ko - p2 - 2V(g1, g2)]112, then
Iko . J
[2ko - p2 - 2V(g1, q2)] -112 dql dq2 dP2
(6.1.9)
k0
is an integral invariant, if proper account is taken of the reflection of flko(t) at the
Rotation and winding numbers
9
`boundary' of the energy manifold (where pl = 0). In the present case this yields a very simple result for the Poincare map. Consider an infinitesmal region da = dp2 dq2 on S. In a time dt it sweeps out a volume v dt d a where v = q, When this volume crosses S again it sweeps out the .
volume v' dt' do' (Fig. 6.10). What (6.1.9) tells us is that, if these volumes are weighted by Fig. 6.10
the local values of [2E - pz - 2V(g1, q2)] - 112, they are equal. But this denominator is simply the local value of 41(- v), so v dt da/41 = v' dt dc'/4', or dq2 dp2 = da = do'= dq2
(6.1.10)
dp2.
In other words, the area on S is preserved by this Poincare map. The program in the remainder of this chapter is to first review and extend the
concept of rotation and winding numbers, which arise so frequently in various contexts. Then we will use the distinction between nonconservative and conservative maps to attempt to bring some order to this general subject. Thus the road map is
Rotation and winding numbers Section 6.2
Conservative maps Sections 6.5-6.14
Nonconservative maps
0
Sections 6.3, 6.4
Dynamic entropies and information production Section 6.15
6.2 Rotation and winding numbers In Chapter 4, the concept of rotation/winding numbers was introduced in connection with the circle map. Here we consider this concept in connection with flows in the extended phase space, and the associated Poincare map.
Models based on second order difference equations
10
If the system is not the coupled nonlinear form of (6.1.8), but rather the simple pair of harmonic oscillators (see Exercise 2.26)
H=
z(pi
+ p2 + (4q
+0)2 q 22)
(6.2.1a)
then the motion is indeed very simple. If we introduce the usual action-angle variables, (I k, Ok) (k = 1, 2), Pk = (2Ik(ok)112 COS (0k),
qk = (21k/wk)112 sin (6k),
(6.2.2)
the Hamiltonian is simply
H=0)111+0)212,
(6.2.1 b)
and the equations of motion are ik
= - aH/a0k = 0,
6k = aH/aIk = 0)k,
(k = 1, 2).
Since I1 and 12 are constants, and the system is periodic in the variables 01, 02, because of (6.2.2), the dynamics is confined to a two-torus, T2, which is commonly represented in R3, as illustrated in Fig. 6.11. Note that while this figure looks similar Fig. 6.11
to the above extended phase space, D x S', the present system is autonomous and confined to a two-dimensional manifold, T2. A Poincare map can now be defined by the repeated intersections of a trajectory with any circle on the torus which can not be contracted to a point. The resulting map yields a series of points that look quite similar to the figure for the Henon-Heiles Hamiltonian in Section 6.1.
Exercise 6.4 Because Fig. 6.11 is commonly found in articles, it is worth while examining it in more detail. Indicate what 11 and 12 represent in this figure, and therefore how (p1, q1) and (p2, q2) are related to the figure. What is the basic difficulty
in representing all values of (11,12) in a single figure? Under what restrictions does this figure uniquely represent the state of the system? When the trajectories are on T2, the Poincare map takes points on a (topological) circle onto itself P: C - C, where the circle will be taken to be the `minor meridian'
Rotation and winding numbers
11
Fig. 6.12
(0 = constant), illustrated in Fig. 6.12. An important feature of this map of the circle onto itself is that it preserves the orientation of the points on C. Thus, if M:
0k+1 =Ok+A(Ok)
(6.2.3)
A(0) is called the angular function, and satisfies dA/d0 > - 1 if the map preserves orientation. In particular, when (0, 4)) satisfy equations of motion
0=J(0,¢),
4)=K(0,0),
where J(0 + 2n, 4)) = J(0, ¢ + 2n) = J(0, 0), and similarly for K(0, 0), then the trajectories are on T2 and satisfy dO/d¢ = F(0, 0)
(F(0, 0):0 0)
(6.2.4)
F(0 + 2ir, 4)) = F(0, ¢ + 2iv) = F(0, ¢). Assuming that the solutions of (6.2.4) are unique, then the map (6.2.3) does preserve the orientation. In this case Poincare introduced the concept of a rotation number (also frequently called the winding number) P = lim 0(¢, 00 )/4)
(6.2.5)
where 00 is an initial point, 0(0, 00) = 00. The rotation number represents the average slope of the trajectory in Fig. 6.13. The important feature of orientation-preserving Fig. 6.13 k
0 E
m
00
¢ mod 2a
maps is that p is independent of the initial point, 00. In other words, p is solely a characterization of the map, and is not dependent on the initial point. Recognizing
12
Models based on second order difference equations
that the map (6.2.3) corresponds to 0 increasing by 2n, the rotation number can also be written in the form
p = - lim -
21rn-oo n k=0
A(00
(6.2.6)
(see Fig. 6.14). Fig. 6.14
If some power of the map (6.2.3), M', has a fixed point, M'00 = 00, then p is a rational number. This follows because the solution of (6.2.4) must then satisfy 0(21rl, 00) = 00 + 21rm, for some integer m. Then, setting n = IN in (6.2.6) p=
1 lim 1 (2nmN) = m/l. 21rN-oo IN
(6.2.7)
Conversely if p = m/1 for primed integers m, 1, then M'00 = 00 for some 00. On the other hand, if p is irrational and 0(21c, 00) has continuous first and second derivatives in 00, then the trajectory is ergodic on V. Moreover p(F) obtained from (6.2.4) varies continuously with a change in F, provided that all functions are Lipschitzian in 0. In other words, for any s > 0 and F there is a b > 0 such that I p(F) - p(G) I < e if the maximum of I F(0, G(0, 0)l < 6 (for 0 <, 0, 0 < 21r). For more details see Hartman (1973).
Exercise 6.5 (a) What is the rotation number of d0/d4) = sin 0? (b) If I f (0)I < 1, 0 < 0 < 21r, Lipschitzian, and f (0 + 2n) = f (O), what is the rotation number of d0/d4) = sin 0 + f (O)? (c) What is your best estimate of the rotation number for the Henbn-Heiles Poincare map illustrated in Section 6.1?
We illustrate in Fig. 6.15 several rational rotation numbers and their relationship to flows on a two-dimensional torus, T2. The left figures show the maps for the cases P = 2, 3, and 3. The right figures illustrate the corresponding flows on the extended
phase space torus, where only two transverse sections are shown for clarity. Note the topological difference between the p = 3 and p = 3 orbit (see a knot?). It is also important to note the difference between the above orbit on T2 with p = 3, and
Rotation and winding numbers Fig. 6.15
13
Flows on TZ
Maps
P
P
P
2
3
3
Fig. 6.16
P-3
the following orbit in D x S' in Fig. 6.16 (see Section 6.1), also with p = 3. Are they TOE (Chapter 2)? We will return to such flows in Chapter 7 (the Lorenz system). Systems with such rational rotation numbers are quite special. In the case of the harmonic oscillators, (6.2.1), rational rotation numbers indicate that the system has a time-independent, algebraic constant of the motion of the form N
K=
Ajkirop'ip2k
91I
gi
dK
- = 0,
(6.2.8)
dt
in addition to the two constants Hk = wkl k = (pk '/Mk + mkwk gk )
(k = 1, 2).
(6.2.9)
14
Models based on second order difference equations
Thus, if p = w1/w2 = m/n, where m and n are relatively primed integers, then
dt
cos (n01 - m02) = - sin (n01 - m02)(nw 1 - mw2) = 0,
because dk = wk. Thus K = cos (nd 1 - m02)
((o1/w2 = m/n)
(6.2.10)
is a constant of the motion. The constant (6.2.10) can be put into the algebraic form (6.2.8) with the help of (6.2.2). Thus, if p = 2, the constant is cos (201 - 02) = cos (201) cos (02) + sin (201) sin (02)
= (1 - 2 sine (01)) cos (02) + 2 sin (01) cos (01) sin (02).
Using (6.2.2), a little algebra leads to the constant K = (P1 - miwiq )P2 + 2m1w1m2w2g1g2P1.
(6.2.11)
Note that K is a homogeneous polynomial of degree (m + n). We will need to generalize the classic concept of rotation numbers to include sets {dk = Mk0o} whose rotation number P(0o) = 1
lim
27[n-oo nk=o
A(dk)
(6.2.12)
may depend on O. Here A(0k) will be taken to be the acute angle between the lines from a given point to xk and xk+ 11 with one sense of rotation with positive A(0k) and the opposite sense negative. These sets may simply be closed curves corresponding Fig. 6.17
to different tori, T2, (e.g., the `nested' set of tori, illustrated in Fig. 6.17)
dd/d4 = F,(0, 0)
(1= 1, 2,.. .)
(6.2.13)
or more complicated sets which do not lie on tori (see the standard map). An interesting and important theorem can be readily proved with the help of the Poincare-Bendixson theorem (see Chapter 5, Section 6):
Rotation and winding numbers
15
Theorem Let a trajectory of (6.2.4) have a rotation number which is a rational number. Then every trajectory of (6.2.4) is either closed or approaches a closed curve as 101 -> oo.
To prove this we need to recognize the fact that, if you cut a torus along a closed curve (4 0), you get an annulus. If your visualization of such matters is as poor as mine, you may find the (0, 4)-plane representation in Fig. 6.18 helpful. Illustrated is Fig. 6.18
FA 0
the case p = 2. If the plane is cut along the trajectory and the equivalent edges are connected, we get a bracelet-like band, as shown. All of the trajectories lie on this band, which, with a little stretching, is seen to be a flow on an annulus in R2. All trajectories are confined to this annulus, which contains no fixed point (F(0, ¢) 0 0). Therefore, by the Poincare-Bendixson theorem, every trajectory must approach (or be) a closed curve, proving the theorem. The closed curve may, of course, be a boundary of the annulus. If the trajectories of (6.2.4) approach a closed curve as 4 -+ co, then that system must be dissipative. On the other hand if all the trajectories of (6.2.4) are closed curves, the system is always conservative. This is because any dissipative effect can only be compensated by an external periodic force for a discrete (enumerable) set of periodic orbits - otherwise the force and the dynamics will not stay in a'compensating phase'. These two physical systems are illustrated in Fig. 6.19, where p = 1 for the dissipative case. Note that, in the case of an attracting periodic orbit (stable limit
Models based on second order difference equations
16
Fig. 6.19
p=1
cycle), there must also be an unstable limit cycle on the torus. Two orbits are shown in the nondissipative case (p = to illustrate the periodicity of all the orbits. 2)
6.3 The Cartwright-Littlewood, Levingon and Levi analyses We now return to the second half of the Cartwright-Littlewood and Levinson's study of the self-exciting oscillators Ex + (D(x)x + Ex = b sin t
(6.3.1)
for small $, which we began in Chapter 5, Section 5.14. The purpose of this section is to illustrate the remarkable properties of the dynamics which can be deduced by analysis (albeit, tedious at times!), which is not accessible by computers (nor observable??).
Recall that, if b is contained in certain intervals, Bk, then this equation has two stable and two unstable subharmonic solutions with periods (2n + 1)T and (2n -1)T, where n = o(1/E), and the period is T = 2n for (6.3.1). Call these solutions P1 and P2 respectively. In addition, Levinson proved that there is also an unstable periodic solution, PO, of period 21r. Finally, it was established that there is a family of solutions,
F, whose members can be associated with any doubly infinite Bernoulli sequence, { ... , +, -, -, +, -, +, +I +, - , ... }, or a coin toss, and whose initial conditions are nearly the same. It should be noted that this correspondence with Bernoulli
sequences is much more `compact' than in the case of the logistic map. In the latter case the correspondence was between solutions in the entire phase space and all Bernoulli sequences. In the present case this correspondence holds for a small region of phase space. The second half of their study concerned the relationship of all solutions of (6.3.1), except PO, with another family of solutions, KO, which has some extraordinary
Cartwright-Littlewood, Levinson and Levi analyses
17
properties. To describe and prove these properties they used the Poincare map in the extended phase space ()'(t), x'(t)) = T(x, x) - (z(t + T), x(t + T))
(6.3.2)
or
xk+ 1 = Txk.
Levinson's equation, (6.3.1), has the simple periodic solution, PO, of period 2ir, x = b cos t, if b< 1. Recall that c(x) = +1 for l x i > 1 and I(x) = -1, if l x l < 1. He then showed that this is an unstable solution (see Appendix J for more details). This means that ()i, x) = (0, b) is an unstable fixed point of the Poincare map (6.3.2). Thus a closed curve CO around this fixed point expands and encloses a larger region under the map (6.3.2), C1 = TC° (Fig. 6.20). Proceeding with higher iterates Ck+ 1 = TCk, Fig. 6.20
and the interior of Ck+1 contains the interior of Ck. Levinson then defined the set
H = U00 Interior Cm,
(6.3.3a)
M=0
which is some open connected set of points, of unknown extent. He next showed (Fig. 6.21) that there is a large curve, CO (note the superscript zero), which after a sufficient number (N) of iterates, lies entirely inside C°; that is,
Interior T"C° c Interior C° (n > N). Next he joined the interior of C" with C" to form closed sets, Interior O u C". He then generated the set which is the intersect of all of these closed sets cc
K = n (Interior C" u C"). n=N
(6.3.3b)
The open set H (6.3.3a) must be contained within this closed set, K. Finally, Levinson defined the set
KO=K-H,
(6.3.3c)
18
Models based on second order difference equations
Tn CO
TK Co
which is a closed set contained somewhere in the shaded annular region of the figure. He proved that the area of this region goes to zero as n and k go to infinity. Therefore the area of the set Ko is zero.
Hence, Levinson established that there is an attracting set of points in the phase space, K0, with the following remarkable properties: (1) It is a closed set with zero area, which is connected (K0 is not the union of two compact, disjoint sets). (2) It is the boundary between the open interior set, H, and the open exterior set (the complement of H - K0, equal to R2 - K). Therefore any s-region about a point of Ko contains both interior and exterior points. So far it looks like Ko might just be a closed curve, but: (3) Any periodic solution, with the sole exception of (z, x) = (0, b), must be contained
in K0, because it either attracts all other solutions or else contains them. In particular Ko must contain the (2n + 1) and (2n - 1) points corresponding to the stable periodic solutions, P1 and P21 of period (2n + 1)2n and (2n - 1)2n respectively. But this leads to the conclusion that Ko cannot be a simple closed curve. This follows from considering the rotation number of such an assumed closed curve. Since it contains a set of points with least period (2n + 1)27T, the rotation
number must be p = N1 /(2n + 1) (N1 < (2n + 1)). On the other hand it also has
a set of points with least period (2n - 1)2n, and therefore a rotation number N2/(2n - 1). However, the rotation number of an invariant closed curve is unique for an orientation-preserving homeomorphism (e.g., Hartman, 1973). Thus, under the assumption that Ko is an invariant closed curve, we must have
Cartwright-Littlewood, Levinson and Levi analyses
19
N1/(2n + 1) = N2/(2n - 1), or N1/N2 = (2n + 1)/(2n - 1), and N1 and N2 can be taken to be relatively prime. Since (2n ± 1) are odd numbers, (2n + 1) = N1m, and (2n - 1) = N2m, where m is odd and m >, 3, because of the bounds on N1
and N2 (e.g., 1 > N1/(2n + 1) = 1/m). But then 2 = m(N1 - N2), which is impossible for m 3 3 and integers N1 and N2. Therefore Ko is not a simple closed curve. Levi (1981) has in fact proved that, for every 1/(2n + 1) > p >, 1/(2n - 1), there
is a point in Ko associated with this rotation number. (4) The set Ko contains the chaotic family F, whose members (continuum in number)
can be put in correspondence with a doubly infinite Bernoulli sequence (coin toss). This family exhibits sensitivity to initial conditions. That is, given any finite
accuracy to the initial conditions it is impossible to conclude its itinerary
{:.., +, -, +,...}. (5) Ko is a singular set with points which cannot be approached along a continuous curve which lies entirely in the exterior (or interior) region (the points are neither `exterior accessible' nor `interior accessible'). (6) Ko contains a Cantor set (Levi, 1981). Levi has discussed a number of additional features of the set Ko as well as some aspects of the dynamics which occurs in the `gaps', gk (see Fig. 6.22). He indicates A2
Fig. 6.22 ( g2
gl
g3
b-that within these gaps there is a complicated sequence of bifurcations, an uncountable number of b-values for which the map (6.3.2) has infinitely many sinks (stable periodic points), and a number of other exotic properties. Moreover he notes that hysteresis effects (observed experimentally by van der Pol and van der Mark, and by numerical
studies (Section 6.4)), can be understood as the simple consequence that the only stable periodic solution for beAk has a period (2n(k) + 1)2n, and satisfies n(k + 1) = n(k) - 1.
On the other hand, for bEBk there are two stable periods, (2n + 1)2n and (2n - 1)2n.
Therefore, as b is increased from Ak to Ak+l, and then reduced again, there is a hysteresis effect entering Bk. This is illustrated in Fig. 6.23. It explains the hysteresis effect, obtained in the numerical calculations of Flaherty and Hoppensteadt (1978), to be discussed in the next section. These, however, are not the same hysteresis effects found by van der Pol and van der Mark (as will be discussed shortly). While the above set, K0, is certainly quite exotic, the limiting attractors in this set (i.e., the (o-limit set of all solutions not in KO) are simply the two stable periodic
20
Models based on second order difference equations
Fig. 6.23
(2n + 1)27r_ (2n - 1)27r
Period
//y"/ZZZZ
b
Hysteresis
solutions Pt and P2. The set of points which are attracted to P1 (or P2) are said to be in the basin of attraction of P1 (or P2). The remainder of the set Ko (excluding P1 and P2) contains the boundary between the basins of attraction of P1 and P2. As we have seen (properties (c - e) above) this boundary cannot be a simple closed curve. It indeed may contain a fractal boundary (dimension d, > 1) between the two basins of attraction. Fractal boundaries between basins of attraction of `abstract' maps (not necessarily related to the dynamics of physical systems) also occur in Julia sets, and an explicit `Weierstrass form', obtained by Greborgi, Ott, and Yorke (1983). These will be discussed in the next section. It is, of course, quite hopeless to represent the Ko set realistically, but Fig. 6.24 gives Fig. 6.24
(a)
A Kp
set
9 period 3 ® period 5 x
(b)
Cantor set Hyperbolic , points _,/
Cartwright-Littlewood, Levinson and Levi analyses
21
some sense of its structure. Two periodic solutions with periods three and five are illustrated, wrapped on the `Lienard torus' (see Chapter 5 for a dressed version), and in a cross-cut, indicating the remaining points of K0.
While the results of Cartwright, Littlewood, Levinson and Levi are quite extraordinary, it nonetheless appears that the experimental results of van der Pol and van der Mark remain largely unexplained. Specifically, they observed stable limit cycles with periods NT (N = 1, 2, 3,... ), whereas the theoretical stable limit cycles have periods (2n + 1)T (integer n). Moreover, the above hysteresis is between the two cycles with periods (2n + 1)T and (2n - 1)T, whereas the van der Pol-van der Mark hysteresis is between periods NT and (N + 1)T. It does appear, however, that their hysteresis loops are separated by single limit cycles (corresponding to beA,), and that the theoretical gaps (bEgk) might correspond to their noisy regions. However their seminal experiment appears to be largely unexplained after half a century! As noted on the last chapter, the unusual attracting properties of (6.3.1) are difficult
to observe, using either analog or digital computers. By way of contrast, Shaw's `variant', (5.14.13), readily yields a numerically observable strange attractor. On the other hand, these equations,
z= - y+Acos(f1t);
y=x+µ(1 -x2)y,
(6.3.4)
are not equivalent to (6.3.1), and have not been analyzed in the above detail. If the Poincare' sections of (6.3.4) are taken at four times t = nT, (n + and (n + 2)T where T is 2n/f2 (n = 0, 1, 2,...), then the schematic appearance of the maps (n + on these sections is indicated in Fig. 6.25. The exact form obviously depends on the 4)T,
4)T,
Fig. 6.25
values of the control paramaters (A, f2, µ) in (6.3.4). Thus, for A = 0.93, y = 1.18, f2 = 1.86, 600 iterations on each of the four sections yielded Fig. 6.26. The `flap' formed on the left of Fig. 6.26(a) moves down and is squeezed into the attractor (t = (n + 4)T), while a new flap forms on the right (t = (n + and then squeezes into the attractor Z)T) on the upper side (t = (n + 4)T). Such numerically computed maps of strange attractors
are, of course, subject to the usual caveats concerning numerical accuracy,
Models based on second order difference equations
22 Fig. 6.26
ti (a)
(b)
t=nT
t=(n+ 4)T
(d)
(c)
(n+
2)T
t=(n+
4)T
pseudo-orbits, and the like. The resulting fractal-torus has been illustrated in Chapter 5, Section 14. For other figures, see Shaw (1981) and Abraham and Shaw (1983).
6.4 Some abstract nonconservative maps in R2 Twenty-five years after the discovery of the `curious set', KO, associated with the forced
van der Pol oscillator (Section 3), Henon invented a contracting map which has an even stranger set associated with it. Moreover, in contrast to the set KO, many strange aspects of this set can be observed (computed). Henon considered a quadratic polynomial map T:
xn+ 1 = Yn + 1 -
(6.4.1)
Yn+ 1 = bxn
which has a constant Jacobian corresponding to a contracting flow a(xn+1,Yn+1)=
-b
a(xn, Yn)
T is in fact the most general quadratic map with a constant Jacobian. It is also a one-to-one map, with the inverse
T-1:
xn=b-'Yn+1,
y,
=xn+l-1+ab-2Yn+I.
T has two fixed points at
x=(2a)-1{-(1-b)±[(1-b)2+4a]112},
y=bx
which are real if a > - (1 - b)2/4 - ao. In this case according to Henon one fixed point is always linearly unstable and the other is unstable if a > 3(1- b)2/4 - a1. For ao < a < a1 the mapping is attracted to the stable fixed point. For a > a1 the attractor
is (at first) a set of periodic points (period p) and the value of p increases as a is
Some abstract nonconservative maps in R2
23
increased ('bifurcates' at a = a2, a3, ... with periods p = p21 p3, ...). Such a periodic mapping in a surface of section would correspond to a limit cycle in the phase space, illustrated in Fig. 6.27. As n -+ oo the bifurcation values have a finite limit a - a., < oo Fig. 6.27
p=6
(depending on b) and apparently p -+oo. Henon found 1.06 for b = 0.3.
For large values, a. < a < 1.55 (for b = 0.3). Henon found there was still attraction, but it is no longer a simple periodic solution. Fig. 6.28 represents 10,000 successive mappings using (6.4.1) with a = 1.4, b = 0.3. After about five mappings any initial point
gets `lost' in the set shown in this figure. For example, if (xo, yo) = (0, 0), then only Fig. 6.28 0.4
Y 0.3
0.2
0.I
0
-0.1
-0.2 -0.3 -0.4 -1.5
-1.0
-0.5
0
0.5
Models based on second order difference equations
24
the iterates (x2,Y2) = (- 0.4,0.3);
(xi, yi) = (1,0);
(X31 Y3) = (1.076,0.12)
are distinguishable from the set (not shown in Fig. 6.28) whereas (x4, Y4) _ (- 0.7408864, 0.3228) is lost in this attractor set, as are all successive iterates. The very interesting feature of the attractor set is that, while it looks like a simple one-dimensional manifold, its transverse structure (across the `curves') proves to be quite different than a simple curve. Fig. 6.29 is the enlargement of the square region Fig. 6.29
Y
0.21
0.20 0.19 0.18 0.17
0.16 0.15 0.55
0.60
0.64
0.70
x
in Fig. 6.28 (now for 105 iterates). Obviously the previous three curves are now seen to be at least six curves. The result of enlarging the square in this figure (increasing the iterates to 106, in order to have enough points to define `curves'), is shown in Fig. 6.30(a) (left side). Lo and behold! Those three `curves' are at least six (seven?) `curves'. Yet, once again, enlarging the square (using 5 x 106 iterates), Henon shows (Fig. 6.30(b)) that the three (four?) curves actually are six (seven?) curves. In other words there is no suggestion that this attracting set has a finite transverse structure which is composed of a finite number of curves. Indeed the transverse structure seems
to be reproduced at any scale (size), regardless of how small it may be. This is characteristic of a self-similar Cantor set, which has been discussed in Chapter 2 and elsewhere.
The above figures therefore suggest that the attractor has a local structure (e.g., the regions in the above boxes) which is the same as the cartesian product of a one-dimensional manifold, R', and a Cantor set. By a cartesian product of two sets,
Some abstract nonconservative maps in RZ
25
y 0.1895
0.1894 0.1893 0.1892 0.1891
0.1890
0.625
0.630 (a)
0.635
x
0.640
0.1889 0.6305
0.6310
0.6315
0.6320
x
(b )
X x Y, is simply meant the set of all ordered pairs (x, y), where xeX and yE Y. Thus
this attractor appears to have the local structures as R' x C, where C is a Cantor set. There is, of course, no way that such computations can actually prove that one of the sets is in fact a continuum R', or that the other set is a Cantor set, but physically that is of no real consequence anyway. No observations can verify that a set is either R' or C, but it is nonetheless useful to talk about such sets, even if they are 'over-refined models' of our physical abilities. They are effectively a mathematical shorthand, which avoid (or ignore) the question of experimental errors and limitations. Computations, in fact, have the associated beneficial feature of reintroducing the element of experimental error, or `physical reality,' into our considerations (Chapter 4, Section 11).
An infinite, invariant set, S, of a map F, which contains a dense orbit, and which is the asymptotic limit set (the co-limit set) of most points in some neighborhood, N, of the set S, has been called a `strange attractor' by Ruelle and Takens (1971). The definition of various attractors, strange or otherwise, has been discussed briefly in Chapter 5, and continues to be an area of active discussion (Ruelle, 1983). The above definition for example, does not exclude some very nonstrange attractors, such as a closed invariant curve. Thus, for S to be strange, it would seem reasonable to at least require that S not be a manifold in RZ (or more generally, in R"). In other words, S should have some `wild' structure, which in the above case is the Cantor set component of the cartesian product. These considerations, however, only address the geometric
26
Models based on second order difference equations
aspects of the attractor, and say nothing about the dynamical features near the attractor. Thus, for example, consider again the logistic map, and the Cantor set, X, which occurs at c = cam. By the above definition, this set is a strange attractor but, on the other hand, the Lyapunov exponent of that set is zero, as discussed in Chapter 4. Thus this set does not possess the feature that it is (exponentially) sensitive to initial conditions. Put another way, the set X is geometrically wild, but dynamically tame.
It has been suggested by F. Albrecht that dynamically wild sets might be called `chaotic sets', to distinguish them from the purely geometric strangeness. Again in the case of the logistic map, when c = 4, most points in the interval (0, 1) belong to such a chaotic set, but they are not a strange attractor, or any type of attractor (see again the discussions in Chapters 4 and 5). In 1980, Ruelle included the condition that a
strange attractor must also be sensitive to initial conditions. This brief discussion indicates only a few of many possible refinements which may turn out to be important in the future. A nice, wide-ranging discussion of many other related concepts can be found in Shaw (1981).
The attractor of the Henon map appears to possess both the strange and chaotic behavior, for suitable values of the parameters (a, b) in (6.4.1). For the value b = 0.3, both Curry (1979) and Feit (1978) have computed the Lyapunov exponents for this map over a range of values of a. Some of the results of Feit are shown in Fig. 6.31. Fig. 6.31 Characteristic exponents for b = 0.3. 0.50 r0.25
-0.50 -0.751 0.00
1
0.25
1
0.50
1
0.75
1
1.00
1
1.25
1
1.50
parameter a
Several of Curry's results are shown in Fig. 6.32. He found two different curves for the exponents, depending on the initial condition used in the trajectory (recall the discussion concerning the orbital dependence of the average Lyapunov exponent, given in Chapter 4). He concluded that, for some parameter values, the Henon map can have several distinct strange attractors.
Some abstract nonconservative maps in R2
27
Fig. 6.32 Henon map-characteristic exponents for b = 0.30. 0.14
TTT- I
0.12 I
0.08 0.10
0.04
0.06
0
-0.04
0.02 0
-0.08-
I
I
-0.02
-0.12
-0.16
-0.06 -0.10 1.070
-0.20
(a)
(b) 1
1.072
1.074
1.076
1.078
1.080
1.070
1.072
1.074
1.076
1.078
1.080
a-axis
As an example of coexisting chaotic and periodic attractors, we can use a = 1.07, b = 0.3 and the two initial states (xo, yo) _ (1, 0), and (1.5, 0). The former goes to a strange attractor with four bands, whereas the latter tends to a period-six orbit, as illustrated in Fig. 6.33(a). On the other hand, if a = 1.08, b = 0.3, the initial states (1.4,0) and (1.43,0) tend to different strange attractors, as illustrated in Fig. 6.33(b). Fig. 6.33 (a) a = 1.07. (b) a = 1.08.
I
The basins of these attractors alternate as x0 is varied (e.g., you might explore the region 0 < x0 < 1.5, yo = 0). A very extensive analysis of the map (6.4.1), including a study of the fusion of strange attractors, has been given by Simo (1979). It is also known that the more general map (x', y') = (y + t - ax3, bx) has an infinite number of periodic attractors, for suitable values of a, b, and t (Newhouse, 1979). Such multiple attractors are quite distinct from the single-attractor dynamics found for one-dimensional maps. A study of many distinctions between one- and twodimensional maps can be found in Holmes and Whitley (1984).
Models based on second order difference equations
28
Many other abstract maps in R 2 have been investigated for the purpose of illustrating certain dynamic features. We mention a few examples:
(A) Period-three does not imply chaos in R2 In R 1 we have seen (Chapter 4) that if a continuous map has a period-three orbit, it produces Li-Yorke chaotic dynamics. However, the map xk+1-(axk+byk)(1-axk-bYk);
Yk+1-xk
(6.4.2)
has a stable period-three orbit when a = 1.9 and b = 2.1. Hence it has nonchaotic dynamics for these values of (a, b). This example is due to J.L. Kaplan and R. Marotto. (Exercise: Find the period-three and show that it is stable.) Another example will be seen in (C) below.
(B) A fractal boundary of basins of attractions The map (Greborgi, Ott, and Yorke, 1983)
M:
Xk + 1 = AXk + cos 9k; ek + 1 = 20k mod (2n),
(6.4.3)
when 2 > 2 > 1, has two attractors, at x = ± oo. There are no finite attractors because
the eigenvalues of the Jacobian matrix are 2 and A > 1. Thus MN(xo, 60) = (xN, ON mod (2n)), and XN either tends to + o0 or - oo as N - oo, except for the (unstable) boundary set x = f (O), for which XN remains finite.
To find this boundary set, we note first that ek = 2k00 mod (2n). The map M is two-to-one (noninvertible), but we can select any XN and find one orbit that ends at (XN, ON), by using the above Ok and taking Xk -1 = 2 -1 xk - A given (XN, B0) we find that this orbit started at
cos (2k -190). For the
N-1
x0=2-NxN- E 2-l-Icos(2'Oo). l=0
The boundary between the two basins are those (x0, 60) such that XN is finite as N - oo, so these x and 0 are related by OD
x= - 1 2-l-1cOS(2`e)= f(e).
(6.4.4)
Since A > 1, this sum converges absolutely and uniformly. On the other hand df(O)/dO = 1
(2/A)'+ 1 sin (2'O),
21=o
and this sum diverges, because 2 < 2. Hence f (9) is nondifferentiable. In fact the curve (6.4.4) has a fractal dimension dd = 2 - (In 1)(ln 2) -1 (the proof is not easy; Kaplan,
Some abstract nonconservative maps in RZ
29
Mallet-Peret, and Yorke, 1984). That is, the boundary of these basins of attraction is a fractal curve (a term coined by Mandelbrot in 1975). Such curves were known to Weierstrass, and generalized by Hardy in 1916. The curve (6.4.4) is illustrated in Fig. 6.34, for A = Z. Fig. 6.34
The modification of (6.4.3) to M*:
xk + 1 = axk + xk + /3 cos 0k;
0k + 1 = 20k mod (21r)
(6.4.5)
yields a strange attractor near x = 0, as well as an attractor at x = + oo, when a = 1/2, /3 = 0.04. The boundary between these attractors is again a fractal boundary. As a is increased, the strange attractor makes an intersection with an unstable fixed
point on the basins' boundary (a crisis occurs), at a* -- 0.6. For a > a*, the only attractor is x = + oo, but the time required for many initial states to develop large values of x is very long, if a is only slightly above a*. This leads to the concept of a chaotic transient, namely a set of initial states which take a `large number' of iterates to approach the attractor. The average lifetime of these chaotic transient states is predicted to be
= exp(A/(a - a*)1/2), which is very large for a a*. Another example of chaotic transients, and for flows in R3, will be discussed in the case of the Lorenz system, in Chapter 7.
(C) The Julia and Mandelbrot sets The logistic map Pn+ 1 = 2Pn(1 - Pn),
which was considered in some detail in Chapter 4, can be put in the simple quadratic
30
Models based on second order difference equations
form, (6.4.6)
zn + 1 = z, + c,
where z = A(1/2 - p) and c = (x,/4)(2 - A). In the logistic case p and
.
(hence z and c)
are, of course, real numbers, since they typically represent populations and reproduction rates. Such maps can be formally extended into the complex plane, z = x + iyEC, in the
hope of discovering new dynamic features, and a better understanding of the bifurcations of the classic logistic case. If c is also complex, c = c, + i c;, the map (6.4.6) is xn+ 1 = xn - Yn + Cr,
Yn+ 1 = 2Xnyn + Ci.
(6.4.7)
One can attempt to find some physical rationale for studying such a map (e.g., along the lines of a discrete form of the Lotka-Volterra predator-prey model) or simply view it as a `natural' generalization of (6.4.6). We will take the latter point of view for this short discussion. More general studies involve the complex maps N:
zn+ 1 = zn - f(zn)/f'(zn),
(f' = df/dz)
(6.4.8)
known as a Newton map. Newton's method of obtaining the zeros of f (z), which are fixed points of N, is based on the convergence of the iterates of (6.4.8) to the fixed points of f (z). This method has a long history, and has been studied in the complex plane by Smale (1981), and others (Benzinger, Burns, and Palmore, 1987). The complex
map (6.4.8) is frequently more directly connected with physical applications than (6.4.7). See Peitgen and Richter (1986) for several examples. The map (6.4.7) produces dynamics in the `phase space' (x, y), which depend on the value of c in the `control space' (ce, c;). For fixed c, the set of points in the phase space
which remain bounded for all n is called the filled-in Julia set. Points which start outside this set rapidly tend to infinity. The boundary of the filled-in Julia set is called the Julia set. In some parts of phase space this set may not have any interior points (i.e., the filled-in Julia set may locally equal the Julia set, as in a cloud of points, or a curve).
Thus, if c = 0 in (6.4.7), all points in the complex z-plane either go to z = 0 (if Izol< 1), or limn - . I zn I = oo (if I zo I > 1), or I zn I = 1 (if I zo I =1). In this case, the unit circle I z I = 1 is the boundary between the two basins of attraction (one to z = 0, the other to I z I = oo) (Fig. 6.35). This boundary set is the Julia set, but it is not very interesting in this case (c = 0). However, if c 0, the filled-in Julia set can take on an incredible variety of forms, from fat clouds, to the branches of a bramble bush, to sea horses, etc. The beauty of these sets has stimulated the esthetic senses of mathematicians, scientists, and artists alike. They have been beautifully represented in color codes by Peitgen and Richter
Some abstract nonconservative maps in RZ Fig. 6.35 c = 0
31
Julia
x
(1986), which also contains useful articles and references. We limit the discussion here to a few meager examples. If c - 0.124 + 0.565 i, the Julia set has the fractal structure illustrated in
Fig. 6.36(a) (after Peitgen and Richter (1986)). In contrast to the fractal boundary (6.4.4), we have no explicit expression for this Julia set. However, the present Julia set is self-similar, in the sense that any arbitrary piece can be used to construct the entire set by a finite number of iterations of (6.4.7). We note also that the present Julia set is connected (all in one piece). Fig. 6.36
If c is changed to c = 0.12 + 0.74 i, the Julia set becomes much more involved (Fig. 6.36(b)), but still is connected. In this case the shaded region is the basin of attraction to the stable period-three orbit, indicated by the three dots (another example of (A) above).
Mandelbrot was the first to consider the change in the Julia set as a function of the complex (rather than only the real) control parameter c. Just as we can distinguish two sets of points in the phase space (x, y), two sets can also be distinguished in the
32
Models based on second order difference erence equations
control space. The Mandelbrot set is the set of points { ce., c. } such that zo = 0 yields a bounded sequence {z"}. This is also equivalent to the set of c-points such that the Julia set is a connected set. The Mandelbrot set is a connected set, but it has an amazingly complex structure. It is the dark region in the complex c-plane, illustrated in Fig. 6.37. It is composed
of three types of structure. The large heart-shaped region is called a cardioid. This Fig. 6.37
-1.75
-0.75
0.25
Real c
cardioid has a cusp at c = 0.25. Tangent to it, at c = - 0.75, is a disk region, centered
at c = - 1. Tangent to this disk and the cardioid are smaller disks, etc. (`fleas on fleas'). In addition there are filaments (dendrite structures) which connect this main structure to other smaller cardioids, which in turn have similar tangent disks. One such cardioid can barely be seen with its cusp at c = - 1.75. Others lie off the real axis. For example, one is centered at c = - 0.1565201668 + 1.032347109 i (discovered by Mandelbrot). We note that the points where the disks are tangent along the real axis correspond to the period-two bifurcations of the logistic map. Recall that the first two bifurcations
of the logistic map occur for Al = 3, and 22 = I + 6112 3.45, which by (6.4.7) correspond to c1 = - 0.75 and c2 - 1.25, the tangent points noted above. The 2"-bifurcation limit, A. ^-- 3.75 corresponds to c. - 1.40. What does the cardioid with a cusp at c = - 1.75 correspond to? The largest stable structure in the logistic map, when A > A,, is the period-three window at A = 1 + 8112,
which indeed is c = - 1.75. Thus we might expect an infinite number of smaller
The standard map; twist maps
33
cardioids along the real axis, corresponding to the windows in the usual logistic diagram (Fig. 6.38). This recognition of how the bifurcations and windows (with their
bifurcations) fit into the more general complex control-plane, is certainly a useful perspective. One is never certain where such insights will lead! Fig. 6.38
We now turn from the study of nonconservative maps, with their attracting sets, to some conservative, area-preserving maps, which have their own distinctive chaotic and `sensitive' sets.
6.5 The standard map; twist maps To introduce an important and widely studied conservative map in two dimensions, we will consider several very different physical systems which can be related to this map. Note again that the term `conservative' signifies a certain 'area-preserving' property of these maps, discussed in Section 6.1, which does not necessarily imply that energy is conserved, as we will see. A simple physical example, which yields the map of present interest, is an elastic ball bouncing vertically on a plate which has a velocity VP = V sin (wt)
(see Fig. 6.39). We ignore the vertical displacement of the plate relative to the excursion
of the ball. When the ball strikes the plate the ball experiences a change in velocity
Models based on second order difference equations
34 Fig. 6.39
n r
0 Av = 2vp(t). If v is the speed of the ball prior to the nth bounce, at the time t,, then (6.5.1)
vn+l=vn+2VsinOn where O n = wtn. The phase at the next bounce is Otn+1 = tn+1 - to = 2vn+1/g Setting a = 2w/g, then 0n+ 1 = On + avn+ 1.
0,, + 1 =
0n + wAtn + 1, where (6.5.2)
The equations (6.5.1) and (6.5.2) define the two-dimensional map which is known as the standard map (named by Chirikov (1979)). Exercise 6.6
(A) The derivation of (6.5.1) neglects the motion of the plate, x, = - (V/(O) cos (wt). Let (vn, v*) be the vertical (v > 0, upward) just (before, after) the nth collision. Set pn = 2wvn/g, k = 4 Vw/g, Bn = wtn, and 0 = w(tn+, - Q. Relate p* to (pn, 0n) and to (pn+,, 0). Obtain the implicit equation for 0 in terms of (k, On, p*) and thereby the general map. What limit yields pn+1 Pn - k sin 8n, 0.11 " On -P.11 (note pn < 0 in this limit; why?). (B) A particle moves between two parallel walls, separated by an average distance L (Fig. 6.40). The right wall vibrates with a velocity V = V sin (wt) _- Vo sin (0). Fig. 6.40 L
07V(t)
Neglect the displacement of the wall. Obtain the map for the particle's velocity
vn, and the phase, on, before the nth elastic collision with the right wall, vn+ 1= F(vn, 0n), 0n+ 1 = G(vn, 0n). Note how it differs from the standard map. Is
the map area-preserving? A more sophisticated physical situation where this map arises is in the relativistic motion of an electron in a microtron accelerator (Fig. 6.41). Here an electron goes
The standard map; twist maps Fig. 6.41
35
period = 21r E"+1 eBc
S=SO Sin wt
n+1
through a gap (length d), where there is a uniform electric field .9 = S0 sin ((ot)
changing its energy by an amount DE = e,90d sin
where t,, is the time when it passes through the gap. It is assumed that the particle's velocity is nearly c, and that wd/c << it, so that the variation of '(t) during the transit of the gap can be neglected. Outside the gap the electron makes a circular orbit in a uniform magnetic field, B. The period around this orbit is At = 2n(mc/eB) = (E/eBc)21r
where E is the electron's relativistic energy (radiation losses are ignored). If E is the energy of the particle prior to nth entrance into the gap (at phase B = then Eri+ 1 = E,, + K sin (0.)
(6.5.3a)
where K = ef0d. Moreover the phase at the (n + 1) transit of the gap is
0,+1=0"+fE,+1
(6.5.3b)
where j3 = 2nw/eBc. The map (6.5.3) is seen to be of the same form as (6.5.1) and (6.5.2), again the standard map. For more details concerning the nonlinear dynamics of particle accelerators, see Jowett, Month, and Turner (1985).
Exercise 6.7 An accelerator mode is a dynamic situation in which the energy continually increases, E. + 21rN; where N is a positive integer, and co is adjusted so that fi = 1. This requires that K sin 0 = 2nN, so K is larger than 2n. What must be the injection energy, E0? Set 0,, = 0 mod 2n + 86n, E,, = E0 + n(2nN) +
6E to study the stability of mode N. Obtain the condition on K, in terms of N, for stability of this mode.
Models based on second order difference equations
36
Quite a different physical example of the standard map occurs in the Frenkel-Kontorva model of solid state physics (e.g., see Aubry (1983)). This models the spatial structure of a linear chain of atoms which are deposited on a spatially periodic force field (a lattice). Let x; be the location of the jth particle in the chain (see Fig. 6.42), and assume that the forces between these particles is a linear function Fig. 6.42
periodic potential
of their separation (a harmonic lattice). The total force on the jth particle is then
F;=k(xj+1 -xi-1)-k(x;-xj_1 - 1) + A sin (21rxj/A)
(6.5.4)
where I is the equilibrium separation distance when A = 0. The question of present interest is to determine the spatial structure of this chain when it is in an equilibrium configuration. The dynamic behavior of this system leads to the sine-Gordon equation (Chapter 9). For equilibrium, (6.5.4) requires x3+ 1 + x3 _ 1 - 2x3 + (A/k) sin (27rx;/i) = 0.
(6.5.5)
If we set 0; = 27rx,/A, and rj = (27r/A)(x; -
xi- d,
then (6.5.5) yields
r3+1 =rj+Ksin(0),
6.5.6a)
where K = (- 2nnA/Ak), and (6.5.6) gives 0;+1
=03+r;+1.
(6.5.6b)
Thus (6.5.6) is again a standard map, but in a very different context. Here it has nothing to do with dynamics, but rather the relative configuration of adjacent particles in an equilibrium configuration. We will consider the significance of these physical differences shortly. We will take the following as one of the canonical forms of the standard map: S:
r,+1
0,,+1
(6.5.7)
The standard map in this form, or a similar map, arises in many other situations
involving the interplay between translational and some periodic motion. Two examples are magnetic bottles and toroidal magnetic systems (e.g., a stellarator, used
The standard map; twist maps
37
in plasma fusion experiments), where (6.5.7) refers to the particle's (r, 6) location as it crosses some cross-sectional plane in the system, see Fig. 6.43. Fig. 6.43 Guiding center motion along B field which has a rotational transform (twist)
Magnetic bottle
The mappings (r,,, 1, 0,,+ 1) in these cases are not as simple as (6.5.7), but have many of the same equalitative features. Specific examples of maps in tandem mirror devices have been considered by Ryutov and Stupakov (1978) and by Ichikawa,
Kamimura, and Karney (1983). Other experimental results have been obtained in a stellarator (toroidal system) by Sinclair, Hosea, and Sheffield (1970), which will be discussed after we understand some aspects of the standard map. Note that in these magnetic systems a region in the cross-sectional plane ((I.) maps into another region (11,,+ 1) which has the same flux
fa. Bdxdy=
BZdxdy. fon
(See Fig. 6.44). This follows from Green's theorem ('
f V-Bd3x = v
B-dA, Js
and taking the volume V, to be the tube defined by the field lines from 0 to i2,,+ Since V-B = 0 in the volume, the integral on the right side yields the previous result. Fig. 6.44
Models based on second order difference equations
38
With a suitable choice of a metric (so that dx'dy' = BZ(x, y) dx dy), this invariant flux could be made an invariant area (i.e., a measure-preserving mapping).
What is really important for future arguments is that something be preserved (invariant) under the mapping, which is of the form
J'Jdr d0a(r, 0)
and
oo > a(r, 0) > 0.
(6.5.8)
To simplify matters one usually takes an 'area'-preserving mapping (in some space). This `area' may not be the physical area, but the latter will be of the form (6.5.8). The point is that, if (6.5.8) is invariant, then we can conclude topological consequences regardless of the detailed nature of the positive (nonvanishing) function 6(r, 0). This will be demonstrated below. The area-preserving character of the map (6.5.7) is shown by evaluating the Jacobian of the `transformation' (map) from the region Stn to Q,,+ 1, namely 0(rn//+IIen+1) _ 3(rn, 0n)
1
K cos (0n)
1
1 + K cos (0n)
= 1.
(6.5.9)
Another common representation of the standard map is in the form p,,+1 = pn - 2 sin (2ngn) qn+1 =qn+pn+1
(6.5.10)
which is related to (6.5.7) by the simple transformation 0 = 2nq + it, r = 2np. Thus the 2n intervals of (r, 0) become unit intervals of (p, q).
Depending on the physical application of the standard map, it may be useful to consider the map in different spaces. Thus, in the above magnetic field example, the `natural' representation is in the polar coordinates of (6.5.7). That is, in R + x S1(reR+,
OES'), where the rectilinear axes are the (x,y)-axes in the plane transverse to the magnetic field.
In the above examples of accelerators or the solid state lattice, the variables r (or
p) can take on negative as well as positive values, so the polar representation is inappropriate. One compact representation is to use a cylindrical space, (rER1, 0ES1),
as illustrated in Fig. 6.45. In this case 0 and 0 + 2n (or q and q + 1) are treated as the same point, which however may not be physically meaningful (e.g., in the lattice example).
Finally, particularly for many mathematical discussions, the standard map (6.5.10) is represented on a unit two-torus, T2 = S' x S' (Fig. 6.46). In this case both p and q are taken to be periodic, mod (1). This is a very compact representation, but it is not always appropriate for physical applications, as noted above.
The standard map; twist maps Fig. 6.45
39
y
x
27r
Fig. 6.46
P 1
0
9
1
To make the differences in these representations quite clear, Fig. 6.47 shows the same three sets of points, obtained from three initial conditions, in polar, cylindrical,
and toroidal representations (for K = 0.5). The central curve in T2 and R' x S'
corresponds to the circle in R+ x S'. The smaller loop at the top of R' x S' corresponds to the inside arcs (top and bottom) of S' x S', and the `banana' region of R+ x S'. Such banana regions are of great importance in plasma fusion systems, so this representation is of crucial importance in that case (see the next section). Finally, the large bottom loop in R' x S' becomes the outside arcs (top and bottom) of T2, but this set is improper for polar coordinates because p (or r) has negative values. Therefore when this improper set plotted in (what should be) R+ x S1, the result is interior loops in Fig. 6.47 (because (- r, 0) is treated as (r, 0 + ir)). The physical
implication of this is that the standard map cannot be correct in such regions (if r or p is required to be positive). This is illustrated in Exercise 6.6. This polar representation is very useful to visualize some general aspects of the standard map (Fig. 6.48). Thus, if k = 0, r is constant (r, 1 = re), so the mapping is along circles, and it twists the plane (larger 1 - 0,, for larger r). In this case (k = 0) the rotation number of the map is p(r) = r/2n, according to the definition (6.2.6). There are two types of circular surfaces in this plane: (a) If the rotation number is rational, p(r) = m/n (m, n: relative primes) then every
40
Models based on second order difference equations
Fig. 6.47 K = 0.5. (a) Polar map, R+ x S'. (b) Two-torus map, TZ = S' x S'. (c) Cylindrical
map, R' x S'. pmod(1)
p
0 I
gmod(1) 0 (b)
(a)
gmod(1) (c)
Fig. 6.48 Standard map (k = 0).
point on a circle is periodic with period n, and S":
(r/2n = m/n, 90) -- (r/2,r = m/n, 60)
(for all 90).
These are the rational circles. This set of circles is dense in the plane but denumerable, and hence has zero measure. (b) If r/2n = irrational number then the mapping (6.5.4) ergodically covers the circle,
that is the set of points S'(r, 00) (1- oo) are dense on the circle r. This set of irrational circles is nondenumerable, and has measure one in the plane.
Near-integrable systems
41
The standard map is a particular example of the general twist map TE:
r"+ I = rn + Earn, 0n) 0n+ I = 0,, + 2np(r,,) + E$(r,,, en)
(6.5.11)
where p(r) is the rotation number if E = 0. The functions a(r, 0), /3(r, 0), and p(r) satisfy the area-preserving condition a(en+I1rn+I)=
1,
(6.5.12)
0(0n, rn)
and a(r, 0 + 2n) = a(r, 0), /3(r, 0 + 2n) = /3(r, 0). The map T. has the same properties as
the standard map, provided that E is small enough (for example, so that it
is
orientation-preserving, E0/3/00 > - 1). We will therefore continue to use the standard map as the specific example of (6.5.11), since it is widely used and discussed.
6.6 `Near-integrable' systems Now we return to the standard map and consider what happens if k 0 0, but is `sufficiently small'. Note that this is not the parameter range of interest in the microtron
accelerator (Exercise 6.7). We will consider a region near some rational surface r/2n = m/n (call it C) and the map S" (so all points on C are fixed points of S", if k = 0). In the region of a nearby circle C+(r/2n > m/n) all points rotate counterclockwise under the map S", whereas near another nearby circle C-(r/2n < m/n), all points rotate clockwise under S" (see Fig. 6.49). Then by continuity, if k is small,
at each 0 there must be a point between C+ and C- which does not change its
Fig. 6.49
C+
42
Models based on second order difference equations
angular location under the map S". Thus there exists an r = R(0), such that S":
(R(0), 0) - (r'(0), 0).
(6.6.1)
The set of points r = R(O) defines a continuous curve, which we will call D. The points
of D are mapped only in a radial direction by S". If S" has any fixed points (r0, 00) they must lie on D, and satisfy r' = R(00) __ r0 in (6.6.1). Next consider the map of the curve D, as given by (6.6.1). The essential point about the map S" is that it is area-preserving, so that the deformation of the region enclosed by D must be to another region of equal area. This yields limited possibilities, several of which are illustrated in Fig. 6.50. Aside from possible degenerate cases (see below), Fig. 6.50
S"D
the curves D and S"D must intersect at an even number of points. Moreover, these points of intersection are fixed points, so the area-preserving map S" has an even number of fixed points. Actually, this conjecture by Poincare was proved by Birkhoff (1913) after Poincare's death. This Poincare-Birkhoff fixed point theorem is known as Poincare's last geometric theorem (so called by Birkhoff). Some controversies concerning the proof of this theorem existed for many years (Brown and Neumann, 1977), but the present discussion suffices our needs. The above is true for any n, but more can be concluded about the n dependence.
Near-integrable systems
43
Let (ro, go) be a fixed point of S" S"(ro, Bo) _ (ro, Bo),
then clearly (1 = 0, ... ,)
S"+'(ro, 00) = S'(ro, 00)
so the points S(ro,00),S2
(ro,00),...,5n
(ro,00)
are also distinct fixed points of S". Hence every fixed point of S" on D belongs to a set on n distinct fixed points of S". Moreover it will become clear shortly that the adjacent fixed points in Fig. 6.50 cannot belong to the same invariant set. Therefore
S" generally has at least 2n distinct fixed points in the region of the curve (C:r/2n = m/n). Exercise 6.8 Show that the number of fixed points of S" can only be 2nk (k = 1, 2, ...),
and determine the meaning of k. What are the possible values of n for the second figure illustrating S" D?
On the other hand, because of the area-preserving character of this purely radial map, it might seem reasonable from the above figures that there can only be a finite number of fixed points for the map S". If S were a polynomial map we could easily prove this fact, for it could be related to the number of zeros of a pair of polynomial equations (such as (6.6.2), below). In other cases we can establish that the fixed points are nearly always isolated. To show this, consider an area-preserving map which has a fixed point at the origin (x, y) _ (0, 0)
x"+1 =F(xn,Yn);
(6.6.2)
Yn+1 = G(xn,Yn)
with F(0, 0) = G(0, 0) = 0. To investigate the map near the fixed point, we consider the linearized form
(-F)(OF) ax
o
ay o
Yn+1
=(az)oxn+(Ox
0Yn
or simply
(Yn+1/ -(M21 M22)\y /
(6.6.3)
The fact that it is area-preserving implies that a(xn+1'Y.+1) a(xn, Yn)
M11 M12 = 1. M21 M22
(6.6.4)
Models based on second order difference equations
44
Next consider the eigenvectors of this map
Mil M12 \M21
X = M22/ \Y/
(6.6.5)
I\y/
The eigenvalues A are given by the determinant
IM11-A M12 I M21
I=0
M22-Al
or, using I M I = 1, we obtain the so-called characteristic multipliers of the map at this fixed point:
A = i Tr M ± i [(Tr M)2 - 4]";
(Trace M =_ Tr M =- M11 4- M12).
(6.6.6)
Therefore the two eigenvalues satisfy A122 = I M I = 1, and either (a) 21 and 22 are real with 121 I % 1 and 122 I (= 11 /211) < 1, or else (b) they are complex conjugates of the form 21 = exp (i9), 22 = exp (- i9). In case (a) the fixed point is a (saddle) hyperbolic
point, whereas in case (b) it is a (center) elliptic point. These fixed points and some sets of points which are mapped into themselves, and hence are invariant sets, are illustrated in Fig. 6.51. Fig. 6.51 (a) l Tr M I > 2. (b) I Tr M I < 2.
(b)
(a)
It is very important to note that these curves are not the same as flow lines in phase space, because the points are related by a discrete map, (6.6.3). We will call these invariant sets, the invariant curves of the map. Some details of the map applied to points along the curve are illustrated in Fig. 6.52. In particular, the outgoing separatrix, So, is the set of all points whose a-limit point is the fixed point, whereas the incoming separatrix, Si, is the set whose w-limit point is the fixed point. Fig. 6.52 Invariant
TPZPT3P I_1P QTQ
curve
T2
T3Q
45
`Near-integrable' systems
Exercise 6.9 What is the difference in the maps Tr M > 0 and Tr M < 0 (called `direct' and `inverse' respectively)? Clearly in these cases there cannot be any other fixed point of M in the neighborhood of (0, 0). The only possible exception is the case where Al = 22 = ± 1 (Tr M = ± 2). In this exceptional case the point (0, 0) is called a parabolic point, and the neighboring points are either fixed points of M or M2 (if Al = 22 = - 1). Hence the only possibility for the fixed point (0,0) not to be isolated is if it is a (direct) parabolic point (A1 = 22 = ± 1). Since this is a `measure zero' possibility it is usually not important. The fact that the area-preserving map usually has either hyperbolic or elliptic fixed points is made particularly clear by considering the standard map again. Recall that there is a closed curve D which is mapped radially by S", preserving the enclosed area. The fixed points of S" lie on D, and are the intersections of D with S"D. Since the `flow' of the map S" is counterclockwise (and, of course, also radial) outside these curves and clockwise inside, it is clearly seen that the flow generated by S" near the fixed points is either hyperbolic or elliptic (as illustrated in Fig. 6.53). Note moreover they occur in alternate pairs - this is a global feature which is not predicted by the above local analysis near the fixed points. Finally we note that, because elliptic and Fig. 6.53
S" D
r 2a I
m n
46
Models based on second order difference equations
hyperbolic flows are not topologically equivalent, it follows that elliptic fixed points are not mapped into hyperbolic fixed points. Hence there are always 2nl fixed points of S", where l is an integer, as was claimed above. Before considering more details let us restate the results so far:
All rational circles (r/2n = m/n) which were invariant curves under S, if k = 0,
do not go into other invariant curves if k # 0. They are `destroyed,' in the sense that S" has only isolated fixed points, and therefore any perturbation method for calculating the change in C(r/2n = m/n) for k:0 must diverge. This by no means completes the story of what can be deduced for area-preserving maps. To proceed, we will first discuss such maps near hyperbolic fixed points, then consider elliptic fixed points, and finally put them together, as required above.
Hyperbolic fixed points (h.fp.) Consider a hyperbolic fixed point of a map S (e.g. replacing S" above). We might use the standard or twist map as an example, but only the area-preserving property is of importance in the following. The eigenvectors of the linear approximation (the full lines in Fig. 6.54) are tangent at the h.f.p., x0, to curves which are invariant under the full nonlinear map. These curves are simply the sets of points which are either Fig. 6.54 i
WS
W°
mapped asymptotically to xo by S" or S-" as n tends to infinity. Specifically the unstable manifold, W' , is defined to be the set W" = { x: l
Jim S-"x n-oo
= xo }
(6.6.7a)
whereas the stable manifold is the set
w = {x:
lim S"x = xo}
(6.6.7b)
`Near-integrable' systems
47
where x0 is a hyperbolic fixed point. These manifolds are tangent to the eigenvectors with I A I> 1 or I A I< 1 respectively (see Fig. 6.54).
What is the character of these manifolds away from the immediate neighborhood of the hyperbolic fixed point? That is, if xe W', where does S"x go as n -+ oo or if xe Ws, where does S-"x tend as n - oo? The uncommon ('nongeneric') possibility is for two fixed points of S, a and b, to share their stable, W5, and unstable, W", manifolds as illustrated in Fig. 6.55(a). Of course what is stable for one is unstable for the other.
The second nongeneric possibility is for Wa = Wa for a single fixed point a, as illustrated in Fig. 6.55(b). Fig. 6.55 WU=WS b a (a)
a-fixed points-b
An impossible situation is for a manifold to cross itself. Fig. 6.56 illustrates W' crossing itself. Such a crossing violates the condition that nearby points map to nearby points (by continuity). To see this, consider the crossing point a, and the points b, c, d. Their mapped points Sa, Sb, Sc, Sd follow in that order by continuity around the loop. But d is next to a, so Sd should be next to Sa, again by continuity. Hence there is a contradiction, and the crossing at a cannot occur. Fig. 6.56 h.f.p.
48
Models based on second order difference equations
Instead, the generic situation is for WS to cross a manifold W' which is connected with either the same hyperbolic fixed point, or with another one. These are possibilities illustrated in Fig. 6.57. If the manifolds Wb and Wu, associated with the same fixed point b, have a point in common, it is called a homoclinic point. On the other hand,
if Wa crosses W6, involving the two fixed points a and b, then their intersection point is called a heteroclinic point (see Fig. 6.57). The figure however illustrates more than one homoclinic and heteroclinic point.
Homoclinic points
Poincare indeed recognized that if there is one homoclinic or heteroclinic point there is an infinite number of such points. This follows from the fact that if xE(Ws and W°), then
S"xe(WS and W")
(n = 0, 1, 2,
... J.
(6.6.8)
This is so because all points of WS map onto WS and similarly all points of W° map onto W' . Moreover, if the map S is one-to-one, there cannot, be a `first' homoclinic point. This is, if xE(Ws and W"), then
S-"xE(Ws and W")
(n=0,1,...,)
(6.6.9)
(note that it is S` rather than S"). In other words the inverse map of a homo (hetero)clinic point must also be a similar point. Note finally that this series of maps cannot have a last point, for this would require that it ends up at a fixed point*, but *S"x might also end up on a periodic set (x is eventually periodic), but this is also excluded for one-to-one maps.
`Near-integrable' systems
49
Fig. 6.58
S
then the inverse map would not be off the fixed point. Thus Fig. 6.57, to be more (but not totally!) complete, should look as illustrated in Fig. 6.58. As bad as this looks it does not begin to do justice to the complexity, because not only must there be an infinite number of homoclinic points, (6.6.8) and (6.6.9), but W" and WS must oscillate more and more wildly (in phase space) between the points
S"x(n- ± oc). Indeed, between any two homoclinic points on W", say xo and xo, there must be a dense set of intersections of W° and Ws! (see Fig. 6.59.) Similarly a dense set of intersections exist on WS between xo and x,. By a dense set, we mean that in any portion of W we can find an intersection. Fig. 6.59
To see that this must be so, assume that there are no such intersections. Then all points in AO must map to Al = SAO. Similarly, we would have A,2 = SA1. Note that
adjacent loops do not map into each other because the mapping is orientationpreserving. Thus Sxo = xi, rather than xo*, because only the former retains the
50
Models based on second order difference equations WS
Fig. 6.60 xo
M110.
wu
WS
/' WS
}
not xo**
-Wu
1) orientation of W° and WS, as illustrated in Fig. 6.60. Thus we see that all would be confined in a finite region bounded by WS (since we assumed there are no further intersections) and Wu (since W° cannot intersect Wu). However this map is also area-preserving, so there is not room enough for an infinite number of equal areas, A,,, in a finite region. Thus we conclude that between xo and xo there must be a dense set of intersections of WS with Wu (a few are illustrated in Fig. 6.61(a)).
Fig. 6.61 0
(a) S
`Near-integrable' systems
51
If there were any finite arc on W° not containing an intersection, the above argument could be repeated for that finite-area arch (shaded in the figure). A similar situation exists, of course, between xo* and x1, and so on.
The consequence of this is illustrated more globally, but with less detail, in Fig. 6.61(b). Note where (A, B, C) have mapped after three iterations, (A', B', C').
Exercise 6.10 This complexity is best appreciated by deducing for yourself the mappings which are implied by such `secondary' homoclinic crossings. Consider the rather simple situation shown in Fig. 6.62. Determine the possible locations for the maps, S, of the homoclinic points a and b. Specifically do this for a1 = Sa, a2 = Sa1, b 1 = Sb, and b2 = Sb 1. Show how these points lie on the invariant curves. Note in particular what must occur between points b and 2, or 1 and a, or a and b (etc!). Fig. 6.62
Finally, there is yet another feature about the region near homoclinic points, which was pointed out by Birkhoff (1927), as illustrated schematically in Fig. 6.63. Consider a point po inside the loop formed by the crossing of W° and WS at the homoclinic point. The point is mapped into a series of points, p. = Spo, and any arc connecting (po, p 1) maps into arcs connecting (p,,, p,, + ), forming some curve. If we consider the limit where po approaches the hyperbolic point, xo, and the arc connecting
(Po, P1) is simply taken to be a straight section which becomes parallel to the appropriate eigenvector (6.7.5), then it is clear that the connecting arcs form a curve
which crosses the similarly generated `backward' curve, between the points (p _ p _,._ 1)(n = 0,1, 2, ...), and that this crossing is in the neighborhood of the homoclinic point. As po approaches xo the density of mapped points, p,,, on these
52
Models based on second order difference equations
Fig. 6.63
curves increases continuously, because the density of mapped points on WS and W°
goes to infinity as x approaches x0. In particular the density of points near the homoclinic crossing region, S"po, S-"po (where n N, a finite number), increases continuously. Because of this type of continuity, as po - x0 there are an infinite number of points Po = Pok such that Pk = S"kpo = S-"kpo (k = 1, 2,...), where these `crossing points,' Pk, are near the homoclinic point. These points are clearly periodic, S2"kpok (and similarly for Pk), so that in any neighborhood of a homoclinic Pok = point, or its associated hyperbolic point, there are an infinite number of periodic points of the map. A few of the characteristics of a flow in R 3 which could produce the Poincare map near a hyperbolic fixed point is illustrated in Fig. 6.64. The trajectory which passes through the hyperbolic point has nearby trajectories which fan out from it in one direction, and converge toward it in another direction. Only a small portion of these unstable and stable manifolds are indicated. This will be discussed in greater detail in the next chapter. Also only three homoclinic points are illustrated. The fact that WS and W' consists of discrete points from the map generated by many trajectories, Fig. 6.64
`Near-integrable' systems
53
is illustrated by showing points produced by one trajectory on WS n W°, and another on only WS. Beyond these details, the picture becomes very complicated indeed. Considering the above complexity near a hyperbolic point, it is not surprising that this dynamics can be put into correspondence with any Bernoulli sequence (Hirook, Saito, and Ford, 1984). To show how this can be done, label the regions (0, 1) around
a hyperbolic point, as illustrated in Fig. 6.65, and consider the shaded regions in Fig. 6.66. Consider starting from a state near the hyperbolic point in the figure, and returning to its neighborhood. Fig. 6.65
0 1
1
0
Fig. 6.66
S-2x
W°
(1) If we start in region 0, and also in (a) the region ® , then we can return to region 1 (b) the region , then we can return to region 0. (e.g., consider S-'x in Fig. 6.66, and follow it around to near S+Sx). (2) If we start in region 1, and also in: (a) the region , then we can return to region 0 (b) the region , then we can return to region 1. However
® overlaps both
and
in 1
overlaps both 0 and
in 0.
near
Models based on second order difference equations
54
Moreover this overlapping is dense near the hyperbolic point. Thus a solution in ® may be in either or 0 in 1 after cycle-one, and therefore go to either 0 or 1 in cycle-two. Following this reasoning, we see that in two cycles we can go through any sequence of regions. 000, 001, 010, 011,100,101,110,111.
Continuing this process, we find that in any neighborhood of the hyperbolic point we can find a solution corresponding to any Bernoulli sequence.
Elliptic fixed points The above discussion gives a vague idea of the complexity of area-preserving maps
near hyperbolic fixed points. What occurs near an elliptic fixed point? It has been shown by Arnold (1961) and Moser (1962) that around the elliptic fixed points, for sufficiently small k in the standard map (6.5.7), ore in the twist map (6.5.11), there are closed curves which are covered densely (ergodically) by the iterates of the map,
S"(n = 1, 2,...). Fig. 6.67 repeats an earlier figure, showing two elliptic and two
Fig. 6.67
hyperbolic fixed points, and one of the closed curves about each elliptic point (there may be many others). Fig. 6.68 shows how such invariant curves can be generated by the map of a flow inside a torus. This particular illustration is for the case where
the rotation number of the elliptic points is 1 (they are fixed points of S2). This contrasts with the simpler case in Fig. 6.67 for the hyperbolic point which is a fixed point of S (rotation number 1). Each invariant curve about the elliptic points represents the intersection of a plane with a double-looped torus, which is covered ergodically by a trajectory. For simplicity, the two hyperbolic points and their associated flows
in R3 are not drawn in the same figure. These invariant closed curves about the
`Near-integrable' systems
55
Fig. 6.68
elliptic points are isolated from one another. Zehnder (1973) has shown that between any two such invariant curves are hyperbolic and elliptic fixed points of some higher
order map. That is, if the elliptic fixed point is a fixed point of S", then around it (between any pair of closed curves) are elliptic and hyperbolic fixed points of (S")'". What this means is that we obtain a `microcosm' of what occurs for the original system of rational and irrational curves. Around each of these secondary elliptic fixed points is another family of closed curves, between which is yet another set of elliptic and hyperbolic fixed points of still higher iterates of the map (e.g., ((S")'°)P). This process of building microcosms continues indefinitely, and is illustrated in magnified steps in the figure. Thus, around the original fixed point of S, the circles with rational rotation numbers
break up into hyperbolic and elliptic fixed points. However, for sufficiently small values of k, some of the circles with irrational rotation numbers are only slightly perturbed, and remain closed curves which are densely covered by iterates of S. These `preserved' closed curves are now known as KAM (Kolmogorov-Arnold-Moser) curves illustrated in Fig. 6.69. A characteristic example of such preserved curves, produced by the periodically forced conservative oscillator, was discussed briefly in Chapter 5,
Section 14. In that case the trajectories on the preserved KAM tori produce the present KAM curves in a Poincare map. As these KAM curves fill up most of the plane (their measure goes to unity), but for any k :0, the rational surfaces become a dense set of `broken' rings, consisting of the hyperbolic and elliptic fixed points with their associated homo(hetero)clinic points and microcosms within microcosms! For greater details see Moser (1973), Arnold and Avez (1986), and Zehnder (1973). This incredible complexity was recognized by Poincare, and led him to observe (Vol. 3, p. 382, Les Methodes Nouvelles de la Mecanique Celeste): The intersections form a kind of lattice, web, or network with infinitely tight loops; neither of the two curves (the `outward' or `inward' of a hyperbolic point)
must ever intersect itself, but it must bend in such a complex fashion that it intersects all the loops of the network infinitely many times.
Models based on second order difference equations
56 Fig. 6.69
Irrational p
k ='small'
One is struck by the complexity of this figure which I am not even attempting to draw. Nothing can give us a better idea of the complexity of the three-body problem and of all problems in dynamics where there is no holomorphic integral and (the canonical perturbation) series diverge.
Despite Poincare's reluctance to draw this complicated figure, a number of people
have ventured to give a characterization of this situation, similar to Fig. 6.70. Fig. 6.70
The breakup of KAM curves
57
Obviously this figure is not to be taken seriously, since there is no way to illustrate dense sets of preserved and broken (k = 0) surfaces. The preserved surfaces, illustrated in Fig. 6.70, as closed circles, although they are
generally distorted, are a special subgroup of the k = 0 irrational surfaces. This subgroup is characterized by the fact that their irrational rotation number is not closely approximated by rational numbers, m/n, with small values of n. The specific condition is given by Moser's twist theorem (1962), namely that if the rotation number satisfies the inequality m
P
>
n
C > 0, and all integers m, n, then the surface is preserved for sufficiently small values of k:0.
6.7 The breakup of KAM curves The magnitude of k which is required to destroy a particular KAM curve apparently depends on `how irrational' is the rotation number associated with that curve. All irrational numbers can be approximated to any accuracy by rational numbers, m/n, provided that n is sufficiently large. A classic result along this line is the 1891 theorem due to Hurwitz (e.g., Olds, 1963), which states that any irrational number a has an infinite number of rational approximations, m/n, which satisfy the inequality m < 1/(5'12n2) n
(n > 1).
(6.7.1)
Moreover the number of rational approximations is generally finite if 5'/2 is replaced by any larger number. In other words, if we want to approximate a to any arbitrary accuracy, so that n has to be taken arbitrarily large, then the value 5" is the largest number which can generally be used in (6.7.1).
On the other hand, all irrational numbers, a, aside from a set with measure of order C, satisfy the opposite inequality
a--mn > C/n2
+E
(8 > 0)
(6.7.2)
for sufficiently small C. Because of this, it is possible to show that the `small denominators,' Am,n/(mil, - ndd2), which arise in perturbation theories involving coupled oscillators, (52,, i22), do not necessarily decrease more rapidly than their numerators, A.n, in the higher order terms (larger m, n). In particular, as Kolmogorov
suggested, a perturbation theory based on Newton's method can be proved to converge for most irrational rotation numbers, provided that the coupling of the
58
Models based on second order difference equations
oscillators is sufficiently small. This is analogous to small values of k in the present case. The convergence of such perturbation methods can be used to prove that KAM surfaces exist in such cases. More details on this topic will be found in the notes on the KAM theorem, in Appendix L. If the inequality (6.7.2) is not satisfied, there is no proof that the KAM curve p = a is preserved when k 0. On the other hand, the failure of the convergence proof of the perturbation series does not prove that the KAM curve is in fact destroyed. The interesting and important problem is to determine the values of k which destroy the various KAM curves. The motivation for this type of study is not based solely on abstract interests, but
is also tied to a variety of physical questions related to the possible irreversible, diffusive, or ergodic behavior of various physical systems. In the present case, where
the KAM surface is simply a closed curve in R2, the influence of such preserved surfaces on the possible behavior of a system is relatively clear (much clearer than in higher dimensional cases, as will be seen in Chapter 8). The mathematical idea is quite simple. If there exists a closed invariant curve (i.e., a closed, one-dimensional, continuum of points which maps onto itself), then this curve divides R2 into two disjoint regions, namely the inside and outside regions produced by this closed curve.
The dynamical importance is that a point in the interior region cannot map onto a point in the exterior region, or vice versa. There are several ways to see that this is true. One way is to look back at the illustration of the R3 flow in a torus which gives rise to the two KAM surfaces around two elliptic points (Fig. 6.71). In order for a point to be mapped across a KAM curve,
it is clear that the corresponding trajectory in R3 would have to go through the Fig. 6.71
double looped torus at some point. However, this torus is covered densely by the trajectory which is mapping the KAM curve in R2. Thus this would require two trajectories to intersect, and this is not possible for unique solutions. An alternative approach, which does not require the consideration of the flow in R3, is to simply use the continuity of the standard (or twist) map as a function of k (ore). For k = 0, all interior points map into the interior, so for the contrary to happen for some k # 0 for which that KAM surface still exists, requires an intermediate value
The breakup of KAM curves
59
of k such that the interior point maps onto the KAM curve. However, since the KAM curve is invariant, and has a unique inverse map, the interior point cannot map onto it. This again proves the feature.
Exercise 6.11 You can also show that an interior point can not be mapped into an exterior point, by using the continuity of the map with respect to x = (r, 0). Try it. Therefore, while a point in the phase space may be mapped in a very erratic fashion around hyperbolic points, as described above, this erratic motion will be confined to be inside, or outside of any existing KAM curves, as schematically illustrated in Fig. 6.72. This means that the dynamics, while erratic, is highly restricted in the phase
space. In particular, if we consider a group of nearby initial states, they can only
spread out in a diffusive fashion between preserved KAM curves. Such a physical system retains a significant amount of correlation with its initial state, and therefore does not behave like a system approaching a `universal' equilibrium distribution (such as a Gibbsian distribution in statistical mechanics). On the other hand, when a KAM curve is destroyed (breaks up) and ceases to be a continuum, then there is no longer an inside-outside restriction on the mapping dynamics. This is crudely illustrated in Fig. 6.73. Different regions of initial conditions, which were previously on different sides of a KAM curve (for smaller k), now may Fig. 6.73
Possibly!
Initial
distribution
KAM
`Broken' KAM
become distributed over phase space in a similar fashion and loose the correlation with their initial state. This would indicate some form of relaxation, or `irreversible' behavior of the physical system. The fact that may happen is no guarantee, of course,
Models based on second order difference equations
60
that such dispersion will happen, as we will soon see. However, once the last KAM curve is destroyed, there is at least some opportunity for dispersion to take place. To be more specific, let us consider again the standard map, and the fate of some `KAM surfaces', as k is increased. We call them `KAM surfaces' (in quotation marks),
because they look like smooth surfaces (lines) on a computer screen. However many may not be connected lines, but a series of islands, etc., too small to numerically resolve. But let's not worry about that at first, and just see what a simple computation produces.
Figures 6.74 and 6.75 show the behavior of eight arbitrarily selected initial conditions po = 0.97 - 0.12n,
qo=0.49
(n=0,...,7)
each iterated 500 times and for three values of k = 0, 0.02, and 0.25 we obtain the first three diagrams (Fig. 6.74). Fig. 6.74 ........................................................
------------......................................................
..................................................
k=0
k=0.02
k = 0.25
When k = 0, we see clearly that a number of initial po are low order rationals. Hence po = 0.85 has period 20, and po = 0.25 has period 4. All the other po are also periodic, but are only barely resolved on the graph. These orbits lie on the KAM curves defined by same po, but all 0 < q < 1, { p, q I p = po, 0 < q
11.
Changing k only slightly (k = 0.02), takes the initial state po = 0.97 off the above set of KAM curves, because it no longer maps across the entire interval 0 < q < 1. Note again that the top and bottom arcs are connected (on T2). All the remaining initial conditions now appear to ergodically cover continuous lines extending from q = 0 to q =1(= 0), hence, they iterate on KAM curves. Raising k to k = 0.25, we can see that n = 2 now resides on a set of four islands, whereas n = 4 iterates on only two islands. Thus these orbits clearly do not reside on any KAM surface. Moreover we now have three single islands corresponding to
The breakup of KAM curves
61
Fig. 6.75
k = 0.60
k = 0.75
k = 0.85
n = 0, n = 1, and n = 7. Note that, on V, the island n = 7 lies between n = 0 and n = 1.
Raising k to k = 0.6, causes the initial state n = 6 to iterate along 16 islands, but two initial states (n = 3, 5) still appear to be on KAM surfaces. When k = 0.75, only the initial state n = 3 appears to remain on a KAM surface. The state n = 2 has moved onto a set of six islands, which are difficult to resolve, since they are next to the chaotic set from n = 6. Fig. 6.76
62
Models based on second order difference equations
Finally, when k = 0.85, the initial state n = 3 iterates on three islands. Hence none of these initial states remain on a KAM surface. This, of course does not mean that there are not KAM surfaces associated with other initial states. To finish these numerical examples, we consider a large value, k = 1.1, but represent
the mapped points in the polar representation Fig. 6.76. There are several reasons for doing this, as will become clear later. One point to note here, is that the chaotic set is all one orbit (3,500 iterations). It can be seen that, even though there is no remaining KAM surface from the original set, this chaotic set does not wander freely over the plane, as schematically illustrated in Fig. 6.73. Thus the global properties of chaotic motion is not simply a question of the preservation of the original KAM surfaces. Indeed, as the figure shows, the new `banana' tori, are as regular as the origin KAM tori (k = 0). They are, in fact, often referred to as KAM surfaces, because of
this feature. The only distinction between the two KAM sets is that they do not continuously deform into each other as k is varied. That, however, is a physically irrelevant fact, since the banana tori can again separate regions of space, preventing
dispersion (the above chaotic set does not enter any new `KAM torus'). We will return to these features in the next section. Having some numerical results, let us now consider briefly some methods to determine the value of k which destroys a KAM surface. To show that a KAM curve exists (has not been destroyed) by numerical methods entails a search for a sequence of mapped points (an `orbit'), which never repeats, and which continues to fill up a smooth curve in the (r, 0) plane. It is clear that no numerical method, with its finite precision and finite time constraints, can establish that such a curve exists. Greene (1979) has suggested an alternative approach which is based on the stability property of certain finite (periodic) trajectories generated by the standard map. He
suggested that we should consider a sequence of periodic orbits, with rotation (winding) numbers PN = PNQN which converge to the irrational rotation number, p, of the particular KAM curve of interest NI mPN=Ni mPN/QN=P
(6.7.3)
In this case, for any N, we are considering only finite (periodic) orbits. However the length of the orbit equals PN iterates, and becomes increasingly longer in the limit (6.7.3), so the analysis is not trivial. A systematic way to develop a sequence of rational approximations of some p, satisfying (6.7.3), is to employ the unique continued fraction representation of the irrational p, p = ao +
1
1
al + a2 +...
= [ao, a,, a2, ... ],
(6.7.4)
The breakup of KAM curves
63
where the a,, are positive integers. Thus ao is the integer part of p, whereas a1 is the integer part of 1/(p - ao), and so forth. For example, e - 1 = 1.718281828459... _ [ 1,
whereas
(e-1)/2=[0,1,6,10,14,18,...]. By truncating the continued fraction at the Nth term, we obtain the rational approximation PN = [a0,a1,..., aN] = PN/QN,
(6.7.5)
which is called the Nth convergent of p. The rational number, PN/QN, is the `best' approximation of p, in the sense that I P - PN I < I P - P/Q I , for any integers (P, Q), if Q < QN (or P < PN). Put another way, the periodic orbit with the rotation number PN has the minimum separation from the chosen KAM curve among all periodic orbits of a given length. The differences, p - PN, alternate in sign as N is increased, so PN converges onto the KAM curve from both sides. To illustrate this, the first five convergents of e -1(above) are [1] = 1, [1, 1] = 2, [1,1, 2] 1.6667, [1,1, 2, 1] = 1.75, [1, 1, 2, 1, 1] = 1.7143 whereas the first five convergents of (e - 1)/2 are [0] = 0, [0,1] = 1, [0, 1, 6] = 0.85714.... [0, 1, 6,10] 0.859155, [0,1, 6,10,14] = 0.859140859.
The convergence of the approximations is much more rapid in the second case (IP - P51/P ^' 6 x 10-8) than in the first case (IP - P51 /P 2 x 10-3). This reflects the fact that the values of the aN are much larger for the irrational number (e - 1)/2.
Exercise 6.12 Obtain ao,..., a4 for the irrational numbers n, 2112 and 3112 . Note an interesting property of the continued fraction representation of 2112 and 3112 (examples of `quadratic irrationals'). Obtain the first four convergents of both n and 21/2, and note the difference in the convergence to it and 2112.
The group of irrational numbers which are most difficult to approximate by rational numbers are those with the continued fraction form [ao, a 1, ... , a,,1,1, ... ], that is a,, = 1 for k > m. This forms an equivalence class of irrationals, and it is because of them that 5112 must be used in (6.7.1) (that is, if we exclude this class of irrationals, 5112 can be replaced by 8112 in (6.7.1)). The most famous member of this class is
P*=[1,1,1.... ]=(1+5112)/2
(6.7.6)
known as the golden mean. It is the irrational number least easily approximated by rationals, and therefore it presents the minimum difficulty to perturbation methods (because the small denominator problem is most easily avoided). It is perhaps not surprising therefore, as noted and established by Greene, that the last KAM curve to be destroyed as k is increased, is the one with the rotation number p*, (6.7.6).
Models based on second order difference equations
64 Fig. 6.77
L
A
1 B
-J
C
3X5
Exercise 6.13 (Fig. 6.77). If you would like to see a simple geometric occurrence of p*, consider a line segment AB, with an interior point C. Show that if the length AC is to CB as CB is to AB, then CB = p*AC. Also, it is widely felt that a rectangle is most esthetically pleasing if its sides are in the ratio p*. Hence the popularity of the 3 x 5 index cards Q= 1.03p*).
The rotational approximations (6.7.5) of (6.7.6) are pN = 1 + FN/FN+ 19
(6.7.7)
where FN is the Nth Fibonacci number, defined by FO = 0, F 1 = 1, and the recurrence relation FN+1 = FN + FN- 1
Greene considered the stability properties of the periodic orbits with rotation
(winding) numbers equal to the convergents (6.7.5) associated with the KAM surface of interest. As we have already seen, these periodic solutions are either of the elliptic or hyperbolic types, depending on the value of
R=4(1-TrM).
(6.7.8)
According to (6.6.6), the perturbations near the periodic points are hyperbolic in character if R >, 1 or R < 0, and elliptic if 0 < R < 1. Moreover, according to the Poincare-Birkhoff theorem, these types of periodic cycles occur in pairs. Greene found
that, for sufficiently small values of k, there is in fact only one hyperbolic and one elliptic periodic cycle for each pN. He then examined the possible relationship between the bifurcation of the periodic cycles with rotation numbers (6.7.5), as k is increased,
and the breakup of the associated KAM surface (i.e., p =
limN-. pN). The bifurcation in question involves the change in stability of the elliptic points as k is increased. This stability is determined by RN(k), given by (6.7.8), evaluated for the periodic orbit with rotation number PN, and considered as a function of k. Based on a variety of analytical and numerical results, Greene made a number of `assertions',
or conjectures, concerning the breakup of the KAM curves associated with the standard map. Because R, (6.7.8), appears to increase exponentially with the length of the periodic orbit, PN of (6.7.3), Greene defined a `mean residue' by f (PN) = (41 RN I )11IN.
(6.7.9)
Physical regularity in mathematical chaos
65
Greene's principal assertion is that the KAM surface with the rotation number p,, = limN. J pN exists if and only if
f(p.) < I. He also concluded that
f (p*) < f (p.)
for all p,,, : p*
(6.7.10)
so that the KAM curve p* is the last curve to break up as k is increased. The critical value of k, where this curve breaks up, Greene found to be C
= 0.971635....
(6.7.11)
For reasons discussed above, if k > k*, the system may exhibit widespread stochastic dynamics throughout the phase plane. A proof that no invariant circles exist for k > 3 was give by Mather (1984), and for k > 63/64 = 0.984375 by MacKay and Percival (1985).
Greene's method has also been applied to other KAM curves by Shenker and Kadanoff (1982). They obtained, for example, the critical values kc 0.834365, if p = [0, 3, 1, 4, 1,1, ... ] and k, _- 0.9744 if p = [0, 2, 2, 2,... ]. A renormalization method has been proposed (e.g., Lichtenberg & Lieberman (1983; section 4.5), and Escande (1985)) which indicates how every KAM surface breaks up as a function of k.
6.8 Physical regularity in mathematical chaos One physical system which is frequently associated with the twist map is a toroidal magnetic field used in plasma fusion research. The vacuum magnetic field, produced by external current-carrying coils, have twisted magnetic field lines, as they progress
around the torus (as schematically illustrated in Fig. 6.78). This is done to help confine a plasma, by eliminating some instabilities associated with the toroidal Fig. 6.78
curvature. Such field lines can be used to define a map on some cross-section surface, which is then a twist map. The fact that such vacuum magnetic fields, B, produce an area-preserving map follows from V - B = 0, as discussed in Section 6.5. We therefore conclude that the field lines should have the properties of such area-preserving twist maps.
Models based on second order difference equations
66 Fig. 6.79
To study such field lines, a charged particle can be injected into the magnetic field, in which it tends to spiral around a field line, as illustrated in Fig. 6.79. This leads to the so-called guiding center approximation, which is valid provided that the field strength varies very little over the gyroradius of the particle. In this case the moving center of the gyromotion (the guiding center) `closely follows' a magnetic field line. Therefore, following the map produced by the return of an electron on a cross section where it was injected, yields an approximate determination of the magnetic twist map. This is nicely illustrated in Fig. 6.80 (after Sinclair, Hosea, and Sheffield, 1970), in which both the experimental results (right) and theoretical predictions (left), from Fig. 6.80
Top
Top
aperture
Bottom
I
aperture
Bottom
the known coil currents, are illustrated. We see that the agreement between theory and experiment is quite good. These tori bear some resemblence to the `banana tori', obtained from the standard map in the last section. As discussed there, these smooth surfaces can also be referred to as KAM surfaces. In the present case we find both `preserved' (simple) KAM tori, and the cross section from a torus which encloses a magnetic field line with rotation number 6. The outer KAM torus is badly distorted, but nonetheless appears to be preserved.
What is conspicuously absent from such empirical maps are indications of the
Chirikov's resonance-overlap criterion
67
chaotic magnetic field lines, which we know must exist in any such area-preserving twist map. The field lines are well-defined mathematical concepts, but we are not
surprised that charged particles, with their finite-sized gyroradii, do not follow individual field lines, particularly if they are chaotically entangled. There is, however, the widespread belief that if electrons move in a region with chaotic field lines, then they necessarily will experience some form of 'random-walk' dynamics, and thereby exhibit `diffusion' of some sort. This belief frequently centers on the mathematical chaos of field lines, coupled with guiding center intuition. It should be noted, however, that the particle dynamics is governed by the Lorentz force my = (q/c)v x B(r)
and, moreover, not only does V-B = 0, but also Maxwell noted that VxB
=---. cat 4nj
+
18E
C
The point of interest at present is simply that V x B exists. In other words, not only does B(r) vary smoothly along B (for V-B exists), but it also varies smoothly normal
to B (for V x B exists). Thus, even in regions where field lines are chaotic (a mathematical chaos), the physically observable field strength, B(r), must vary smoothly. We therefore have degree of smoothness of B(r), even where field lines are chaotic. Whether particles will in fact 'random-walk' in such situations involves a larger scale determination of B(r) than afforded by field line maps.
Recalling the discussion in the last section, and the fact that field lines are as unique as trajectories, it follows that field lines can not map across KAM surfaces, and are therefore confined by such surfaces. Of course particles can gyrate across KAM surfaces, but if there are regions with many KAM surfaces (a gyroradius thick), then charged particles can be efficiently confined by such KAM onion-like surfaces. Particle collisions unfortunately also break down the simple guiding-center picture, and reduce the effectiveness of KAM-surface confinement. Conversely, as we saw in the last section, a breakup of some original KAM set, does not imply that the field lines will wander freely through space. It is clear that the physical problem of magnetic
confinement is much more complex than the mathematical features of chaos and KAM surfaces.
6.9 Chirikov's resonance-overlap criterion In this section we will consider a simple estimate, due to Chirikov (1979), for the occurrence of chaos in a periodically forced nonlinear oscillator. The approach is heuristic but it has a good deal of physical appeal, even though it is mathematically rather rough.
68
Models based on second order difference equations
As discussed in Section 6.2, it is useful to introduce action-angle variable (1, 0) for the study of oscillator dynamics. If the Hamiltonian is nonlinear, for example In
H
2m
p2 + 2 co2 q2 + eq4,
(6.9.1)
the introduction of action-angle variables, (6.2.2), yields z
H = coal + e
sin4 0.
21 mwo
(6.9.2)
In this case the action is not a constant, 1= - 8H/a0 = 4x(21/m(o0)2 sin3 0 cos 0,
where I = 0 in the case of the harmonic oscillator (c = 0). The unraveling of these complications, when there are many coupled nonlinear oscillators, but `small e', is the problem addressed by the KAM theorem. This is discussed further in Chapter 8.
Here we consider only one oscillator, but with the addition of a periodic perturbation. To make matters simpler, we consider the Hamiltonian (Channell, 1978) 00
H=w01+H1(1)+e E 8(t-27r1/Sl)cos0
(6.9.3)
rather than perturbing the Hamiltonian (6.9.2). (6.9.3) is a periodically `kicked' oscillator, with a measuring the strength of the force, and At = 2n/0 being the period of the kicks.
Exercise 6.14 Show that standard map, (6.5.7), can be obtained by integrating Hamilton's equations I = - 8H/00, 9 = 8H/8I, where 00
H=
8(t - 27r1/fl).
212 + e cos 0 00
Identify (r, K) in terms of I, a and i2. Note that this Hamiltonian has no harmonic oscillator limit (even when e = 0).
The effect of kicking the oscillator is to introduce all the harmonics of il, since 00
6(t - 2irl/i1) = 1 + 2
cos (lilt).
(6.9.4)
!=1
Therefore (6.9.3) equals H = w01 + H 1(1) + e{ cos 0 +
[cos (0 + lilt) + cos (0 - lilt)] }.
(6.9.3)
Chirikov's resonance-overlap criterion
69
Now the equations of motion are
B=wo+(8H1/81)-w(1),
(6.9.5)
[sin (0 + lilt) + sin (0 - lilt)].
1= E sin 0 + E
(6.9.6)
1=1
We note that 1 is proportional to E, so it is a slow variable, relative to the rapid variable, 0. This system can therefore be readily studied using averaging methods, such
as the Krylov-Bogoliubov-Mitropolsky discussed in Chapter 5. For a given 0, some term of (6.9.6), say sin (0 - kilt) = sin ((o(I)t - kilt + 00), is slowly varying, provided that I satisfies
w(1,) - M.
(6.9.7)
This defines a resonance amplitude 1,(k). If I is initially near 1,(k) and does not vary too much due to this slowly varying force, then w(1) will remain within the range of values
(k - 1)il = w(I,) - it < w[I(t)] < w(1,) + it = (k + 1)il.
(6.9.8)
If this is satisfied then all the remaining terms in (6.9.6), will vary rapidly in time, and hence essentially time-average to zero, and can be ignored. This situation is illustrated in Fig. 6.81, together with a few slightly influenced trajectories. We want to determine, first, what happens in the shaded band near 1,(k), assuming that (6.9.8) is valid. Fig. 6.81
Ir(k + 1) I, (k) I,(k - 1)
-A-0 2a
Therefore, if we set 0=0-kilt
(6.9.9)
(6.9.6) can be approximated by 1 ^- E sin (0)
and
i = (w(I) - w(1,)).
(6.9.10)
Models based on second order difference equations
70
Setting
1=1,+81
(6.9.11)
then, to lowest order in 81, these yield = (d(g/dI)1,81,
d(8I)/dt = E sin (i/i);
(6.9.12)
so
= E(d(o/dI)I, sin ', or
2(,//)2 + E(dcw/dI)/, cos >/i = E.
6.9.13)
(See Fig. 6.82.) In other words sli(t) behaves just like a pendulum, with its associated separatix. The significance of this separatrix in the present context is that it indicates
that a state with an initial action I, (i.e., 81= 0), can vary to a value
1=1,(k) ± Max
I/(du)/dI)I,.
= (dw/dI)I, SI
Fig. 6.82
92
=0-k2t
0
it
2a
Thus the width of this separatrix is a measure of the maximum influence of this force.
This separatrix is given by E = E(dcw/dl),, in (6.9.13), and therefore it has a total width (at >/i = n), 4[E(d(w/dI)I,] 1/2.
(6.9.14)
This range in j means, according to (6.9.12), that some 81 can vary by an amount 0(81) = 4[E/(d1)1,]'12
(6.9.15)
Thus we see that the term I = k in (6.9.6) causes some states, initially at I =1,(k), (6.9.7), to vary by an amout 0(81), (6.9.15), provided that (6.9.8) is satisfied. But there are, of course, other states initially near I,(k + 1), I,(k - 1), etc., which are in resonance with other terms in (6.9.6) (i.e., I = k + 1, k - 1, etc.). These states likewise can vary over similar bands (6.9.14) (Fig. 6.83). If these bands do not overlap, (6.9.8) is satisfied,
and we might hope at least that states will in fact only vary within these bands. Chirikov suggested that if such resonance bands overlap, this indicates that states
Chirikov's resonance-overlap criterion
71
Fig. 6.83
0
may migrate over extended regions of the phase space, in a chaotic fashion. Roughly speaking, a system can migrate into regions where it is temporarily in resonance with
one frequency, k52, and later wander to a region with a quite different temporary frequency, M. This is `chaotic behavior'. Chirikov's criterion for this `stochasticity', or `chaos' is that the half-band width is half the separation of the I,(k), or 20(81) ? 2[I,(k + 1) - I,(k)
In terms of Vii, this is
Using (6.9.14), Chirikov's criterion for stochasticity becomes 4[e(dco/dI)] 112 > 52
(6.9.16)
dco/dI = d2H1(1)/d12.
(6.9.17)
and, for (6.9.3)
Thus, if the strength of the force, e, does not satisfy (6.9.16), we expect to find that most states (perhaps not all!) behave in a regular fashion, but if (6.9.16) is satisfied, `chaos' is likely to occur for all initial states. This depends on the region of the phase space, because of (6.9.17). Channel] numerically investigated the case H 1 = S I"`, coo = 2, and 52 = 2n in (6.9.3).
The critical I,, which satisfies (6.9.16), then can be put in the form im-2 =
57r 2
4m(m - 1)e
(6.9.18)
He considered various values of (m, e), and determined the smallest value of 1(0) = I, for which chaos could not be found for the actual equations of motion (6.9.5), (6.9.6). The observed thresholds are compared with I,, (6.9.18), in Table 6.1. We note that (always) I, > observed threshold. Put another way, the system is more sensitive to the force than predicted by (6.9.16), so (6.9.16) gives an upper bound on e, (for given I).
The Chirikov estimate should be taken in the spirit intended; as a rough estimate (upper bound on e) for the onset of chaos (see Chirikov (1979) for more details). It
72
Models based on second order difference equations Table 6.1
m
E
10
0.1 0.1 0.1
5
4 4 4 4
0.05 0.02 0.01
IC
1.04 1.83 3.21
4.53 7.17 10.14
Observed threshold 0.81 1.30
2.19 3.24 4.84 7.56
uses only the lowest order approximation of (6.9.6) in (6.9.10), and yet retains the full nonlinear solution (separatrix) of the latter equation. This is, of course, not consistent. The harmonics of 0 (or 0) implied by (6.9.12), introduce other harmonics, and hence resonances, in (6.9.6). Hence we can expect that there are in fact an infinite number
of such `resonance bands', of finite but diminishing width for higher resonances, nco(I) - m0 = 0. For any finite e some of these resonances will `overlap', destroying all regular motion between the resonances. This is just what we found with the standard and twist maps - namely all tori whose trajectories have rational rotation
numbers, w(1)/S2 = m/n, are destroyed by any perturbation. Some irrational rotation-numbered tori will survive, for small forces (here E), and these are the KAM
tori. Chirikov's estimate, (6.9.16), predicts when there certainly can not be any remaining KAM torus, since the most separated resonance bands now overlap. Thus it gives a rough upper bound on the critical (chaotic) force, EC.
6.10 The numerical Poincare map and discontinuous dynamics When an oscillator experiences a periodic force, with period T, the Poincare map (first-return map) in the extended phase space is simple to obtain, because it amounts to recording (x(t), y(t)) at the specified times t = nt (n = 0,1.... ). When a dynamical calculation is performed numerically, say with the use of a Runge-Kutta iteration method, the time step, At, can be taken to be some fraction of the period T, so At = T/N, say N = 100. Thus recording (x(t), y(t)) when t = nNAt (n = 0, 1, ...) yields the Poincare map. When the dynamics is autonomous x = F(x)
(xeR")
(6.10.1)
the process of numerically obtaining a Poincare map is somewhat less obvious, because it is defined by the fact that x(t) is on a surface of section S[x, (t), ... , x"(t)] = 0.
(6.10.2)
Numerical Poincare map and discontinuous dynamics
73
The difficulty is that (6.10.2) is not usually satisfied for any of the discrete values, x(nAt), obtained in a numerical integration. The most obvious way around this difficulty is to wait for S(x) to change sign, then return to the last time step, decrease At by some amount, and repeat the process until the desired accuracy is obtained. Henon (1982) pointed out that there is a much better way, once S(x) changes sign, which in fact places x(t) `precisely' on the surface (6.10.2) in one iteration step. Henon's method is based upon replacing the unknown time step, required to end upon S(x) = 0, with a known step of some variable, say xm. To illustrate, consider first the simple case when S(x) is
S(x) - xm - A
(A, some constant),
(6.10.3)
so S(x) is like a plane perpendicular to the x;, axis. If we find that S(x(tk)).S(x(tk+ 1)) < 0,
(6.10.4)
we know that the trajectory satisfies (6.10.2) for some tk < t < +I 1. Then we know from (6.10.3) that what is required for S(x) to vanish after t = tk, is that xm must change by an amount Axm
S(x(tk)).
(6.10.5)
Thus if we use xm as the independent variable, and replace (6.10.1) with the system dxi/dxm = F,(x)/Fm(x);
dt/dxm = 1/Fm(x)
(l = 1, ... , n; l
m),
(6.10.6)
starting from (x,(tk), tk), and take Axm to be (6.10.5), the next iteration will place the system `precisely' on S(x) = 0 (that is, to O((Axm)5), for a fourth-order Runge-Kutta iteration). Since the surface of section may not be as simple as (6.10.3), although it frequently is, it is useful to have a genral formulation. Define a new variable (6.10.7)
x" + 1 = S(x)
and add to (6.10.1) the additional equation
+F,,
k=1
Fk(x)(aS/8xk).
(6.10.8)
Now, when x, changes sign (which does not involve integrating (6.10.8)), we 1
introduce the equation (6.10.8), and replace (6.10.6) with the same set, with m replaced by (n + 1). The desired step in the new independent variable, x,,+ 1, is Ax,,+ 1 = - S(x(tk)) As Henon pointed out, this same method can be used to numerically solve various
discontinuous types of dynamics. Thus (Fig. 6.84) a linear array of hard-sphere pendulums (diameter D) collide and exchange momenta. The collision occurs when
R,-x,-x1_1-D=0.
Models based on second order difference equations
74 Fig. 6.84
Therefore, the change in sign of any R, (1= 1, ... , n), can be treated as above, selecting xn+ i = R,, and using (6.10.8), with S = R1, and (6.10.6). Once the collision time has
been thereby established, the iterations can proceed. There are many interesting models of this nature, such as concentric shells of matter (star clusters), where the gravitational force changes discontinuously when they cross. Another important example is infinite electron sheets in a neutralizing fixed positive background (one-dimensional nonlinear plasma oscillation). Here, by Gauss' law the force is proportional to the sheet displacement until the sheets collide, when they experience a discontinuity in the force, given by the (charge/area) on a sheet. A high speed sheet can be sent through this collection, leaving a wake of oscillating sheets (as will the pendulums, if they are not close-packed).
6.11 The Henon-Heiles and Toda-Hamiltonian systems We now leave the realm of periodically perturbed conservative systems, with their near-integrable limits, and turn to autonomous Hamiltonian systems with two degrees of freedom. In 1964 Henon and Heiles considered such a model Hamiltonian system, in order to investigate the possible existence of isolating (global, smooth) integrals of the motion, aside from the Hamiltonian. There is a long history in the search for such integrals, particularly in the field of astronomy, and Henon and Heiles' interest in this question likewise grew out of searches for a third isolating integral of galactic dynamics (the angular momentum being a second such integral). To examine this question, they temporarily abandoned the astronomical problem, and instead considered the model Hamiltonian
H=2(p2+p2)+2(q;2 +q22 +2g12 g2-3q2)
(6.11.1)
which is analytically simple, but sufficiently complicated to give nontrivial solutions. It was subsequently discovered (Lunsford and Ford, 1972) that this Hamiltonian also describes the dynamics of a particular one-dimensional triatomic molecule, which will be discussed in Chapter 8, Section 7.
Henon-Heiles and Toda-Hamiltonian systems
75
The potential in (6.11.1) has a ternary symmetry, as is more clearly shown by introducing the polar coordinates q 1 = r cos (0), q2 = r sin (6), in which case the potential energy becomes V = 2r2 + (3)r3 sin (30).
It is readily found that when V(q1, q2) = 6 (q2+z)(q2+3112g1
- 1)(g2-3112g1 - 1) = 0.
as illustrated in These three straight lines cross at (q1, q2) = (0,1), (± 3112/2, Fig. 6.85, which shows other equipotential lines V(ql, q2) = constant. Inside this triangle the motion is bounded, provided that H < 6. However, if H > 6, or if the 2),
Fig. 6.85
0.5
q2
0
-0.5
0
0.5
q
motion starts outside of this triangle, then the motion is generally unbounded - it escapes to infinity along one of the potential `valleys' 0 = n/2, 4n/3, or 5n/3. Physically this corresponds to the dissociation of the triatomic molecule, which is a process of considerable interest, but will not be pursued here. From the equations of motion of (6.11.1) dq1 /dt = pl;
dp1 /dt = - q1 - 2q1 q2
dq2/dt = P2;
dP2/dt = - q2 - qi + q2
(6.11.2)
we notice that dp1/dt only changes sign if q1 or (1 + 2q2) changes sign. For motion
in the capture triangle, the latter never occurs, so we know that the trajectory repeatedly passes through the plane q1 = 0 (because dp1/dt must repeatedly change sign for bounded conservative motion). Thus the plane q1 = 0 is well-suited to be used as a Poincare surface of section, as was done by Henon and Heiles. We could
likewise make use of the surface where dp2/dt = 0, that is q1= ± (q2 -
(so
76
Models based on second order differential equations
q2 < 0 in the capture triangle), or other surfaces which computations show to be suitable.
As has been discussed briefly in Section 6.1, Henon and Heiles investigated the Poincare map on the surface of section q 1 = 0, and p l > 0. For the energy E = 1`2 they obtained, for example, the series of points numbered in Fig. 6.86(a). The points all appear to lie on a smooth curve. Other initial conditions produce different curves, all of which appear to be smooth. This is shown in Fig. 6.86(b) which contains the curve in (a). The outer curve is the boundary of the accessible region on this surface of section 2
2
p2+q2-3q23 <2E=1
P2
0.2
0
-0.2 0
0.2
0.4
q2
P2 0.4 0.3
0.2 0.1
0
-0.1
-0.2 -0.3
-0.4
-0.4-0.3 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 q2
Henon-Heiles and Toda-Hamiltonian systems
77
It might be noted that the equations of motion (6.11.2) are invariant under the transformation (p 1, q 1) - + (- p 1, -q1), so that, if we consider all of the points crossing the plane q1 = 0 (both p, > 0 and p1 < 0), we obtain two Poincare maps of the above type, which are related by the above transformation. This illustrates the nonuniqueness of the dynamics in the space (q 1, q2, P2 ), unless we also impose p 1 > 0 (or p 1 < 0). While these curves are all apparently smooth, some of them clearly represent rather severely distorted tori. Therefore, for energies below 11-2, the dynamics of this system is essentially the same as if there exists another isolating integral which, together with
the energy, defines a smooth two-dimensional manifold in the phase space (P 1, q1' P2' q2)- The intersection of this manifold with the Poincare surface of section would then result in smooth curves, as observed here. `But here comes the surprise' (as Henon and Heiles put it). For a somewhat higher energy (E = 0.125), which is still well below the escape energy (E = = 0.167), they 6 which again obtained the Poincare map shown in Fig. 6.87. There are trajectories
Fig. 6.87
P2 0.4 0.3
0.2 0.1
0
-0.1
-0.2 -0.3
-0.4
-0.4 -0.3 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 42
appear to move on smooth surfaces, but now there are trajectories which clearly do not move in such an orderly fashion. In fact, the scattered points in the figure are all produced by one trajectory! This indicates quite clearly that this system does not have any smooth global integral of the motion, aside from H. However, limited regions of the phase space do have dynamics with regular behavior, even when E = 0.125. The large `islands' are clearly regular behavior `remnants' of those found at the lower energy, but there is also visible new forms of `regular' motion, illustrated by the five smaller islands. This `island chain', which is very reminiscent of an atoll, is produced by the map of a single trajectory. Henon and Heiles found other island chains (not indicated), containing various numbers of islands, Ni. The dimensions of
78
Models based on second order difference equations
Fig. 6.88
P2 0.5
0.4 0.3 0.2 0.1
0
-0.1
-0.2 -0.3 -0.4 -0.5
i I I 1- I' I I V 7 M N ^- O - N -T Vl \O r 00 O\ q2 T
I
I
I
I
I
OOOOO000000000 I
I
I
I
I
1.0
(b) 0.9 0.8 0.7
0.6 0.5
0.4 0.3
0.2 0.1
0
I
I
I
I
I
I
I
-I
I
O O O O O N .-. O O O O C e O C O Energy
the islands generally decrease very rapidly as Ni increases. They therefore found `coherent islands' in a `chaotic sea', for this Poincare map. As shown in Fig. 6.88(a), when the energy (E = 0.1667) is just below the escape energy (0.167), the chaotic sea `floods' nearly all visible atolls and islands. They made a study to determine the fraction of the available area which is covered by smooth curves, as a function of the energy. They determined this by finding out
whether nearby initial states in various regions tend to separate linearly or exponentially with the number of iterates. Their calculation amounted to determining
whether or not there is a positive Lyapunov exponent. The figure of their results (6.88(b)) shows a dramatic decrease in the island area only when E > 0.11. At present there is apparently no explanation why there should be such a dramatic change only
when E Z' . At this point it might seem clear that all nonlinear oscillator systems with two 9
Henon-Heiles and Toda-Hamiltonian systems
79
degrees of freedom must have chaotic motion, when their energy is sufficiently large. Thus it was initially quite surprising when Ford, Stoddard, and Turner (1973), showed that the Toda-Hamiltonian system
H=
2(Pi
+p2)+ 3(eXP(g2 +311281)+exp(g2
-3'12g1)+exp(-2q2))-
1
(6.11.3)
does not have this property. The origin of this Hamiltonian involves a one-dimensional, triatomic molecular model, in which the interaction potential is the so-called Toda potential V(r) = ar + (a/b) exp ( - br)
(ab > 0).
(6.11.4)
This of course is not intended to be a realistic potential for large separation distances, r, but it leads to remarkable dynamics in lattices, and in the present molecule, (6.11.3). The details of the relationship between (6.11.4) and (6.11.3) can be found in Chapter 8, Section 7. Here we simply want to note some of Ford, Stoddard and Turner's results for this Toda-Hamiltonian. Like Henon and Heiles, they obtained the Poincare map, using q1 = 0, pl > 0 as the Poincare surface of section. The results of their numerical calculations for E = 8 and E = 2048 are illustrated in Fig. 6.89. It can be seen that all trajectories intersect the surface of section along smooth curves, and there are no indications of hyperbolic points, even for large values of the energy. Fig. 6.89
P2
P2
I
E=8
These results can be understood from the fact that this system has another global analytic constant of the motion, as they noted
K = 3(p; - 3P2)P1 + [(p1 + 3112P2)eXP(g2 - 3112 q1) + (P1 - 31"2P2) x exp (q2 + 3112g1) - 2p1 exp (- 2q2)].
(6.11.5)
Models based on second order difference equations
80
Hence the dynamics is confined to lie on two smooth manifolds in R4, H(P1,P2,g1,g2) = E;
K(P1IP2,91,g2) = K0,
which makes chaotic motion impossible (because any flow in R4 - R2 = R2 cannot be chaotic).
We see, in this example, that the smooth curves of a Poincare map indicated that an additional smooth constant of the motion exists for this system. Ford, Stoddard, and Turner used this same method to predict the integrability of the N-body Toda lattice, which we will discuss in Chapter 8.
6.12 Abstract area-preserving maps on R2 and T 2 As we have seen in the last two sections, the process of obtaining a Poincare map from integrating autonomous Hamiltonian systems with two degrees of freedom can be quite time consuming. However the rewards of such efforts are illustrated by the Henon-Heiles association of mapping features with the physical concept of energy, and the important discovery of `nongeneric' physical systems, by Ford, Stoddard, and Turner (1973), discussed in the last section. In this section we return to more abstract area-preserving maps, which not only speed up numerical computations of `dynamics', but also can shed `new light' on methods of analysis. This `new light' dates, in part, from Birkhoff's inventive studies, but the following explicit examples add considerably to the understanding of us less talented individuals.
In an attempt to better understand the properties of a Poincare map for Hamiltonian systems with two degrees of freedom, Henon (1969) investigated the
simplest nonlinear area-preserving polynomial map in R2. He began with the quadratic map (x, y) - (x', y') given by
x'=atox+ao1Y+a20X2+a11xy+a02Y2
(6.12.1)
y'=blox+bo1Y+620X2 +611XY+b02Y2 where the origin has been taken to be a fixed point. If the origin is stable, and (6.12.1)
is area-preserving, then he showed that it can be put in the form =cosaxn-sina(yn-xn)
(0<01 <7r)
T: Yn+1
(6.12.2)
=sinaxn+cosa(yn-xn)
where the restriction on a eliminates trivial or redundant maps. Exercise 6.15 By evaluating the Jacobian 8(xn + 1,yn + 1)/0(xn, yn ), show that, if the map Xn + 1 = COS a f (xn, Yn) - sin a g(Xn, Yn ) Yn+ 1 = sin a f (Xn, yn) + cos ag(xn, yn)
Abstract area preserving maps on R2 and T'
81
is area-preserving for one value of a, it is area-preserving for all a. What relationship
must be satisfied by f (x, y) and g(x, y) for this to be both area-preserving and orientation-preserving? Verify that (6.12.2) satisfies these conditions. For small values of (xn, y"), the linear approximation of (6.12.2) xn+1 =COSxX" - Sincyn (6.12.3)
L:
yn+1 =sin Or xn+cosLX y"
has the rotational solutions
x"=rcos(na+0),
yn=rsin(na+0).
If a = (m 1 /m2 )2n, where m 1, m2 are relatively primed integers, then the solution has
period m2 for all (r, 0). In other words all points are fixed points of L'"2. If a is irrational, the only fixed point of L" is (x, y) = (0, 0). Despite this linear rotation, the map (6.12.2) does not produce rotation throughout the (x, y)-plane, since it has a fixed point at P, P:
z = 2 tan (a/2),
q = 2 tan 2 ((x/2),
(6.12.4)
in addition to the one at the origin (Fig. 6.90). In contrast to the elliptic origin, the fixed point (,2, 9) is a hyperbolic point (unstable). Fig. 6.90
x
Exercise 6.16 Substitute x =z + u, 9 = y + v into (6.12.2) and obtain the linear map for (un, vn). Recalling equations (6.6.2)-(6.6.6), obtain the Trace of the present matrix. Show that it is larger than 2, because of the condition 0 < a < it. While T 2 has no new real fixed points, T 3 has six new fixed points which are real only if cos a < 1 - 2112. These are always elliptic whereas the other three are elliptic only if --I < cos a I - 2112. These, and the fixed points of T4, were explicitly obtained by Henon. Henon noted that the T-map can be written as the product of two simpler maps T = RS,
(6.12.5)
Models based on second order difference equations
82
where S is a shearing-map parallel to the y-axis S:
x = xn,
Yn=Yn-x,2
(6.12.6)
followed by a rotation about the origin R:
xn+l =cosax;,-sinay;, (6.12.7)
Yn+ 1 = sinax + cos ay;,
These maps are illustrated in Fig. 6.91. Fig. 6.91
y
x
The inverse of these maps are easily obtained .S
1:
Xn = Xn,
Yn = Y, + (Xi)2
(6.12.8)
and
R 1:
x;,=Cosaxn+1 +sinayn+ (6.12.9)
y;,=
+cosayn +1
Thus T' - S -' R -1 is also a quadratic polynomial (W. Engel, Math. Ann. 130, 11-19, (1955); proved that the inverse of any polynomial area-preserving map is also a polynomial area-preserving map).
The decomposition of T in terms of R and S, (6.12.5), might appear to be only worthy of passing notice, but we will see how this can be developed into a very interesting perspective of such area-preserving maps. Henon noted that, if we write T-1 in a similar form to (6.12.5), namely
T-1_S-'R-'=R-'(RS-1R-1)_R-'§-1,
(6.12.10)
this defines a new shearing-map S. Moreover the map S-1 produces the mirror-image
dynamics to that produced by S, relative to the mirror-line, M, which is a straight line, passing through the fixed points (0, 0) and (z, P), (6.12.4). This is illustrated in Fig. 6.92. Similarly R -1 produces the mirror-dynamics of R, and therefore T -' P* generates the mirror-dynamics of TP.
Abstract area preserving maps on R2 and T 2 Fig. 6.92 (TP)* = T-'P*.
83
Mirror- line M
These statements require some algebra to prove, based on the definition of the mirror-image point P* of P. In cylindrical coordinates if P = (r cos 0, r sin 0),
then P* = (r cos (a - 0), r sin (a - 0)),
(6.12.11)
the claim is that, for any P,
(TP)* = T -' P*.
(6.12.12)
The proof of (6.12.12) is left as an exercise for those who are industrious. There are several important consequences of (6.12.12). The fact is that any sequence of P, TP, T 2 P, ... , T"P has a mirror image set of points P*, T -' P*, T - 2 p*,.
.., T - "P*. Since Tk(T -"P*) (k=0..... n) covers the same set in reverse
order, this trajectory is symmetric to the trajectory TP*, relative to the mirror-line M (Fig. 6.93). This symmetry of the dynamics is clear from the numerical results in Figs. 6.94 and 6.95. In particular, if there are an odd number of hyperbolic or elliptic fixed points of Tk, one of each type must lie on the mirror-line, as in the case of (0, 0) and (z, y). Fig. 6.93
T2P TP X P X
x
/ P* T -2p*
T-1P*
This is also illustrated in Fig. 6.94, after Henon (1969) for the case cos a = 0.24. It shows the five elliptic and hyperbolic points of V. Aside from the scattered points on the outer edge, this figure does not reveal any chaotic motion, particularly near the hyperbolic points. Henon 'llustrated that feature with an enlargement near a hyperbolic fixed point. In addition to the scattered points, obtained from only a few
84
Models based on second order difference equations
Fig. 6.94
0.5
Y 0
-0.5
-0.5
0
0.5
x
Fig. 6.95
0.175
y
0.150
0.125
0.550
0.575
0.600
x
initial states near this fixed point, we see the wonderous world of islands (of second and third order) as well as another island chain with its hyperbolic companions. All of this from the polynomial map (6.12.12)! Many other figures can be found in Henon's paper. We now return to Hbnon's observation (6.12.12). This suggests that we consider the mirror-map, 11,
P*=I,P
(6.12.13)
Abstract area preserving maps on R2 and T 2
85
given by
Xn+1 =COSaXn+sin ay (6.12.14)
yn+,
Since the reflection of a reflection is the original image, it is not surprising that
I; = I (identity map).
(6.12.15)
A transformation, or map, which satisfies (6.12.15) is said to be an involutive transformation (e.g., see Arnold, 1978). We also note that all points on the mirror-line
M1={x=rtan (a/2),y=rtan (a/2)IreR;0
(6.12.16)
are fixed points of I,. Now if we decompose T in terms of 1, and a second map I2,
T=1211,
(6.12.17)
rather than the decomposition (6.12.5), then because I, is involutive
12=TI1 1=TI1.
(6.12.18)
We find that xn+ 1 = COS (2a)Xn + sin (2a)y + sin a[cos ax + sin a yn]2 I2:
(6.12.19)
Yn+ 1 = sin (2a)xn - cos (2a)y - cos a[cos ax + sin
The map I2 is not generally involutive, but that is not necessary for the following considerations.
We next determine the fixed points of 12- Since P, (6.12.4), is a fixed point of T and 1,, then it is also a fixed point of I2. More generally, from (6.12.19), the fixed points of I2 of present interest are on the curve M2:
x cost (a)/sin (a) = 1 - cos (a)y ± [ 1 - 2 cos (a)y] 1/2
(6.12.20)
which is illustrated in Fig. 6.96. This can be compared with the line of fixed points of 1,, (6.12.16).
To simplify the expression, without altering the concepts, we set a = n/2, in which case Xn+1 =
Xn+1 = Yn
2
Xn+Yn
(6.12.21)
12:
1,:
Yn+1 =Xn
Yn+1 =Yn
Now the mirror-line for I, is simply M1 = {x = y}, whereas the fixed points of I2 lie on the curve M2 = {x = 2y2}. Note that I2 can be written in the form 12:
-(2Yn+1
(_ly2
Yn+1 =Yn
86
Models based on second order difference equations
so 12 maps horizontally equal distances on the two sides of M2. This map is therefore involutive, IZ = 1, as is I, (Helleman, 1977). 1, and I2 are illustrated in Fig. 6.96(a), obtained from the initial point on M2, (0.32, 0.8). Figure 6.96(b) illustrates the concave solution, starting from y = 0.3, and the unstable solution, starting from y = 0.93, both
on M2. The unstable solution outlines 21 `islands', (associated with period-21 hyperbolic and elliptic solutions), that are outlined by over 8200 iterations of (6.12.21), before the solution leaves this region (Henon illustrated the 21 islands). Maps which are composed of the product of two involutary maps, T =121 It (J2 = 1, I2 = 1), were studied by Birkhoff, and particularly by De Vogelaere (1958), in the investigation of periodic solutions. Greene (1979) also employed this method to study the periodic solutions of the standard map (Section 6.7), S:
rn+1
6n+1 =On+rn+,
in which case, S =121,, where
I,:
0'= -0;
I2:
0'= -0+r;
r'=r-Ksin(0) r'= r.
By considering the fixed point curves of I, and I2, the periodic solutions can be obtained (numerically) more rapidly and accurately.
Exercise 6.17 Assume that T =121,, where I, and I2 are involutary maps. Assume that P0 = (xo,yo) is a fixed point of I, and PN = (XN,YN) = TN(x0,y0) is also a fixed point of I,,
1,Po=Po;
1,PN=PN.
Abstract area preserving maps on R2 and T 2 Fig. 6.97
87
TNPo
That is, these points both lie on M,, the curve of fixed points of I, (Fig. 6.97). The problem is to show that P0 is a periodic point of T, with period 2N, T2NPo = P0. If M, = {y = F(x) }, then by varying only x, one can numerically search for the roots of y" - F(x") = 0 obtained from (xo, F(xo)), so the problem has been reduced to a one-dimensional search (Greene, 1979). Fig. 6.98
A very different area-preserving map involves a map on a unit two-torus, T2 (Fig. 6.98)
(x,1-(1 11(xolmodI-A(YO) X0 mod 1. Yi/!
1
(6.12.22)
2J YoJJ
This map is widely known as Arnold's cat map, due to the illustrations in Fig. 6.99 (after Arnold and Avez, 1968). Beginning with a cat in a unit square, it is mapped by A on R2. It is then recollected in the unit square, using the mod 1 operation. This amounts to carrying out the A-map on the unit torus, T2, as illustrated. The final figure shows the poor cat after only two such maps. Note that the cat is all in one piece on V. It only appears broken up when represented in the unit square. This map is very different in character from the previous twist, or shear-and-twist maps we have considered. As we will see, all of the fixed points of A" are hyperbolic fixed points, rather than the mixture of hyperbolic and elliptic fixed points associated with near-integrable systems. The only fixed points of A" are rational values of x and y. We determine the fixed points by noting that the map A cannot increase (x, y) by more than (1, 2) respectively (before we take mod 1). Thus, after n iterations, (xo,yo) will be a fixed point if
x"=xo+k,
Y"=Yo+l
(k=0,...,n;1=0,...,2n)
where (x", y") = A"(xo, yo), without the mod 1 operation. Thus for A, the fixed points satisfy
xo+yo=xo+k,
x0 +2Yo=Yo+1
(k=0,1;1=0,1,2).
Models based on second order difference equations
88 Fig. 6.99
Y
a 3
2
Cat 1
A-cat on R2
X 1
1
2
A-cat on-T2
A2-cat on T2
If we require 0 < xo < 1, 0 <, yo < 1, the only solution is the origin, (0, 0), and
k=1=0. For A2 we have the fixed points
x2=2xo+3yo=xo+k;
Y2=3xo+5Yo=Yo+1
Then xo = (31 - 4k)/5
and
yo = (3k - l)/5.
Allowable solutions are
(k= 1,1=2)
(xo,Yo)=(5,5)
(k=2,1=3)
(xo,yo)=(5,5)
(k<2,1<4).
Abstract area-preserving maps on R2 and T'
89
as well as the A-map of these points, (5, 5) and (5, 5) respectively. The eigenvalues of A satisfy
x+y=Ax,
x+2y=Ay,
which yields
A=i(3±5'/2).
(6.12.23)
Therefore the eigenvectors associated with A have components proportional to (1,A- 1). The eigenvector (1,A+ - 1) is along the unstable invariant curve from the origin, W", whereas (1, 2 _ - 1) is along the orthogonal stable (contracting) invariant curve, W. The angle between W° and the x-axis satisfies tan 0 = (a+ -1) = 2(1 + 51/2) or 6 = 58.28°. We have encountered this number, 2(1 + 5112), the so-called golden mean, in the discussion of irrational numbers in Section 6.7. This irrational number is the most
difficult to approximate by rational numbers. Because of the linear form of the A-map, Wo and Wo are everywhere straight lines in the unit square. Fig. 6.100 illustrates both invariant curves for two encirclements Fig. 6.100
0.618
x
of the torus. These curves do not intersect any other rational points (fixed points of A"), but they clearly intersect each other an infinite number of times. Each such intersection is a homoclinic point, and between any two of them lies an infinite number
of other homoclinic points along either Wo or Wu. Thus, for this map, homoclinic points can be found with ease, which is rarely the case. The eigenvalues of A" are 2" and A', where A is given by (6.12.23). Thus all fixed points of A" are hyperbolic. The unstable invariant curves of all these fixed points,
90
Models based on second order difference equations
are parallel in the unit square. However all W intersect the stable curves, W of all other fixed points, producing a dense set of heteroclinic points. We see that the orbits of this map are highly unstable, since all nearby points diverge exponentially.
This system has the mixing property discussed in Chapter 2. Thus a small cat, of area C (which is invariant) will ultimately cover any region R, of area k, in a dense manner. Therefore the probability that a `cat state' is found in R satisfies lim P(C n R) = P(C)P(R) - C x k, which is the condition for mixing (see Fig. 6.101). Fig. 6.101
W-
C
O
Thus, within the realm of continuous mathematics, the Arnold map describes a highly chaotic form of dynamics. We will see in Section 6.14 that the above chaotic features are dramatically altered when the map-dynamics is constrained to take place on a lattice (i.e., an integer lattice (m, n) mod (N, N)).
6.13 Maps of sets Another perspective of the character of the chaotic motion of a periodically forced oscillator is obtained by following the dynamics of a set of points, rather than a single solution. More specifically, the set will be selected from the invariant set (manifold) of the unperturbed system, so that the change in the set will be due to the forcing. This can be done numerically for any system, as was seen schematically in Chapter 5, Section 14, for the forced van der Pol oscillator (Levi's representation of the `folding' action). It is, however, generally very time consuming to follow many
91
Maps of sets
solutions, so there is an advantage in considering a simple system which can be piecewise integrated algebraically. We therefore consider a bouncing-ball oscillator, between two fixed walls at x = ± L (Fig. 6.102). To this ball is applied a periodic square wave force, of amplitude A and V
Fig. 6.102
L VO
0.
x f
-V0 Fx
A 4T t
Li
-A
period 2T(Mistriotis and Jackson, 1987). Providing that the ball does not hit a wall n
(6.13.1)
AT vn+1
(6.13.2)
=vn+(-1)"AT
where x,, - x(nT), vn = v(nT). We can of course take T = 1 and L = 1, if we like. As with all piecewise-linear dynamics, the price we pay is matching the pieces together. In the present case it is the relationship between F(t) and the bouncing on the walls which produces the nonlinear effects. If I x" + 1 I > L, as given by (6.13.1), then
we must `back up', and solve for the collision time, T.,
L=xn+vnT,, +(-1)"ZATC
(6.13.3)
obtaining the velocity at collision,
v,=vn+(-1)"ATE, then set v = - v, and continue, to obtain ±L-vc(T-TT)+(-1)"2A(T-T.)2
xn+1 = vn+1 =
-v,+(- 1)"A(T-To).
(6.13.4)
(assuming I xn+ 11 < L, otherwise we must repeat the process). Except for very special initial conditions, depending on A, the general solution cannot be explicitly obtained.
For general forces, such collisions with walls can be accurately treated by Henon's method (Section 6.10), but it is still time consuming for many points.
92
Models based on second order difference equations
We now select a set of points from the unperturbed invariant manifold (when A = 0) { ± vo, I x I < L}, and obtain their location at every half-period t = nT, using (6.13.1)-(6.13.4). If vo is not too large (2L/vo > T), then the behavior of this invariant
set is schematically illustrated in Fig. 6.103. The two `kinks' coming from the discontinuous force. As time progresses, these kinks develop into `folds', where the manifold has more than one (positive or negative) velocity at a point x. Fig. 6.103 A
0, t = T.
-V0
To illustrate this, we follow 200 points, initially uniformly distributed between x = ± L, and with v = ± vo. To keep track of the initial spatial ordering, we connect initially adjacent points on this phase space manifold with a straight line. We do this for all times. Initially these straight lines are essentially on the new invariant manifold, but as folds develop, and collide with the walls, initially adjacent points become
located at distant points, with opposite velocities. In this case the straight lines crisscross the phase space, and no longer lie near the invariant manifold. However this gives a good sense of the mixing which is taking place. Fig. 6.104
t=6T
t = 30T
t = 16T
t = 100T
Maps on a lattice
93
Figure 6.104 shows the case (vo = 15, A = 1), taken at the times t = 6T, 16T, 30T, and
100T. The walls are not shown, so the nearly vertical lines come from the straight line connection of adjacent points. Note in particular that the invariant manifold is bounded within narrow velocity-bands about ± vo, but develops an infinite number of folds in this region. This is one of the characterizations of chaos in this conservative system (the mapped area enclosed by this manifold, 4Lvo, is constant, according to Liouville's theorem). It is also interesting to compare this with Levi's representation of the van der Pol strange attractor (Chapter 5, Section 14). In the present case the set is certainly `strange', presumably asymptotically developing a fractal, Cantor-like cross section. However, it is not an attractor, because the system is conservative.
6.14 Maps on a lattice In Chapter 4, Section 11, we considered a number of issues concerning mathematics,
computations, and empirical sciences. Here we again look briefly at the issue of computations of maps in T2, and their relations to mathematical chaos. One question, which was first addressed by Rannou (1974), was whether computer round-off errors `seriously affect' the study of maps on T2. More specifically she compared computer solutions of several (continuous) maps on TZ and their `analogous' discrete maps on a m x m periodic lattice. The dynamics on the lattice of integer values is of course exact, whereas the computer introduces round-off errors of unknown importance. The original motivation was to study the (near) standard map, in the form
xi+1 =xi+yi
mod2ir
Yi + 1 = yi - A, sin (xi + 1)
mod 2n
To:
(6.14.1)
,
and to compare it with the discrete map on a m x m (periodic) lattice ai+ 1 = ai + bi
mod m
To :
,
ll
bi+l=bi-[2mcsinIlr(ai+1)t
I
(6.14.2)
modm
where (ai, bi) are integers. The notation [X] in (6.14.2) represents the nearest integer to X. The map (6.14.1) is related to the standard map, (6.5.7), by (xi, Yi, A) - (Oi -1, ri,
- K).
The staggered index clearly does not affect the qualitative features of the map. In addition to comparing a map and its discrete counterpart, she also compared the
Models based on second order difference equations
94
discrete maps with an ensemble of maps, consisting of all possible maps on the m x m lattice. A one-to-one map of M = Mz points involves a permutation of M points, so
there are M! possible maps. It is assumed that the probability of each map in this ensemble is the same, namely 1/M!, in which case the ensemble is called a random-map ensemble.
All trajectories of one-to-one lattice maps are necessarily periodic, with a maximum
period of M. The following properties of the periods in a random map can then be established: (1) Starting from a given point (a0, bo), the probability that the trajectory has period n is 1/M
(n= 1,...,M).
(6.14.3)
Thus the probability is independent of n. If n = M the map is ergodic, since every state maps to every other state during its period. Exercise 6.18 Show that (6.14.3) is true. Label the points of the lattice 1 through M, and start at site 1 (e.g., a period-three might be 1 to 10, to 7, to 1.). (2) Starting from a given point, the average period is (M + 1)/2. This follows from (6.14.3), M
M
average period = Y np = (1/M) Y n = (1/M)M(M + 1)/2. n=1
n=1
(3) The average number of cycles with different lengths is approximately In M + y, where y 0.5772 is Euler's constant (e.g., see W. Feller (1966) An Introduction to Probability Theory and Its Applications, Wiley, New York).
Rannou noted that T* did not behave like an `average map' from this random ensemble. Thus, for various size lattices she obtained the values in Table 6.2 from computations (L = 10). Hence T** has many more short cycles than expected, based on random mappings. Table 6.2 Total number of cycles m
longest cycle
Average period (B)
T*
(C)
400 600 800
2,058 2,400 3,945
80,000 180,000 320,000
768 1164 1566
12.56 13.37 13.95
T*
Maps on a lattice
95
The cause of this is due to symmetries on the map, which can be written as T* = SOR0, where
a"=a'
So:
a'=a+b
Ra:
b'=b
b"-b'-[-sin2na'
I 2n
(6.14.4)
m
More generally, if
a"=a'
S:
b"=b'+g(a')
R:
a'=a+ f(b) b'=b
(6.14.5)
the symmetry of (6.14.4) comes from the fact that f (- b) = - f (b) and g( -a) g(a). She attributed the nonrandom character of T* to these symmetries. To obtain a map more representative of the average member of the random-map, she added even parts to the map To, (6.14.1), to obtain T:
xi+1=x;+y.+1-cosy;
mod 21r
Yi+1 = yj-2(sinx.+1 + 1 -cosxi+1)
mod 21r
(6.14.6)
The discrete form of this was taken to be
M /'
/2n. l1
_
T*:
(6.14.7)
bi+1-bi-[2M(sin(2mai+1)+1-cos(lmai+1M Now, for A = 10, the following comparison between T* and the random-map ensemble was found to hold,
Table 6.3
Average period
Total number of cycles
m
T*
(B)
T*
(C)
400 600 800
77,663 190,215 355,709
80,000 180,000 320,000
12
20
12.56 13.37 13.95
12
Note that T* is sometimes more nearly ergodic than the average of the random-maps (e.g., when m = 800, both the average period of T* is longer, and the number of cycles
96
Models based on second order difference equations
is fewer than the random mappings). For m = 800 the longest cycle is 435,570, or 58% of the total 640,000 points.
The difference which such hidden symmetries can make is illustrated in the two figures computed by Rannou (see Fig. 6.105), both for A = 10, but using only m = 400 Fig. 6.105
.7S
for T*, and m = 700 for T. Despite the fact that T* maps on a lattice which is three times larger (more dense, in the figure) than the one used for T*, the latter gives a much higher density of points from a single cycle (period 104,037 vs 3,218 for T*). This study indicates that the random character of `generic' maps is probably not seriously affected by discretization. However, if the map possesses symmetries, these can have a significant influence on the lattice map. Rannou's introduction of the average properties of the maps in the random ensemble, serves as a useful benchmark
for comparisons with other lattice maps. It would be very nice to be able to theoretically predict the properties of such lattice maps (maximum periods, distribution of periods, etc.), but that is not simple even for relatively simple maps, as we will now see.
We now return to the famous Arnold's Cat map, discussed in Section 6.12, but now constrained to a n x n integer lattice, A*:
xi+ 1 = xi + yi yi+ 1 = xi + 2yi
mod n mod n
(6.14.8)
where (xi, yi) are integers 0, 1, ... , n - 1. The present lattice constraint produces many
Maps on a lattice
97
features which are either not present, or else well hidden, in the continuous A-map
on V. We shall see that the A*-map does not behave like the average maps of Rannou's random ensemble, despite its origin from the mixing (hence ergodic) A-map.
In particular, since the A-map is ergodic, and n2 is the maximum period of the A*-map, it might be expected that the A*-map would have some trajectories with periods of length en2 (c< 1). However, the special (linear) form of this map appears to severely limit the maximum periods, as we will soon see. While the periods of states are difficult to establish, it is relatively easy to establish some relations between the periods of certain initial states. Before doing this, we first establish a general expression for the iterates, Ak, of Arnold's map. To do this we consider the Fibonacci sequence, {ak}, defined by the recurrence relation ak+2 = ak + ak+ 1 (k = 0, 1,...), with ao = 0, and a1 = 1. We now will establish that the iterates Ak are related to this sequence according to: if A
C1
1)2
then Ak -
(a2k a2k-1 (6.14.9)
Since a, = 1, a2 = 1, and a3 = 2, (6.14.9) clearly holds for k = 1. Moreover, if it holds for any k, then Ak+1
a2k-1 +2a2k
a2k+2a2k+1)- (a2k+2 a2k+3/
which follows because of the above recurrence relation. This establishes (6.14.9). Thus, for the map (6.14.8), (A*)k = (a2k-
1
a2k
a2k+, J/
1\a2k
mod n.
(6.14.10)
We will now use this result to prove a theorem first obtained by S.J. Chang. Theorem 1 (S.J. Chang) The states (x, y) = (0, 1) and (1,0) have the same period under the map A*, (6.14.8). If p is the period of (0, 1), (6.14.10) requires that
(A*)P 0
Thus a2p =mod n, and a2p+ 1 = 1 mod n. Hence a2P- = a2P+ 1 - a2P = 1 mod n, and we therefore obtain for the p'th iterate of (1,0) 1
l a2P-' mod n = P
o
98
Models based on second order difference equations
showing that (1,0) repeats after p iterates. The converse reasoning also clearly holds, so the states (0, 1) and (1,0) have the same period, proving the theorem. This theorem now yields the following interesting result, Theorem 2 (S.J. Chang) If p is the period of (1, 0) and (0, 1), then (1, m) is periodic in p iterations, (A*)p(l, m) = (1, m) mod n, although p may not be the minimum period of (l, m).
This follows simply from
(m')-(1 2)\m/-\1 2)[l(0)+m(O1)], so (mod n throughout)
(1 'A') = l(1
2)p(0) + m(1
'AOHI).
As discussed in Chapter 2, in the case of continuous flows, the recurrence of a system to a region of its initial state is referred to as the Poincare recurrence time. In the present lattice system the recurrence is exact, but how the recurrence depends on the initial state or on n is presently unknown. In any case Theorem 2 shows that the maximum Poincare recurrence time, for any initial state, is the period p of (0, 1) and (1, 0). Despite the fact that the above sequence {ak} in (6.14.10) is given by Binet'sformula
ak=51/2L(1
+51/2)k_(1 -51/2)k] ,
2
(6.14.11)
2
the period of (0, 1) is not generally known. That is, the value of p(n) which satisfies a2 p = kn,
a2 p+ 1 =
1 + In
for some integers l >, k, and arbitrary n, remains to be determined. However the following histogram shows that the maximum period is not proportional to n2, as expected of an ergodic-like system, but is more nearly proportional to n (at best). The histogram in Fig. 6.106 records the number of lattices NL vs p(n)/n for n < 2,000. For n < 3,750 the maximum value of p(n)/n is 3, when n = 10, 50 and 1,250. The most probable p(n) goes as an, where a < 1, and very short periods can occur for large n (e.g., p = 24 when n = 11,592). This very short Poincare recurrence is illustrated very nicely in Fig. 6.107, due to
Crutchfield, Farmer, Packard, and Shaw (1986), which an A*-map is applied to a famous picture of Henri Poincare, rather than to Arnold's cat. As can be seen from the inverted shadows on Poincare's face, the A* map is not
Maps on a lattice
99
Fig. 6.106 80 r_
60
NL 40
20
2
1
p(n)/n
the A*-map, but they are simply related. The A*-map is A*:
with its fixed point at (2,
x'= y; 2),
y'=2+x+y
1 (hence the inverted shadow).
and a(x', y')/a(x, y)
Shifting the fixed point to the origin, A =
1
1
mod!
/
and (A*)2 = A*, where we now use
mod n. The lattice used in the figure was 230 x 230, which has an A*-Poincare recurrence of 120 (240 iterates of the A*-map). The recurrence in the figure is not sharp because noise was added during the iterations. In addition to the short recurrence time of this map, a new feature of such lattice maps is revealed, namely the five `ghosts' of Poincare at iterate 24 of A*. Such an accidental discovery is a prime candidate for Ulam's synergetic research; that is, the interplay between computer results and the enquiries they stimulate, which yield other computed results, which stimulate... . Exercise 6.19 Formulate some questions which this phenomenon suggests for other computations. Can these questions be answered on a small personal computer? If not, find some that can be studied on a PC. To briefly illustrate, the effect of these 24 iterates are similar to sequentially dealing out 230 x 230 cards to five people; each receiving 46 x 230 cards. From (6.14.10) and
100
Models based on second order difference equations
Fig. 6.107 0
I8
240
Dynamic entropies and information production
101
(6.14.11) one obtains A24 mod (230) _
(493
6
139) mod (230),
(6.14.12)
and it is easily verified that the set 1(5k, 5k), k < 46} are fixed points of (6.14.12) whereas (1 + 5k, I + 5k) maps to (1391 + 5k, 1851 + 5k) mod (230), for I = 1,2,3,4. These
sets are associated with the one fixed, and four shifted ghosts in the figure. They reappear every 24 iterates, and have a `reunion' on the fifth reappearance (5 x 24 = 120 = Poincare cycle). Such phenomena may be present in a more subtle form in the original A-map, or at least in all A*-maps, even for arbitrarily large n.
6.15 Dynamic entropies and information production We have seen examples of systems whose nearby initial states behave very similarly
(coherent behavior), and other (chaotic) systems whose nearby states have very different behavior. One of the ways in which we can quantify these differences is by determining the Lyapunov exponents of these systems. More details concerning their evaluation will be given in the next chapter. In this section we will discuss several closely related methods of describing these different types of dynamic systems. While these concepts are closely related, they present different conceptual perspectives of
the nature of such dynamics. The three perspectives are Lyapunov exponents, measure-theoretic and topological dynamic entropies, and information `production' (positive and negative). Fig. 6.108 Lyapunovexponents
KS-entro
T opo I ogica
1
py
Information production
The concept of dynamic entropy, due to Kolmogorov (1958) and Sinai (1959) (Fig. 6.108), was the first method developed to describe a `mixing rate' of deterministic systems. It is concerned with the rate of change in our ability to specify the microscopic
state of a system as time progresses, given the limitations of our empirical measurements. As discussed in Chapter 4, Section 11, the importance of the empirical limitations of real-life measurements, in contrast with mathematical precision, is of fundamental importance. The KS-entropy (Kolmogorov-Sinai-entropy) was the first
precise method proposed to describe this effect. Later the concept of topological entropy (T-entropy) was introduced by Adler, Konheim, and McAndrew (1965). This entropy, as its name implies, does not make use of length scales (a metric), as does
102
Models based on second order difference equations
the KS-entropy. Yet more recently, another characterization of both chaotic and coherent (attractor) dynamics was proposed by Shaw (1981), which is very close in spirit to Lyapunov exponents, but introduces an informational interpretation, somewhat related to the above dynamic entropies. Thus these perspectives form a rather nice complementary set of concepts. We will discuss these in a mixed-historical order.
We begin by considering the dynamics described by a continuous map, f, which takes a compact space, X, onto itself
f:X->X,
(6.15.1)
but generally in a many-to-one fashion (a surjective map). Therefore, if y = f (x) (x, yeX), several points x may yield the same y. The purpose of introducing a dynamic entropy is to have a measure of how the dynamics tends to `mix up' the regions of the space X. This involves considering a sequence of maps, { f "(x) }, where f "(x) =f (f n -' (x)), in the limit n --> oo, or `long times'. What is of concern here are sets of points, rather than particular solutions (trajectories),
so the space X can be left quite general (a topological space) which may or may not have a metric defined on it. However, in most of our applications X can be thought of as a phase space. From a physical (empirical) point of view, the regions of phase space can be viewed as arising from our inability to obtain data with infinite accuracy (see Chapter 4, Section 11). If a measure, p, is introduced it is assumed that it is invariant under f. Recall from
Chapter 4 that if the regions A and B map onto C (Fig. 6.109), then the inverse Fig. 6.109
image of C is the union of A and B, and written
f `(C) = A u B.
(6.15.2)
(Note that the inverse map is not defined in this case.) The measure µ is said to be invariant under f if
µ(C) =µ(f -'(C))
(6.15.3)
for all C. In other words the measure of a region equals the total measure of all
regions which map onto this region. Thus the inverse image occurs in the definition - it's a `backward looking' definition, as discussed in Chapter 4. An explicit example will be given below.
Dynamic entropies and information production
103
The entropy introduced by Kolmogorov and Sinai is based on such invariant measures, and is called the measure-theoretic entropy of a mapping f. We will refer to this as the KS-entropy. The concept of a topological entropy off was introduced
in 1965 by Adler, Konheim, and McAndrew. It does not require a metric for its definition, and hence is less detailed, but easier to calculate than the KS-entropy. We
will call this the T-entropy, and will consider it first. Both of these entropies characterize the average rate of `shuffling' of the regions in the space X. To describe this `shuffling' of regions, we first must introduce a partitioning of X into regions ai. This collection A = fail is called a cover if every xeX is in at least one ai (X (-- Uiai). A partition is a cover of X, A = {ai}, in which the ai are all disjoint (ai n a3 = 0, i 0 j). We will let N(A) denote the number of regions in the partition A (Fig. 6.110). Fig. 6.110 A cover
(a,,
, as)
A partition N(A) = 12
One partition of a phase space might, for example, represent the division of the space into cells which correspond to the empirical resolution of some experimental equipment. One oft-quoted example (for Hamiltonian system) is a partitioning into cells with sides Api and Aqi, satisfying ApiAgi > h, as required by the uncertainty principle. This is a highly simplistic and over-optimistic estimate of our ability to measure the microstate of macroscopic systems. Since we cannot locate particles which are hidden by others in a `crowd' (and 1019 molecules per cm3 is certainly a crowd!), we are necessarily vastly more ignorant about the state of macroscopic systems, than is suggested by the uncertainty principle. This basic macroscopic uncertainty principle has yet to be carefully formulated. (See Fig. 6.111.) However we
will refer to this unspecified (but basic) macroscopic uncertainty partition as a U-partition.
To describe the `shuffling' which a map produces, we need the concept of the `product' (or `join') of two partitions: The product of two partitions, A = fail and
Models based on second order difference equations
104 Fig. 6.111
Macro uncertainty
B = {b;} of X, is another partition, defined as
AvB=C-{c;=a;nbk,all j, k}.
(6.15.4)
An example is illustrated in Fig. 6.112. Generally N(A)N(B) > N(A v B) >, N(A) or N(B), so the space is usually more finely
partitioned by the product partition. Fig. 6.112 bl
AVB
at
N(A) = 4
N(B) = 3
N(C) = 9
Exercise 6.20 Sketch two partitions in X (Fig. 6.112), A
B, such that N(A v B) _
N(B).
Now a map, f, takes a partition A into some set B,
f(A)=B-{f(a;)}. Provided that f maps X onto itself, the set B is also a partition of X. The product, A v B, gives an indication of how f has reduced our ability to predict the outcome of this `dynamics'. Thus, if {a;} is a U-partition, and the map changes it to the partition Fig. 6.113
B
A
77T/// AV.f(al)
Dynamic entropies and information production
105
B (illustrated in Fig. 6.113), the new knowledge where some initial state (shaded) is
located after the map, is determined by f (a;) v A. The product A v B is a global indication of these mixing processes. If the map, f, may be many-to-one, we consider the inverse map of a partition
f -'(A) _ { f -'(a,) }.
(6.15.5)
If A is a partition, f -'(A) is also a partition, for surjective (onto) maps. The `shuffling'
generated by the map f is then described in terms of the products A v f -'A and A v f -'A v f -2A, or generally in terms of the partition n-1 A"=Avf-'Av...vf-n+'A=
V f-kA,
(6.15.6)
k=O
where we are interested in the limit n - oo. Now the T-entropy of a partition A is defined to be H(A) = log N(A).
(6.15.7)
There is no dynamics at this point. The T-entropy of a mapping f with respect to a partition A is defined to be n-'
h(f,A)= lim 1H( V f-kA n-ac n
(6.15.8)
k=O
Now dynamics has been introduced through the map f. To complete this scheme, the topological entropy, h(f), of a mapping f is defined as the maximum (supremum) of h(f, A) over all partitions A, h(f)=SUPAh(f,A).
(6.15.9)
Note that, unless the partition associated with A", (6.15.6), grows exponentially in number (i.e., N(A") A"), the entropy h(f, A) = lim"_ (1/n) log A' = log A would vanish, so h(f) vanishes unless there is a partition with this dynamic property. The prospect of examining the limiting partitions A" of all partitions A, in order to determine h(f), is clearly not very appealing. Fortunately that can be avoided by using the idea of a refinement, B, of a cover A. The partition B = {b;} is a refinement of A = {a1} if, for each i, b1eaj for some j, in which case we write B > A. For example, see Exercise 6.20, and the prior example. In that example
C> A,
C> B
(in general A v B> A and B),
because each set c, `fits inside' some member of {a;} and {b1}. Now a sequence of partitions { A" I n = 1, 2,...j (not to be confused with A") is said to be a refining sequence if (1) An
106
Models based on second order difference equations
(2) for every partition B there is an An > B. Using the property that, if B > A, then h(f, B) > h(f, A), a refining sequence can be used to replace the supremum in (6.15.9). It has been proved that, if {A"} is a refining sequence, then
h(f) = lim h(f, An)
(6.15.10)
n-oo
(Adler, Konheim, and McAndrew, 1965). To see how these concepts can be applied, we will now consider a few examples: Fig. 6.114
AP = AP V .f-1AP
AP: p = 4
Example I (Fig. 6.114) Let X be the unit circle, {x 1, x2 I x i + xZ = 1 } and let f be a mapping which simply involves a rotation by an angle 0. Let AP be the partition of
X by intervals of length 2n/p. Then A" =p A vP f -1 A vP. . . v f -"+ 1 AP
is a
partitioning, and it can be easily seen that the number of intervals in AP cannot be greater than nN(AP) = n - p, so N(A p) < n- p (when does the equality hold for all n?). Therefore we first obtain for the entropy of f with respect to the partition AP h(f, AP) '< lim 1 log (nN(AP)) = 0. n-oo n
Secondly, if we consider the sequence of partitions {AP} as p-> oo, then clearly this is a refining sequence, since AP+ 1 > AP, and ultimately the elements of AP are subsets
of any other partition one might choose. Therefore the entropy off is h(f) = lim h(f, AP) = 0. P_ ao
This example also shows that the dynamic entropy is not always related to ergodicity. A measure-theoretic dynamic system, consisting of a map, f, and an invariant measure, µ, is called ergodic if every set A which is invariant under f (i.e. A = f -'A) has either measure zero or 1. In the present example, an invariant measure is simply the arc length of the set of points. If 2ic/O # 1/m (m: integer) then f is ergodic,
whereas f is nonergodic if 2n/0 = 1/m (an invariant set consists of the m arcs, A0, centered about 0 = k2n/m, k = 1, ... , m). Both of these maps have zero dynamic
Dynamic entropies and information production
107
entropy. It is necessary, however, for the dynamic entropy to be nonzero to ensure that the system is ergodic (see Rohlin, 1967).
Example II We now will consider an example with nonzero entropy, based on the tent map illustrated in Fig. 6.115. Let the partition A consists of the single set 0 < x < 1. Fig. 6.115
Then
A2=Av f -'Avf-2A={c1,c2,c3,c4},
Al - Av f -1A={61,62},
as illustrated, and so forth for A". Clearly N(A") = 2n. Therefore h(f, A) = lim
i log 2" - 1 =log 2.
n-.oo n
Moreover Ak = Ak can be used as a refining sequence, and h(f, Ak) = lim
i log (k2" -1) = log 2
n-oo n
(for any k),
so h(f) = log 2. Note that this topological entropy is the same for all tent maps, regardless where the peak is located. As we will see below, other measures of this mixing process distinguish between different tent maps. An example of generalization of this result has been obtained by Misiurewicz and
Szlenk (1980), for continuous maps, f, of an interval, I, onto itself. Let c" be the minimum number of intervals in I on which f "(x) is monotone. Thus, in Fig. 16.116 c, = 12, where I = [0, 1]. The theorem then states that
h(f) = lim ilogc" "-aDn
(6.15.11)
and moreover (1/n) log c" 3 h(f) for any n. Many other special results can be found in the references.
108
Models based on second order difference equations
Fig. 6.116
J'(x)
x 1
Exercise 6.21 To emphasize a feature of dynamic entropy, consider the above theorem, (6.15.11). Assume that f has a T-entropy h(f) = h*. Determine the T-entropy, h(g), of g = f k (k: positive integer). Comment on the significance of this result.
Example III We next consider another example with nonzero entropy, which is a generalization of Arnold's Cat map, discussed in previous sections. The map is again on a torus, T2, which is equivalent to a map on a plane, in which points that differ by an integer in either coordinate are taken to be identical, i.e., (x, y) - (x + m, y + n) (m, n: integers), as indicated the crosses in Fig. 6.117. Fig. 6.117
We consider the map of the unit square (0 < x < 1, 0 < y < 1) onto itself, such that the area and orientation are preserved
f:
a
b
x
c
d
y
_
x'
y'
; ab - be = 1,
(6.15.12)
and (x', y') are identified with points inside the square by the above mod (1) association. The eigenvectors of this map satisfy I a
b )I
,0I' 1
) =A 1
\c d \'Y2/
All A2 =
i{(a + d) ± [(a + d)2 - 4]1/2)
and have eigenvalues (6.15.13)
Dynamic entropies and information production
109
and 1. Thus, if (a + d) > 2 there are real eigenvalues corresponding to expansion (I).1 I > 1) along one axis, a, and contraction (I A21 < 1) along the orthogonal direction, /3.
As an explicit example we can think of Arnold's Cat map,
A-':
A:
Az(3±51/2)_
(2
I
2.618=A1. 0.382 = .,2
However, we can just as easily consider the more general map f, (6.15.12), for the following discussion. We note that, if U is the unit square, then N(U v f U) = 10 in the illustrated case in Fig. 6.118, compared with 4 in the Cat map. The inverse map, f -1, expands along /3 and contracts along a in a similar manner. Fig. 6.118
x
c+d d
a+b
To obtain the topological entropy of such a map. We can proceed by several routes,
some more clever than others. Let us not be too clever but, on the other hand, use some intelligence. We first generalize the meaning of N(A) in the definition of entropy, to include those cases where the cover, A, may not be composed of disjoint regions
4a1} - that is, the cover may not be a partition. A minimal cover is some smallest set of regions of a cover A which can be used to cover X. N(A) is now taken to be the number of sets in this minimal cover, A. For example, in Fig. 6.119, the four sets (a 1, ... , a4) of the cover A = {a; I i = 1, ... , 5} form a minimal cover of X, so N(A) = 4. However, this minimal cover has overlapping
Models based on second order difference equations
110 Fig. 6.119
regions, so it is not a partition of X. Nonetheless we can define the T-entropy of a cover in terms of this minimal N(A), (6.15.7), and similarly generalize (6.15.8) and (6.15.9).
Now, rather than selecting a simple (x, y) grid, we use a cover which can be more easily followed (or estimated) when the f -map is applied. To do this, we consider the set of regions: {SP}: squares with sides parallel to (a, /3), and of length l/p, covering the entire
plane R2. The vertex of one square is at the fixed point (0,0). p is an arbitrary integer. Fig. 6.120
These are illustrated in Fig. 6.120 for the case p = 2. The advantage with such a set of regions is that the squares map into parallelograms with sides of length A", 1p, and A "/p, after n maps, f" or f - ". Now we use these regions to form a cover for the torus (i.e., unit square), making use of the equivalence (x, y) - (x + m, y + n). This takes a little care, since the resulting cover, AP, of the unit square involves overlapping regions. Thus, if p = 2, it can be
Dynamic entropies and information production
111
Fig. 6.121
seen from Fig. 6.121 that it takes five squares from {S2} to cover U, because there is overlap. Nonetheless, this is a minimal cover based on the squares in {S2}, and we see that N(A2) = 5. Clearly for arbitrary p, since it takes p2 squares to cover a unit area, N(AP) > p2.
Now consider what happens when the map f -' is applied to these squares. As illustrated, it changes the squares to parallelograms with one side reduced by a factor 1/),,. Thus it requires at least A, of these regions to cover the original (minimal) p2 squares, and hence at least A,p2 regions are in the collection AP v f -'AP. Continuing in this fashion N(AP v f -'AP v ... v f -"AP) > Ai p2.
On the other hand if only N(AP) squares from {SP} in fact covered U, then certainly no more than ),,N( A,,) parallelograms from f -'{SP} are required to cover U (why?), hence N(AP v f -'A,,) < A,N(AP). Therefore, for arbitrary n
),N(AP)>, N(APv f -'APv ... v f-"AP)>Eip2. Then the topological entropy off with respect to the covering AP is h(f, AP) = lim
1
n- ccn
log N(AP v f - ' AP v
v f -"A,)= log I A,1,
because both of the above bounds have the same limit. Finally we again use the fact that the sequence of covers {AP}, p = 2,4,8.... is a refining sequence of covers, we
112
Models based on second order difference equations
obtain
h(f) = lim h(f,AA^)=logI),I If X is an n-dimensional torus and f is a mapping represented by an n x n matrix with real characteristic values A,,, n linearly independent eigenvectors, and it preserves the volume (determinant equal to one), then
h(f)= E loglA;I.
(6.15.14)
IA+l> I
The quantities log I A,; I in (6.15.14) are the Lyapunov exponents for this map, because nearby points along the eigendirections separate proportional to ).k in k iterates. Since Ak = exp (k log I A; 1), log (Ai) is the corresponding Lyapunov exponent. Only the positive
exponents, IA,; I > 1, contribute to the topological entropy. Hence h(f) = 0 if all Lyapunov exponents are negative. In contrast with the topological entropy, the KS-entropy does not simply involve counting the number of regions in a space, but takes into account a weight (measure) which has been assigned to these regions, and which is invariant under the mapf, i.e.
µ(a) = y(f -'a)
for all aeX.
The KS-entropy is therefore more detailed and informative than the T-entropy, but it is generally also more difficult to calculate. Now assume that A = {a;} is a partition, consisting of disjoint elements (i.e., X = U, _ , a;, p(X) = 1, and p(a; n aj) = 0 for all i 0j). The KS-entropy of this partition is by definition N
H,(A)
Y µ(a;) log µ(a;).
(6.15.15)
=1
The KS-entropy associated with the measure-preserving map f for the partition A is 1
h,,(f,A)= lim _H,(A v f-1A v ... v f-n+'A). n-oo n
(6.15.16)
Finally, the KS-entropy of the measure-preserving map f is defined to be hµ(f) = supA hµ(f, A),
(6.15.17)
that is the maximum value of hµ(f, A) over all finite partitions of X. As in the topological case, this can also be obtained by using a sequence of refining partitions An+ 1 > A. which ultimately refine any given partition, in which case h,,(f) = lim hµ(f, An). n
ao
(6.15.18)
Dynamic entropies and information production
113
It was conjectured by Adler, Konheim, and McAndrew and proved in Dinaburg (1971) that there is a relationship between the topological entropy, h(f), and the KS-entropy, hµ(f ). The theorem states that if X is a compact space of finite dimension, and f : X is a homeomorphism of X onto itself, then
h(f)=sup, hµ(f), where the supremum is taken over all normalized Borel measures which are invariant under the map f. Indeed the topological entropy is never less than the KS-entropy h(f) ? h,,(f )
This can be interpreted to mean that the KS-entropy is generally a more `critical' judge of the dynamic mixing process than is the T-entropy. When the usual Lebesgue measure (i.e., u(dx) = dx) is invariant, as in Example III above (the area-preserving map on T2), then h(f) = hµ(f ). We will see an explicit example of this below (Exercises 6.22 and 6.23).
Adler, Konheim, and McAndrew also conjectured that, if f, is a one-parameter group flow on a compact space X, then h(ff) = I t I h(fl)
(6.15.19)
and this has also been proved to be correct (see Dinaburg, 1971). It is important to note the distinction between the entropy of a flow (i.e., a dynamic situation) and the previous entropies. In the previous cases we considered a map f, which can be thought of as a flow over a unit time period, which then results in a certain change in the
entropy, h(f). Continuing another time interval results in a total change of h(f2) = 2h(f ), and so on. Indeed, the result (6.15.19) is a generalization of the observation made in Exercise 6.21, in regards to (6.15.11). Finally, we turn to another perspective of the `shuffling', or mixing influence in dynamics. This perspective, due to Shaw (1981), emphasizes the rate of change of our ability to obtain information about the microstate of a system. One method of representing the information obtained in an experiment, given as set of a priori probabilities of the possible outcomes, was discussed in Chapter 2. At
present, however, we are not concerned with the information obtained in one experiment, but rather the rate of change in the possible experimental information in a sequence of experiments. From the present viewpoint, if our measurements at a later time give `significantly' more information about the detailed state of the system than we can obtain at present, we say that this system has a positive `information production'. A complementary viewpoint is that, if the `uncertainty' about the state of the system increases rapidly with time, so that we can obtain more information in a later measurement, then the system has a positive information production. On
114
Models based on second order difference equations
the contrary, if many states are attracted to a region of phase space, so that we lose information about its origin when we make a later measurement, then the system has negative information production. This is a generalization of such classic concepts as `entropy production', which was introduced in the thermodynamics of irreversible process (e.g., de Groot, 1961), and Boltzmann's statistical H-function (e.g., Chapman and Cowling, 1970). In contrast to these concepts, information production relates a macroscopic (empirical) phenomena with the deterministic phase space dynamics of a system. Thus we acknowledge at the outset our empirical limitations for acquiring knowledge about the microstate of a system, and consider only regions of phase space (e.g., the U-partition discussed above), rather than individual trajectories. When the motion is regular, such as periodic, we know (at least roughly) what to expect in the future, even if we do not know the precise initial state. Essentially no information is gained or lost in making subsequent measurements. On the other hand, if the system has an unstable limit cycle, and we initially know its approximate state outside this cycle, we will be unable to predict its future state with much accuracy. Therefore, in this system, a future measurement will greatly increase our information
about the precise state of the system. We will thereby know more accurately its initial state, and in this sense information has been generated (or at least the opportunity has been created). Note that the contrary is the case, if the initial state is inside the cycle (in R2). Fig. 6.122
Empirical grid
At (a)
(b)
This is illustrated in Fig. 6.122, where an `empirical grid', or U-partition,
schematically indicates our limits of resolution in an experiment. In system (a), in which the states in one square spread out over four squares, the uncertainty of the state of the system has increased. Hence our information about the actual state of the system is increased by making the second measurement, since we determine which
Dynamic entropies and information production
115
of the four squares it is actually in. Indeed, given only the knowledge that the system
was in the square on the left, it has an equal probability of being found in any of the four squares on the right; so they each have an a priori probability of Hence, as discussed in Chapter 2, the informational change by the measurement at the latter 4.
time is
AI= - I-log--0=log4. k=1
What is more important is that, during the next time interval, At, the states in these four squares may occupy 16 = 42 squares, so that the information which is gained in a later measurement (given only the initial measurement) is proportional to the time interval between measurements, as illustrated. On the other hand, in system (b), the change in information gained from the two measurements is 4
AI=O+ Y llog=-log4. k=14
4
That is the uncertainty has decreased, so the information gained is negative. Put another way, information has been lost during this time step, because we could have found out more about the system in the first rather than in the second measurement. This is similar to the situation which occurs with a stable limit cycle, or any ordinary attractor (e.g., fixed point, torus, etc.). Fig. 6.123 (c)
(d)
V=0
In the case of system (c) (Fig. 6.123), there is no loss or gain of information in subsequent measurements, as characterized by undamped harmonic oscillators. However, there are other systems, such as anharmonic oscillators for which subsequent measurements do indeed produce new information about the state of the system, but
116
Models based on second order difference equations
only at a rate which is less than proportional to the passage of time. Thus, consider system (d), which consists of a particle bouncing between two walls. The initial uncertainty does grow, but the number of cells in which the particle can be found only increases linearly with time, not as a power of the time, as in system (a). The above considerations can be easily generalized along the following lines. Let pk(f) and pk(i) be the final and initial probabilities of being in a cell k in the phase space. Assume that these probabilities respectively equal 1/N(fj and 1/N(i), where N(f) are the number of final cells into which the initial N(i) cells propagate. Then the change in the information is given by A1= E pk(f) log N(f) - E p,(i) log N(i) = log N(f) - log N(i),
(6.15.20)
k
k
using the fact that the sum of the probabilities equals one, both initially and finally. If the cell size in phase space is 6, the volume in phase space is V = N6, so (6.15.20) can be written Al = log V(f) - log V(i) = log [V(f)/V(i)]
(6.15.21)
or, in differential form
dl = d log V(t). dt
(6.15.22)
dt
Note that the size of the cells does not enter in determining the rate of information
change, but it (or N(t)) does enter into the value of the information. Also, the information `production', dI/dt, is constant only if V(t) changes exponentially with time. For reasons which are not altogether clear, a sharp distinction has historically been made between exponential and polynomial divergence with time (no matter what degree the polynomial may be). The former systems are described as being unpredictable, whereas the latter are said to be `predictable'. It is clear, however, that there are shades of gray in this characterization. We can illustrate these ideas with several maps, f (x), on an interval. We note that the change in the information at x per iteration is, in keeping with (6.15.21), AI(x) = log
d f (x)
dx
which depends on x. The average rate of change of the information is
Al=flog
d f (x)
dx
P(x) dx,
(6.15.23)
where P(x) is the invariant probability density for the map fl x). That is, if we denote
Dynamic entropies and information production
117
the measure of dx about x as u(x, dx), then P(x) dx = µ(x, dx),
(6.15.24)
and y is invariant in the sense of (6.15.3). Referring back to Chapter 4, Section 6, we recognize that the integral (6.15.23) is the average Lyapunov exponent of f (x), so
AI =), - f log
d f (x)
dx
P(x) dx.
(6.15.25)
AX)
Fig. 6.124
Thus, for example, as shown in Exercise 4.11, the illustrated tent map (Fig. 6.124) f:
x' = x/x0
(x
x0);
x'= (1 - x)/(1 - x0)
(xo < x
1)
(6.15.26)
has P(x) = 1, and hence
AI = _ - xo log xo - (1 - x0) log (1 - x0 ).
(6.15.27)
We note that this rate of change of information is never larger than the T-entropy for this map (Example II, above)
h(f)=log2> -xologxo-(1 -x0)log(1 -x0), where the equality holds only if x0 = 2
To complete the comparisons, we can investigate the KS-entropy for the map (6.15.25). We will do this through stages, namely the following two exercises.
Exercise 6.22 (a) Verify first that the usual Lebesgue measure, µ(x, dx) = dx is invariant for (6.15.26), in the sense of (6.15.23), (b) Consider the partition of [0, 1],
A = {a,,a2} where a, = [0,xo] and a2 = [x0,1]. Obtain the KS-entropy of this partition, H,(A), (6.15.15), (c) Explicitly obtain the partition A' - A v f -'(A) (see Example II), (d) Obtain the KS-entropy of the partition A', Hµ(A' ). You should find a simple relationship between H,,(A) and HM(A' ). If this exercise is easy, then we can generalize this result:
118
Models based on second order difference equations
Exercise 6.23 We consider next a more general partitioning of the intervals [0, x0] and [x0,1]. We divide each into (n + 1) arbitrary pieces, separated by the points 0 < xk < xO (k = 1, ... , n) and xO < xk < 1 (k = (n + 1), ... , 2n). We will call this partition of [0, 1], An. The problem is to determine the relationship between KS-entropy of this partition, HU(An), and the KS-entropy of the partition An v f -' (An) = f -'(A,,), Hµ(f -' (An)). If we understand this, then the KS-entropy of the map f, hµ(f ), (6.15.17), will be obvious.
Finally, returning to the information production (6.15.23), we note that in some fortunate cases it is not necessary to know P(x) in order to determine Al. Namely, Fig. 6.125
1'(x )
IF-
A
if f (x) consists of monotone pieces with equal slopes (in magnitude), as in Fig. 6.125, then df df A1= log JP(x) dx = log dx
dx
Finally, we note that the information production rate Al, (6.15.23), can be either positive or negative, whereas the T-entropy and KS-entropy are either positive or zero. This contrast can also be seen in the fact that only the positive Lyapunov exponents enter into (6.15.14), in contrast to the arbitrary sign of !in (6.15.25). Hence
all of these quantities measure somewhat different properties of the mixing rate produced by the dynamics.
6.16 Epilogue: order-order, order-chaos, chaos-chaos in the house! It is important to recognize that, interwoven in the many details presented in this chapter, we have begun to obtain examples of dynamic systems which exhibit the
coexistence of chaos and order, or the coexistence of several types of chaos. Coexistence, of course, refers to the coexistence of different regions of phase space, each with one of these properties, and for certain fixed values of the control parameters. These dynamic examples complement other phenomena observed in Chapter 5. There we saw examples of forced oscillators which can have several stable periodic motions
Epilogue
119
for the same control parameter. The fact that there may also be several types of coexisting strange attractors becomes even clearer with the help of Poincare maps in the extended phase space. Thus the phase space may contain regions of very different dynamics, and we obtain a more holistic perspective of the dynamic possibilities (Fig. 6.126). Fig. 6.126 Phase spaces. Periodic
Periodic
Chaotic
I
I
Orderly
Chaotic
Chaotic Chaotic
Chaotic
Periodic
II
II
In the case of dissipative or autocatalytic systems (e.g., the forced van der Pol oscillator, or Brusselator and other chemical oscillations) the chaotic regions involve basins of attraction in which the solutions tend to be localized strange attractors. In the case of conservative systems (e.g., Hamiltonian systems), the order and chaotic regions are frequently (mathematically) densely interwoven. However, in terms of a
measure ('area'), which is presumably the important empirical concept, different regions of phase space are again frequently dominated by either chaos or order (e.g., KAM-like islands, in a sea of chaos). In conservative systems the `order' is not typically
periodic motion, but rather 'ergodic' on some smooth (orderly) surface, or manifold
(e.g., KAM surfaces, or banana regions). Also the chaotic regions are no longer attractors, but contain solutions most of which continue to wander around regions of phase space, which have the dimensions of the phase space, possibly leading to large-scale 'Arnold diffusion'. The above perceptions can stimulate our frequently limited imaginations to be on
the lookout for other interplays between chaos and order. Of course we already found, in one-dimensional maps, that there are cases of 'false-periodicity' or 'temporary-periodicity', which we call intermittency, and the 'not-quite-periodicity', which we call semiperiodicity (Fig. 6.127). Such important 'interplays' are distinct from the above divisions of phase space, which however may become dynamically linked by the ever present stochastic environment. Fig. 6.127 'Chaotic.
Intermittency Semiperiodicity
Models based on second order difference equations
120
As we move on to consider flows in higher dimensions, we will find interesting examples of 'false-chaos', 'quasi-turbulence', knotted basins of attraction, and will see how order can arise out of diffusive interactions between spatial regions. All of these insights contribute to our growing understanding of the wonderful variety of dynamics we observe about us (and in us!), which is frequently based on some form of symbiotic interplay between chaotic and coherent (orderly) processes. Of course for a long time we have known of examples where long-range interactions between thermally `chaotic' particles can produce coherent dynamics (e.g., plasma oscillations, superconductivity, etc.), or in `crowded' billiard-ball situations (e.g., sound waves, water waves, vortices, etc.). What we will find, in addition to other examples of this type of dynamic interplay, are localized interactions being diffusively communicated over large spatial regions, to produce coherent global dynamic patterns and/or structures (fixed configurations) all of this possibly within a `stochastic' environment. Life itself, of course, depends on something like this. Thus, as we move forward, we should stand back occasionally from the immediate flower we are examining, and view the growing garden which we are cultivating, and the interdynamic patterns it presents to us.
Comments on exercises (6.1) This example was studied by Beau, Metzler, and Ueberla (1987). More details
concerning the bifurcations have been obtained by Kuznetsov (1986), and Schult, Creamer, Henyey, and Wright (1987). (a) x = uD/(1 + 2D), y = vD/(1 + 2D) (b) x, = (0, 0), unstable; x2 = (2, 2), stable if D < 2; unstable if D > 2. Fig. 6.128
(c) A stable period-two pair of points branch away from x2 as D increases slightly above D = 2. This might be viewed as the map of a stable limit cycle (double-looped) in R3. Fig. 6.129
Epilogue
(d) Around D
121
0.6 the period-two points become unstable, and are `circled'
by a stable nonperiodic set of points. This is something like a Hopf bifurcation except the limit cycle in R2 is replaced by a stable limit-T2 surface of a flow in R3. Fig. 6.130
(e) The two circles grow, then produce a series of overlaps, some stable periodic
orbits, and then something like the Eiffel Tower. The map is unstable (unbounded) for D >, 0.686.
(6.2) h(t) satisfies the Levinson-Smith theorem; so does g(x), provided that E > 0; finally f (x; z) satisfies the theorem if A > 0 (taking M > A) and µ > 0. (6.3) The direction which can go to infinity, for a fixed E, is along the q2-axis, because
of the factor -q3/3. There are three real roots of q32 - 3g2/2 + 3E = 0 provided 2 2
that (-9(2)2)3+(2E+2'-(2)3)2<0, or (ZE-g)2<64. Therefore, if 1>E>0 there is a bounded region, and a disjoint unbounded region. (6.4) I1 and I2 are presumably the major and minor radii of the two one-dimensional tori, V. Thus (p,, q,) and (p2, q2) are ('sort of') in two orthogonal planes, in keeping with (6.2.2). The difficulty is that we are trying to represent the phase R4, (p,, g,, P2, q2), in R3, and that cannot be done uniquely. As long as we deal with fixed values of I, and I2, as in the case of uncoupled oscillators, the figure can uniquely represent the state of the system.
space
0
Fig. 6.131
A
0
(6.5) (a) The flow looks like Fig. 6.13 1. The rotation number is zero. There are two limit cycles, one stable (0 = n), the other unstable (0 = 0). (b) Since If (0) I < 1 and continuous, sin 0 + f (O) must be zero for some 0, and change sign across this value of 0. Since f (O) is periodic, there must be two such values, corresponding to a (stable, unstable) limit cycle, at fixed 0. Therefore p = 0. (c) p
30/271.
122
Models based on second order difference equations
(6.6) (A) After the collision, v*
vn + 2vp(tn), so p* = -p, + k sin 0n. Since v, 1 = vn -g(tn+1 - tn), Pn+1 = P+i -20. The next collision occurs when x(t) =
xp(t). If t = to+, - tn, then x(t) = xp(On) + vn t - (2)gt2 = - (V/w) cos 0n+, or p*0_02
= (7)k[cos 0n - cos(0n + 0)]. In the limit k x 0, the last equation yields p* - 0 - 0 or pn+, x -0 = - 0n+, + 0n, and since pn+, x -p*, Pn+, pn -k sin 0n; the standard map for U'nl = -Pn (B) vn+1 = vn - 2 V0 sin 0n ; On+, = On + (2&)L/vn) ; note the inverse dependence
on vn. This is an area-preserving map. (6.7) In order for En+ 1 = En + 27rN, when /3 = 1, we must have E0 = 21mM (M: integer). The map for (60n, 5En) is 5En+ 1 = 8En + K cos 0060n, and 60n+ = l60n + 6E, + 1
K cos 0060n. The characteristic equation for the multiplier ), 5En+ = AbEn, bon+ 1 = A80n, is A = (1 + (K/2) cos 00) ± [(1 + (K/2) cos 00)2 1]1/2 . Hence the mode is stable if, and only if (1 + (K/2) cos 00)2 < 1. Note that n > 00 > 0
-
1
(K sin 00 = 27rN), and the last condition requires that cos 00 < 0, so it > 00 > 7r/2, for stability. We find that mode N is stable if [(2mN)2 + 16]i > K > 27mN.
(6.8) k is the number of invariant groups of hyperbolic-elliptic pairs. n = 1 or n = 3. (6.9) If Tr M < 0, then xn + 1 is on the opposite side of the fixed point, relative to xn.
(6.10) a, and b, lie on WS between 3 and 4. This requires the W°-arc that passes from a to b must return between 5 and the hyperbolic point, pass through the arcs 1 to a, and b to 2, in order to make a loop through WS between 3 and 4 - and so on! (6.11) The point here is to consider x and Sx as x approaches the KAM curve. Since x on the KAM curve maps into it, whereas off the curve it presumably maps across the curve, there would be a discontinuity as x approaches the curve. (6.12) n=[3,7,15,1,292,1,...],2112=[1,2,2,2,2,...],3112=[1,1,2,1,2,1,2,...]. Note that the factor 292, makes it much easier to approximate than 21/2. A `quadratic irrational' is an irrational solution of a quadratic equation. There is a theorem,
due to Lagrange, which states that any quadratic irrational has a continued fraction expansion which is periodic from some point onward. (6.13) Of course AB = AC + CB. (6.14) Straightforward integration. K = (2n/Sl)E and rn = (27r/S2)In. (6.15) (a f /ax)(ag/a y) - (ag/ax)(a f /a y) _ + 1 (= - 1 for orientation reversal).
(6.16) M11 =cosa+4sinatan(/2),M12 = - sing, M21 = sin a - 4 cosa tan (a/2), M22 = cosa; Tr M = 2 + 4 sin2(a/2). (6.17) We assume T =1211,12 = 1,1, = 1, and 11 PO = P0,11 TNPo
11PN = PN. Then T2NPo=TN-1121,PN=TN-112PN=TN-112TH-112Po=TN-%1211 TN-2x
0 = ... = TN-K12TN-K12P0=... =12 Po = Po. Can you generalize this to the case 11 PO = P0, but 12PN = PN? 12P0=Tn`I,TN-212P0=TN-212TH-212'
Epilogue
123
(6.18) Period-1 is 1 to 1. The remaining (M - 1) points can map in any permuted fashion. Since there are (M - 1)! permutations (maps with this period 1) the probability of period 1 is (M - 1)!/M! = 1/M. For period-2, 1 to X to 1, there are (M - 1) possible values of X, and (M - 2)! such maps for each X. Hence
the probability of period-2 is (M - 1)X(M - 2)!/M! = 1/M. Hopefully the generalization is clear. (6.19) Many questions can be raised, such as: (a) For what values of n, or for what Poincare recurrences, does this phenomenon occur? (b) How many patterns are
produced in these different cases? (c) Are there cellular patterns which can disappear and then reappear in a displaced configuration many times before recurring? (d) Could such `bubbling cellular patterns' in any way represent such `turbulent' patterns in fluids (e.g., Benard convection)? Why not? These can all be explored on a PC, using a few lines to outline cells (see text). Fig. 6.132
b2
bl
a2
a,
b4 b3
A
B
(6.20) See Fig. 6.132. The partition B is an example of a `refinement of A' (see the definition below).
(6.21) If c is associated with f, then the minimum number, cn, associated with g is c'n =
Hence, from (6.15.11),
h(g) = lim
1
n-oo n
log c;, = lim
1
n-oo n
log ck = lim
k log cm = kh*.
m-'oo M
That is, the T-entropy is k times larger for g than for f. This illustrates the fact that the dynamic entropies measure the rate at which mixing takes place. Since g is k iterates of f, its rate of influence is clearly k times `faster' (per iteration). Fig. 6.133
f(x)
Models based on second order difference equations
124
(6.22) (a) See Fig. 6.133. A region dx has two pre-images, with total length 2
dx/Idf/dxl1=xodx+(1 -xO)dx=dx. (b) Since µ(a1) _ (xo - 0), and µ(a2) = 1 - xO, HM(A) xo log xO - (1 x0)log(1 -xO). (c) The two pre-images of xo, f (x;) = xO (i = 1, 2), are x1 = xo and x2 = 1 - x0(1 - xO). Thus f -'(A) _ { [0,x1 ], [xt, xO], [xO, x2], [x2, 1] }, so
A'Av f-'(A)=f-'(A). (d) H,(A')= -xolog xo-(xO-xo)log (xO-xo)-(1-xO)2log (1 -x0)2x0(1 - xO) log [x0(1 - x0)]. A little algebra yields H,(A') = 2Hµ(A); where H,(A) is in part (b). (6.23) H,,(An) _ - Y_ (xk+, - xk)1og (xk+ 1 - xk) (with trivial notational changes). As in the last exercise, the pre-images of any Xk are xk = xOxk and xk = 1 - xk(1 - xO). Thus each interval (xk+ 1 - xk) in An has the pre-images of length Ixk+,, -xkI=xOIxk+1 -Xkl and Ixk+1 -xkI = Ixk+, -xkI(1 -XO). Therefore Hµ(f -'(An)) _ - Y-{xO(xk+ 1 - xk) log [xO(xk+ 1 - xk)] /+ (1 - xO) x (xk+
1 - xk) log [(1 - xo)(xk+ 1 - xk)] }. Since Y (xk +I - xk) = 1, Hµ(f -'(A.))=
-XologxO-(1 -xo)log(1 -xo)-(xO+ 1 -xo)y(xk+1 -xk)log(xk+1 -xk)
In the notation of Exercise 6.22 (b), this can be written H, (f -'(A,,)) = H,(A) +
H, (A,,). Hence, for any partition A, the entropy of its pre-image partition f -'(A,,) (which is also An v f -'(A,,)) simply has the basic entropy H,(A) added v f -n+'(A), then to the entropy of A. If we take An to be An - A v f -'(A) v we can explicitly carry out the limit (6.15.16), and (to no surprise!) find that
h,,(f, A) = H,(A) = h(f) (the T-entropy). Since this does not depend on A, it is clear, from either (6.15.17) or (6.15.18), that h (f) = h(f) (as in the area-preserving torus map, Example III).
7
Models based on third order differential systems
There are a number of examples of interesting and important models which can be described by the third order autonomous system z = F(x, c) In addition,
(xeR3).
(7.0)
these physical models have stimulated the creation of various
mathematical models which give a better understanding of what features of F(x, c) in (7.0) are needed to produce particular dynamic effects. Among the models which will be discussed in this chapter are: (1) The Lorenz model: A highly simplified model of a particular hydrodynamic flow (the Rayleigh-Benard problem), which has proved to be of great heuristic value, due to the rich variety of dynamic phenomena which are contained in this `simple' model. This system can exhibit, for various values of one control parameter, either stable or unstable fixed points, a globally attracting periodic or aperiodic ('strange attractor') solution, a homoclinic orbit embedded in a two-dimensional stable manifold, a heteroclinic orbit (between an unstable fixed point and an unstable limit cycle), intermittency effects, bistability and hysteresis, coexistence of stable limit cycles and chaotic regions, and a variety of cascading
bifurcations. Because of this wealth of new phenomena, a rather extended discussion of this model will be given first. (2) Lorenz-dynamic fluid system: A fluid in a vertical circular tube, which is heated
along the circumference, obeys equations that can have a separable set of equations which are the Lorenz system. The advantage in this model is the nonapproximate connection of the Lorenz equations and a simplified physical system.
(3) Dynamo systems: These elementary models of electromagnetic-mechanical dynamo systems were stimulated by the fascinating geophysical observations of the erratic inversions of the Earth's magnetic field, which has occurred many times over the past 150 million years. These models are not serious attempts to explain the geophysical phenomena, but they do illustrate a system which
126
Models based on third order differential systems can produce erratic reversals of a large-scale magnetic field. The simplest model with this property turns out to be equivalent to the Lorenz model.
(4) Rossler systems: This is a collection of various mathematical models which shed light on several specific aspects of nonlinear dynamics. They are not `encumbered' by the need to be justified on physical grounds! (5) The Field-Noyes equations (Oregonator): This model was devised to explain the spectacular Belousov-Zhabotinskii oscillations in chemical systems, in which the color of a chemical mixture changes periodically in time. This basically very complicated system can therefore, under suitable conditions, exhibit very stable, simple dynamics (i.e., a stable limit cycle). These systems may also exhibit
very beautiful spatial-temporal dynamic patterns, but that lies outside of the realm of ordinary differential equations in R3. Before taking up any of the details of these models, or of the linear systems, we should note a few general features of the motion in three-dimensional phase spaces which account for the much richer dynamics than can be obtained in two-dimensional phase spaces. We have already seen some examples of this richness in the case of nonautonomous second-order systems (e.g., the various periodically forced nonlinear oscillators). The nonautonomous system can be viewed as a dynamics in an extended phase space, where the new axis is the time, which raises the dynamics to three dimensions, thereby recovering the uniqueness property of the automomous system. We are presently dealing with a bona fide three-dimensional phase space, of which the extended phase space is a very special case (namely, when one `unidirectional' variable is not influenced by the other two). Clearly, therefore, all of the richness found in the case of forced nonlinear oscillators must be generally contained in the present systems. However, the present autonomous equations can produce other effects which are both more uncontrolled (since the third variable is now dynamically coupled to the other two) and entirely new, such as a homoclinic orbit which must be embedded in a surface, or a heteroclinic orbit which connects a fixed point and a limit cycle. Moreover the nature of chaotic motions becomes richer, if for no other reason than it can now take up more than two physical dimensions. Before looking into these wonderous possibilities, it is important to first establish a few of the facts about linear motions near the fixed points.
7.1 Linear third order equations As in the case of two-dimensional flows, it is important first to describe the possible types of motion in the neighborhood of fixed points, which are usually governed by the linear equations 3
x; = Y. aux; j=1
(i = 1, 2, 3),
(7.1.1)
Linear third order equations
127
where the a. are real constants. Setting x, = Ai exp (),t) yields the usual characteristic determinant I a;j - lb ijI = 0, or
23-T2.2+MA-D=O,
(7.1.2)
where T = Y- aii = Tr aij;
M=
all
a12
all
a13
a22
a23
a12
a22
a13
a33
a32
a33
= sum of the diagonal minors;
D=detla, I An eigenvector of (7.1.1), associated with the eigenvalue 1k, is the vector (A 1k, A2k, A3k) satisfying lkAik = Y_j=1 ai;A;k (i = 1, 2, 3). It is complex, if 1k is complex. In this case
there is a complex conjugate eigenvalue and vector, because the a, j are real.
Exercise 7.1 The behavior of the system (7.1.1) is governed by the three control parameters (T, M, D), and hence this system can be transformed into another system involving only these parameters (instead of the nine aid). Obtain such a system of equations. (This is the generalization of exercise (5.3)).
The cubic equation (7.1.2) has the standard properties for the roots: (Note also
l(D,T,M)= - A(- D, - T,M))
Al+22+23=T h1A2+2113+2223=M
(7.1.3)
211223 = D.
Since D, T, and M are real, one 2i is real, and the other two are either real or are a complex conjugate pair. Moreover all roots are real if
q3 +r2<0, where
q-3M-4T2 and r=6(-TM+ 3D)+21.,T3. Hence all roots are real if
G(D,T,M)-4M3-T2M2-18DTM+27D2+4DT3<0,
(7.1.4)
which defines two regions in the `DT M space'. This can be compared with the simpler
situation, in the `DT space', in the case of second order systems (Chapter 5). The projection of the surface G(D, T, M) = 0 onto the (D, T) plane is illustrated in Fig.7.1.
128
Models based on third order differential systems
Fig. 7.1
Note that, since G(D, T, M) = G(- D, - T, M), only the portion T > 0 is represented in the figure. The possible complex eigenvalues can be represented in the complex plane by using
a circular portion of that plane whose center is the origin, 2 = 0. The use of the circular portion of the plane is simply for esthetic reasons, and its size is arbitrary. We can thereby make the following representation between the eigenvalues and the values of G and D (the further breakdown is left as an exercise). The first group, the nondegenerate cases, contains those eigenvalues such that
0, and no real parts are equal unless they are complex conjugates. The remaining eigenvalues are referred to as degenerate. Some examples are shown in Re (ilk)
Figs. 7.2 and 7.3. The representation of the various flows is considerably more involved in the three-
dimensional phase space than in the two-dimensional case. If we want to represent not only the real vs the complex eigenvalue flows, but also want to show the relative magnitude of their real parts, then a full three-dimensional figure is required. A number of these figures have been given by Arnold (1978), from which Fig. 7.4 has been adopted.
As nice as these figures are, they are also difficult to draw, and contain more information than may be of immediate interest, such as the relative magnitudes of the real parts of A. If this information can be neglected, then there are simpler representa-
tions which nicely convey the remaining features. This type of representation is due to the creativity of C.D. Shaw and of G. Francis, and is illustrated in Fig. 7.5. (For a much richer collection of dynamical representations, see Abraham and Shaw, 1982, 1983, 1984.) Note that a three-dimensional arrow is used to display motion along a
Linear third order equations
129
Fig. 7.2 Nondegenerate cases.
D>O
D
G<0
G>0
Fig. 7.3 Degenerate cases. etc
T=0
M<0 D=0
T>0 M>0 D=0
T<0 M>0 D=0
T>0 M>0
D>0
T<0
M>0
D<0
Fig. 7.4
x2
A3
ReAi =ReX2
Models based on third order differential systems
130 Fig. 7.5
Stable node (Index 3)
Stable focus
(Index 3)
Saddle node (Index 2)
Spiral-out saddle (Index 1)
one-dimensional manifold (trajectory), whereas a flat arrow denotes the motion of the family of solutions on the indicated surface (all, of course, simply to give perspective to the figure). These illustrate four flows, together with the figure for their eigenvalues. The figures, however, do not illustrate the relative magnitude of the eigenvalues. The terminology
`index' is sometimes used to indicate the dimension of the `inset' (that is, the stable manifold of the fixed point), and is sometimes used in descriptive names associated with some of the flows. By the stable (or unstable) manifold (Ws or of a fixed point is meant the collection of all states in the phase space which tend to the fixed point at an exponential rate as t -+ + oo (or t -> - oo) respectively. Such manifolds are tangent to the eigenvectors or eigenplane defined by conjugate pairs of eigenvectors
with negative (or positive) real parts. Such planes are illustrated in Fig.
7.5
by the disks, or `spinning pinwheels'. The full arrows in these figures are tangent to
one-dimensional stable, or unstable, manifolds. If the stable manifold is threedimensional it is called the basin of attraction of the fixed point, and this fixed point is called an attractor. The above figure is not exhaustive, as a number of cases are not displayed. This representation of the flows near fixed points will generally be used in what follows, since the relative magnitude of the real parts is not frequently important. Many imaginative representations of dynamics and surfaces have been created by Francis (1987). These representations show `sheets' of trajectories which may (or may
Linear third order equations
131
not) be parts of a stable or unstable manifold of the fixed point (at the origin). Full arrows may be used to represent various one-dimensional sets, such as those which are asymptotic to the fixed point as t -> + oo or t --> - oo. These ideas have been freely borrowed in many of the following figures. Fig. 7.6 illustrates a saddle node. In this case the fixed point has a two-dimensional stable manifold, WS, associated with two real negative eigenvalues (a node in R2), and a one-dimensional unstable manifold, W. The local portions of WS and W are illustrated in the figure, together with an arbitrary manifold containing local trajectories, which is not part of WS or W. This gives an indication of the flow off of these manifolds. Fig. 7.6 Saddle node.
When W. has a stable focus character near the fixed point, the arbitrary manifold of trajectories takes on a more dramatic spiral structure, as illustrated in Fig. 7.7. This is sometimes referred to as a saddle focus, or saddle-spiral, neither of which indicate the stability or instability of the two-dimensional manifold. `Stable spiralsaddle' or 'Spiral-in saddle' clarifies this point, but is not generally used. In the degenerate case, when the fixed point has a purely imaginary pair of complex eigenvalues, then there may be a local manifold that contains periodic solutions of the
nonlinear equation (7.0). An invariant manifold that is tangent to the manifold that contains centers about a fixed point is called a center manifold, W, W, might contain only periodic solutions, and look something like Fig. 7.8(a), but more commonly it has expanding or contracting dynamics due to nonlinear effects (see Fig. 5.5). If there is no `radial' flow, the uniqueness of the manifold is not insured by the uniqueness of the solutions. In particular, W, can bifurcate into
132
Models based on third order differential systems
Fig. 7.7
Saddle focus or spiral-in saddle
two branch surfaces. This is illustrated by the `pinched cylinder' with a central disk, all of which is a center manifold (Fig. 7.8(b)). This illustrates one global difference which can occur. For more details on center manifolds, see Guckenheimer & Holmes (1983).
Before discussing the details of particular models, we will consider a few other global nonlinear flows - that is, flows which are not necessarily near a single fixed point.
7.2 Nonlinear flows The flows which can occur in three dimensions are much more varied in character than those in two dimensions. Indeed it is only possible to present a few examples, to illustrate the richness of this dynamics. A relatively simple, but important example of such flows are those in the neighborhood of periodic orbits. Fig. 7.9(a) illustrates the case where the periodic orbit is a stable limit cycle, in which the nearby flow spirals around the orbit in the process of approaching it. This limit cycle is called a focus cycle. If the flow near a periodic orbit, F, has a two-dimensional set of solutions which
Nonlinear flows
133
Fig. 7.8
(a)
(b)
is asymptotic to F as t + oo (an R2 stable manifold, Ws), and another R2 set asymptotic to F as t -> - oo (the unstable manifold, W, of F), then it is called a saddle cycle. An illustration is shown in Fig. 7.9(b). Only a small portion of W, and W is shown, for clarity. The remainder connects smoothly with the portion shown (see below), and extends out into the phase space (in some unspecified fashion). If both of the manifolds in the figure are stable (unstable, respectively), then the limit cycle is a stable (unstable) node-cycle.
Another possible flow in the neighborhood of a periodic orbit is illustrated in Fig. 7.10. In this case we illustrate the flow on a set of nested tori, and the orbit is a center cycle. If a solution on such a torus is not periodic, then all solutions on that torus cover the surface ergodically - that is, they come arbitrarily close to any point on the torus.
134
Models based on third order differential systems
Fig. 7.9
(Q)
(b)
Fig. 7.10
Nonlinear flows
135
Exercise 7.2 Obtain an example of some (x,(t), x2(t), x3(t)) that moves on a torus, which is centered at the origin, and contains (inside the torus) a periodic orbit of
radius R, lying in the (x,, x2) plane. The example should contain two arbitrary frequencies (cwt, w2) and a minor radius r. Note a requirement which must be satisfied
if this represents a (unique) solution of some differential equations. Determine for what ((0,, w2) the motion will ergodically cover the torus. Prove that this motion cannot be a solution of the linear equations (7.1.1). Fig. 7.11
The above figures illustrate only a few examples of the possible flows near periodic
orbits. More generally, the flows near periodic orbits in three dimensions, F in Fig. 7.11, can be partially described with the help of Poincare's first return map. This map is defined on a two-dimensional plane, S (a `Poincare section'), which is transverse to the periodic orbit. All points on S near the intersection of F and S (point 0), when considered as initial conditions of the equations of motion (7.0), .z = F(x), will generate
solutions that move in R3 near to F. These solutions will again intersect S in the neighborhood of 0. This produces a map of the points on S near 0 to other points on S near 0, P: (x,, x2) - (x i , X2*), which is known as Poincare's first return map. The figure shows an example of this, where the initial state 1 has a solution which intersects
S at point 2. Near the point 0 this map is linear and can be described in terms of its characteristic multipliers (CM), Mk (k = 1, 2). These are determined by the eigenvector equations
x* = A;;x; = Mx;
(i = 1, 2)
yielding
(A11 -M)(A22 -M)-A12A21 =0. Since the A;j are real, M, and M2 are either real or a complex conjugate pair. An alternative description of this map is to use the characteristic exponents (CE), 2k, which are simply related to the CM by Mk = exp ().k)
There is clearly no profound reason for selecting one description over the other, but the distinction should be kept in mind (thus, in the differential equations of Section 7.1,
136
Models based on third order differential systems
the characteristic exponents were used exclusively, whereas multipliers may seem more `natural' for maps).
Exercise 7.3 Before proceeding, see if you can describe the (real, imaginary) and magnitude properties of the two CM for each of Figs. 7.9(a), 7.9(b), 7.10. If you can do this, then transcribe these properties of the Mk to those of the characteristic exponents 2k. If some I Mk I < 1 (respectively, I Mk I > 1) then there is attraction toward (repulsion from) the periodic orbit in some direction. If both I Mk I < 1, k = 1, 2, (respectively, > 1)
then all nearby solutions are attracted toward (repelled from) F. If they are moreover complex, then the attraction is in the form of a spiral motion (Fig. 7.9(a)). If the Mk
are purely imaginary, the nearby solutions spiral around F on toroidal surfaces (Fig. 7.10), and may not be periodic (Exercise 7.2).
A very interesting and important case is when the Mk are real, such as the case I M11 > 1, I M21 < 1 (the saddle-cycle, Fig. 7.9(b)). Note that the stable and
unstable manifolds in this figure have not been completed all the way around the periodic orbit. The signs of M1 and M2 give additional information about how this completion is accomplished. Just as in the case of one-dimensional maps, a negative multiplier means that the mapped point occurs on the other side of the fixed point (0 in the present case). In the present two-dimensional map, this is again true, but in addition, the signs of M1 and M2 must be related because they are determined jointly by the flow in three-dimensions. Consider first only the unstable manifold, with M1 < - 1. What are the possible forms of W in this case? Several are illustrated. In Fig. 7.12(a) the unstable manifold `rotates' one half revolution about the periodic orbit per cycle. The local unstable manifold in this case lies on the famous Mobius band. The Mobius band has only one side and one edge. This one-edge property is
of basic importance for the process of subharmonic bifurcations, as we will see illustrated in the case of the Lorenz model (Section 7.9).
Fig. 7.12
(a)
(b)
Nonlinear flows
137
In Fig. 7.12(b) the unstable manifold rotates three half-revolutions around the periodic orbit per cycle. Since there is an odd number of half-rotations per cycle, we
again have M 1 < - 1. That is, the Poincare first return map of a solution on this manifold produces a mapping on the other side of the point 0. Again the local manifold lies on a band which has only one edge, and may be associated with a subharmonic bifurcation.
Since the stable manifold rotates in a similar fashion, we have 0 > M2 > - 1 in both of the above examples. In other words M2 is negative (or positive) when M1 is negative (or positive, respectively). The characteristic multipliers are positive when the
stable and unstable manifolds rotate some integer number of complete revolutions about the periodic orbit per cycle. We see, therefore, that the values of the Mk only indicate whether these manifolds belong to sets of manifolds which rotate an even or odd number of half-revolutions about the periodic orbit per cycle. Many other nice illustrations of these flows can be found in Dynamics by Abraham and Shaw (1983). As already indicated, we will return to these considerations when we consider subharmonic and harmonic bifurcations of flows in R3. The global flows which involve several fixed points produce interesting `interactions'
between their respective stable and unstable manifolds. A very simple case is the `repulsion' between the two two-dimensional unstable manifolds of two saddle-nodes. The nonintersections of these manifolds is required by the uniqueness of the solutions. An example of this illustrated in Fig. 7.13.
Fig. 7.13
138
Models based on third order differential systems
Exercise 7.4 What can (must?) occur to the two two-dimensional manifolds WS , W of a stable-node saddle (-) and a coexisting unstable-node saddle (+)? A more dynamic `interaction' is illustrated in Fig. 7.14, where a saddle-node (SN) is joined with a spiral-in saddle. The unstable manifold, is mostly (except for one orbit) `thrown away' by the spiral-in saddle, S-. Fig. 7.14
We end these illustrations with the case of two saddle-spirals, one in, the other out (Fig. 7.15). Now both the stable manifold, W,(S-) and the unstable manifold, have a spiral structure. Three other manifolds, which are asymptotic to orthogonal directions, are shown in one of these spirals, so W,(S-) is seen with three interspersed manifolds. A similar construction could be made about the other spiralsaddle, but this figure is sufficiently complicated! The purpose of introducing these interspersed manifolds will become clear in the discussion of the Lorenz model, where it is necessary to consider such structures.
7.3 The Lorenz model In 1963 E.N. Lorenz published a study of a highly simplified model of a particular hydrodynamic flow problem, which has become a classic in the area of nonlinear dynamics. The importance of this model is not that it quantitatively describes the
Lorenz model
S+: Spiral-out saddle
139
S-: Spiral-in saddle
hydrodynamic motion, but rather that it illustrates how a simple model can produce very rich and varied forms of dynamics, depending on the value of a parameter in the equations. Moreover, Lorenz analyzed the dynamics by employing a new type of `map', distinct from the Poincare map, which gives very convincing evidence that the dynamics can have a `strange attractor' character. Since his initial study, which
was followed by a long period of neglect, he and many other investigators have uncovered a rich variety of additional dynamic features in this `simple' model, which clearly justifies its consideration in some detail.
As Lorenz later recounted (1979), this model resulted from his interest, as a meterologist, to obtain aperiodic solutions of a thermal conduction situation in viscous
hydrodynamics, which could be used to simulate `statistical' wheather conditions. Using a twelve variable system, a group at MIT numerically found solutions which indeed were not only aperiodic, but also proved to be very sensitive to the initial conditions. That is, it was discovered accidentally that solutions with nearly the same initial conditions can behave very differently after some time. Lorenz recognized that if the real atmosphere behaves like this model, then long-range forecasting of detailed weather conditions would be impossible (although he was quite cautious
Models based on third order differential systems
140
about what `long-range' really meant). This group at MIT failed, however, to reduce this model to a more manageable size than the twelve variables they had used. A
break in this problem came from an interaction with B. Saltzman, who had been studying thermal convection with a system of seven ordinary differential equations. Saltzman (1962) had found some solutions which were also aperiodic, and moreover four of the seven variables appeared to tend to zero in these solutions. Lorenz then sought to find such aperiodic solutions using the equations which only involve the remaining three variables. Thus was born the Lorenz model. The approximations of the hydrodynamic equations which give rise to this model are indeed severe, and have caused many questions to be raised about the relationship of the model and real hydrodynamic flows. We will simply present the approximations which yield the model, without any implication that the hydrodynamic conduction is accurately represented by such a model (more refined models will be discussed later). It is noteworthy, however, that there are mechanical and electrical systems which are accurately represented by the Lorenz model, as will be discussed later. The original problem, considered by Rayleigh in 1916, concerns the energy transported through a fluid layer of depth H, when the lower surface is maintained at a temperature AT above the upper surface.* The governing hydrodynamic equations can be written in the form used by Saltzman (1962), aatV20
0(0' V20)
+ a(x,z) 00
0(0,0)
-
4
a0
AT aO2 - KV 0 = 0
(7.3.1)
at + a(x, z) - H Ox
where it is assumed that the flow is only a function of x, z, and t (see Fig. 7.16). The function 0 is the stream function, so that the components of the flow velocity are given by uz = - ac/az, uZ = 00/ax. (7.3.2)
0 is the departure of the temperature in the fluid from that which occurs when there
is no convection present (i.e., 0 = T - To - AT(1 - z/H)). The constants g, a = - p -1(d p/dT), v, and x represent, respectively, the acceleration of gravity, the coefficient of thermal expansion, the kinematic viscosity, and the thermal conductivity. Finally, V4
a4 ax4
+
04 aZ4
and
a(A,B)=aAaB aAaB a(x,Z)
ax OZ
az ax
*See M. Velarde (in Fluid Dynamics, R. Balian and J.L. Peube (eds.), 1977) for a nice discussion,
and some critical comments on the Rayleigh-Benard instability.
Lorenz model
141
As the temperature difference, AT, is increased, the transport of energy from the lower to upper surface by heat conduction becomes unstable, and is augmented by fluid convection, in the form of rolls illustrated in Fig. 7.16. Rayleigh found that this fluid motion represented by the functions 0 = t/io sin (nax/H) sin (irz/H) 0 = 0 cos (7rax/H) sin (7rz/H)
(7.3.3)
would develop if the following inequality is satisfied, R = gaH3AT/vk > R, _ (it4/a2)(1 + a2)3.
(7.3.4)
Here a is related to the size of the rolls in the x direction (see Fig. 7.16), and the minimum critical value of Rc is 27ir4/4, which occurs when a2 = Z. R is now known as the Rayleigh number. Fig. 7.16
x
If AT is further increased, the Rayleigh convection solution becomes unstable, and is replaced by the time dependent dynamics studied by Saltzman. Of the seven spatial Fourier modes which he considered, the three that appeared to persist in the aperiodic solutions, and therefore were retained by Lorenz, are = x(t) 0 = y(t)
212(1 +a 2) a2
sin (nax/H) sin (7rz/H)
2112Rc
---- cos (nax/H) sin (7rz/H) - z(t) R`- sin (21rz/H)
irR
irR
(7.3.5)
which now define the three functions of time, x(t), y(t) and z(t). It should be noted
that these functions have nothing to do with the spatial coordinates. The above solution differs from Rayleigh's spatial form only in the last term of 0, which thereby involves another vertical temperature variation, now containing one full wave length.
Substituting the ansatz (7.3.5) into the governing equations, (7.3.1), and simply
142
Models based on third order differential systems
neglecting the spatial variations which are orthogonal to the ansatz (7.3.5), Lorenz obtained the system of equations (the Lorenz model)
x= - a(x - Y) rx - y - xz
(7.3.6)
i= -bz+xy where the dot refers to the dimensionless time x = n2(1 + a2)xt/H2, r = R/RC, a = v/x (the Prandtl number), and b = 4/(1 + a2). Following Saltzman, Lorenz used the values
a= 10, and a2 = i (so b = 3), which gives the minimum critical Rayleigh number. We will refer to these values as the `canonical case', because they have been so widely used.
When r < 1, the only fixed point of the system (7.3.6) is x = y = z = 0, which physically corresponds to no convection (the energy is transported only by conduction).
The linear motion about this fixed point has characteristic exponents given by the roots of the characteristic equation
(A +b)[(A2+(a+1)2+a(1-r)]=0
(7.3.7)
which has three real roots if r > 0. If r < 1 they are all negative, so the heat conduction
is stable, but one root becomes positive if r > 1, which corresponds to Rayleigh's instability. Thus the condition for the onset of convection in the Lorenz model is the same as Rayleigh's result for the full system of hydrodynamic equations. Exercise 7.5 In the case r < 1, obtain a Lyapanov function for the Lorenz system, and thereby prove global asymptotic stability of the origin. Exercise 7.6 Show that the characteristic eigenvectors at the origin can be taken to be (0, 1, 1), (a, a + A +, 0), and (a, a + .1 _ , 0), corresponding to the eigenvalues A = - b, and A± = i { - (1 + a) ± [(1 + a)2 - 4a(1 - r)]112}. From this, determine the variation of both the stable and unstable eigenvectors in the (x, y) plane, as r is varied from 1 to 100, by determining their angle with the x axis. Obtain the equations of motion for (s, u, z), where s = (a + A±)x - ay and u = - (a + A_ )x + ay, and see how these variables vary due to the z coupling. Note that the z-axis is globally contained in the stable manifold of the origin. In addition to the origin becoming unstable when r > 1, the Lorenz system simul-
taneously acquires two additional fixed points, which we will designate as C + and C (C +, C -): x = y = ± [b(r - 1)] 1/2,
z = (r - 1)
Lorenz model
143
Notice that, as r increases from unity, these two fixed points move away from the remaining (unstable) fixed point at the origin. These new fixed points represent steady (time independent) fluid convection, as well as a deviation of the temperature away from a simple linear spatial dependence. The fluid convection at these two fixed points differs only in the sense of the rotation of the vortex cylinders, as illustrated in Fig. 7.17. Fig. 7.17
-A 0
0
C+
C-
The physical system is, of course, symmetric with respect to these two situations, which is reflected in the invariance of the Lorenz equations to the transformation (x, y, z) - (- x, - y, z). The phase portrait of the equations (7.3.6) therefore always exhibits this symmetry, for all values of r. The characteristic equation for perturbations away from the fixed points C+ and
C- is
A3+(a+b+1)22+(r+a)b1+2ab(r-1)=0
(7.3.8)
and it is understood that r > 1. Since the coefficients of (7.3.8) are real and positive, one root must be real and negative. We will call it 20 < 0. The other two roots are real only if A3 + B2 < 0, where
A=3b(r+a)-9(a+b+ 1)2
and
B=6b(r+a)(a+b+ 1)-ab(r- 1).
For the canonical values a = 10, and b = 3, this shows that all of the roots associated with C + and C - are real if r < 1.34561 . . . - r*,
(7.3.9)
and they must all be negative (again because the coefficients in (7.3.8) are all positive).
Thus, for r < r*, any small perturbation about C+ or C- damps out without any oscillations.
Exercise 7.7 Obtain the characteristic equation (7.3.8) and determine the real roots for r = 1.05, 1.1, 1.2, 1.3, and 1.3456.
If r is slightly above r*, there is one real and two complex conjugate roots of (7.3.8).
In this case the characteristic equation can be written in the form
(A-1)(1-A,-i11)(2-A,+i2;)=0
Models based on third order differential systems
144
which, when compared with the form (7.3.8), yields the relations
(a+b+ 1)= -1'0-21r;b(r+a)=1.2 +1.; +21o1r 2ab(r-1)_ -1o(11 +1i)
(7.3.10)
For small r, the real parts are negative, and the fixed points C + and C - remain stable (2 < 0) as r is increased until there are roots with A,r = 0, at some r - r, Using the last expression, we readily find that this critical value of r is given by
r,=a(a+b+3)/(Q-b- 1).
(7.3.11)
Exercise 7.8 Obtain .1o and an eigenvector at C + associated with )o when (7.3.11) holds. The inset of C + (those points which tend to C +) becomes tangent to this vector at C +.
Note that, since r, must be greater than one, this instability can only occur if a > b + 1 (as in the canonical case). For r > r, all fixed points are unstable, and Lorenz concluded that r = r, is the critical value of r for the instability of the steady convection (7.3.5). While this conclusion is correct for infinitesimal perturbations, it does not give any indication of the global flow pattern in the phase space. It will be shown later that, indeed, there are important global bifurcations which occur for r < rt,
and these are characteristically not detected by considering only the stability properties of the fixed points. Fig. 7.18 (after G. Francis) summarizes the bifurcations associated with the change in stability of the fixed points, as r is increased from r = 0 to r > r, It is important
to emphasize that the figure does not contain (nor imply) any information about global bifurcations or the global connection of these local flows. This subject will be discussed in Section 7.6. Fig. 7.18
1
Lorenz chaotic dynamics
145
Before considering any details of the dynamics which has been obtained by Lorenz and others, we will first note two general global properties of the dynamics established by Lorenz: Property P1 All volume elements contract in the Lorenz flow.
This follows (recall the proof of the general Liouville theorem) from the fact that the divergence of the velocity vector in phase space is everywhere negative. That is v
ax(x)+a (y)+az(i)=-a-1-b<0.
(7.3.12)
Y
Note also that this contraction is uniform in the phase space, since it does not depend on x, y, or z. Property P2 All solutions of the Lorenz system remain bounded in phase space for all times.
To show this, let u = z - r - a, so that the equations in (x, y, u) are
y= -y-x(u+a);
x= -a(x-y);
u= -b(u+r+a)+xy.
Then we readily find that
2dt/x2+y2+u2)= -aX2-y2-b(u+Z(r+a))2+qb(r+a)2. Since the right side is negative everywhere outside of an ellipsoid in phase space, shown in Fig. 7.19, it follows that the distance, s = [x2 + y2 + U2]1 12 , decreases for all states outside a sphere which contains this ellipsoid (also illustrated). Therefore all states are asymptotic to this spherical region as t goes to infinity. While a systematic approach to the general dynamics might suggest that we now consider the global aspects of the Lorenz flow, even below the critical value where all the fixed points become unstable, r = r, we will instead first follow the historical discovery of Lorenz, and consider the dynamics which he discovered when r is greater than r, Later, we will return to discuss the global bifurcation which occurs for r < r,
7.4 Lorenz chaotic dynamics For the canonical values, a = 10, and b = 3, the critical value of r at which the time independent solutions become unstable, (7.3.11), is r, = 24.7368....
In his computations, Lorenz used the somewhat supercritical value r = 28. The
146
Models based on third order differential systems
Fig. 7.19
x
aperiodic behavior of the dynamics he found is illustrated in Fig. 7.20, which shows y(t) over 3,000 iterations In this form, the results are simply complicated, and not particularly interesting (except, of course, for somebody looking for complications!). The dynamics, shown in Fig. 7.21, is more revealing, but actually appears deceptively simple. The motion
has been projected onto the (x, y) and (z, y) planes, for the case where the initial condition is (0, 1, 0). The points (C +, C -) are the fixed points (6 x 2112, 6 x 2112, 27) and (-6 x 21 12, - 6 x 2112, 27), which indicates the scale in the figures.
It will be noted that the dynamics now takes the system in a growing spiral away from one C-point, then it drops into the neighborhood of the other C-point, and proceeds to spiral away from it also (recall that the unstable manifold of the C-points is two dimensional). Thus the dynamics flip-flops between the neighborhood of these two fixed points. Physically, this would presumably mean that the vortical cylinders Fig. 7.20
Lorenz chaotic dynamics
147
Fig. 7.21
Y
x
change both their rate of rotation in an increasingly oscillatory manner in time, and then completely reverse their sense of rotation to a relatively stationary state, and then increasing oscillate their rotation rate once again. As shown above (Property P1), the flows in the phase space uniformly contract, so that the volume into which the trajectories tend must be zero (when based on the usual three-dimensional measures - more details are in Appendix B and below). This suggests that the trajectories tend asymptotically to some two-dimensional surface in the phase space. The approximate nature of this `surface' was illustrated by Lorenz in Fig. 7.22. This surface has a `butterfly' structure, with its wings going in and out
Models based on third order differential systems
148 Fig. 7.22
z
40
20
0
=1.
- 20
I 0
1
,y
20
of the (y, z) plane of the figure. The lines with numbers indicate the intersection of this surface with planes of constant x values, corresponding to these numbers. With improvements in computer graphics, other representations of this asymptotic `surface' have been published, notably Fig. 7.23 obtained by Lanford (1977). Here the solution starts near (0,0,0) and moves out along the unstable direction, and loops immediately to the neighborhood of C -, spirals outwards, then flips over to the region of C +, to spiral outward again. The trajectory below the plane z = r - 1 = 27, which contains C + and C -, is represented by the dotted curves. Lorenz recognized that this asymptotic set (the (o-limit set) cannot simply be a surface, if the trajectory continues to spiral, because the continual spiraling of the trajectories alternately about the points C + and C - makes it impossible for any trajectory to lie on a surface (because it would have to be self-intersecting, and hence not unique). He described this asymptotic set as an `infinite complex of surfaces', but it is now referred to as a fractal set, since it has a dimension (capacity) between two and three (Russel et al. (1980) and Lorenz (1984) have obtained d, = 2.06 ± 0.01). As nice as the above graphics may be, they do not really establish the complexity of this limiting set of points (the w-limit set) of the trajectories. In particular, there is nothing in these results which rules out the possibility that the trajectories might ultimately settle down to some limit cycle in the phase space. What Lorenz did after finding the above results, was to introduce a new type of `map', generated by the dynamics, and to show, with the help of this `map', that such stable limit cycles most likely do not exist for his value of r. His result, which again depends on numerical
Lorenz chaotic dynamics
149
computations, is not a mathematical proof, but nonetheless is a very convincing and very imaginative approach to this question. While Lorenz now (1979) refers to this `map' as a `form of Poincare map', it is quite distinct from what Poincare suggested, and has moreover stimulated other variants of this Lorenz map, which have further enlightened our understanding of complex dynamical situations.
To make this point clear, we should recognize from the outset that Lorenz's approach to the investigation of the dynamics does not introduce a real map, as in Poincare's method, because, to the value of some variable, Xk, it does not associate a unique value of this (or any other) variable, x,. Instead, Lorenz introduced what we will call a match, to distinguish it from a map. When some given conditions, Q(x, x) >, 0, are satisfied, a match associates with each value of some dynamic variable, Xk, the set of values, {x,}, which x, acquires
when the condition is satisfied the next time. Thus a match is a one-to-many association between these variables Xk and x,, M(Q, k, 1). This concept of a match is presumably most useful when the set of values, {x,}, is
not very `scattered', so that the match can be approximated by a smooth functional relationship, x,(n + 1) F(xk(n)), as in Fig. 7.24. If, moreover, the approximating function F(xk) is unique, as in the second figure, then the match is essentially the
same as a map (the nonunique case will be discussed below). In the case of the Poincare map, the uniqueness and continuity of the function is guaranteed by the
Models based on third order differential systems
150 Fig. 7.24 xl
x1
X1
-__./ A map
Xk
A match
A match
Xk
Xk
uniqueness of the dynamics, and by the restrictive conditions which define the map (which were carefully discussed by Birkhoff (1922)). This uniqueness and the continuity
may not be retained in a match, because the conditions, Q(x, z) >, 0, may not be sufficiently restrictive.
This point is made clear by considering Lorenz's original match. He considered the association of the values of z(t), when it is a maximum (i.e., {Q I z(t) = 0, z > r}), and its value the next time it is a maximum. If we let M represent the nth maximum value of z(t), then he considered the association of Mn+ with M. His motivation for considering such a strange association was expressed by him 1
as follows (Lorenz, 1963): ... we find that the trajectory apparently leaves one spiral only after exceeding
some critical distance from the center... . Moreover, the extent to which this distance is exceeded appears to determine the point at which the next spiral is entered, this in turn seems to determine the number of circuits to be executed before changing spirals again. It therefore seems that some single feature of a given circuit should predict the same feature of the following circuit. A suitable feature of this sort is the maximum value of z.... The new feature here is that he is suggesting that this complicated motion may have a predictive (i.e., nearly unique) association in only a `single feature'. The motion, after all, is in a three-dimensional phase space, so that any Poincare map (which is unique) must associate two features at one time with their values at another time, rather than a single feature. However, Lorenz's conjecture proved to be warranted, as is shown in his following justifiably famous feature relating M,,+ 1 to Mn: Fig. 7.25
clearly shows that there is essentially a smooth functional relationship connecting the values Mn, 1 and Mn. Moreover, Lorenz noted that an essential feature of the above curve is that it has a slope whose magnitude exceeds unity everywhere. The important consequence of this is that all periodic trajectories of this system must be unstable, as we saw in Chapter 4. Therefore the w-limit set of this system cannot be a limit cycle, at least for the value of r used by Lorenz. This, of course, can change if r is changed, and in fact limit cycles do occur at larger values of r, as will be discussed later. The fine structure of the Lorenz match has been studied by Richtmyer (1986). The above conclusions of course depend on the assumption that Lorenz's
'Lorenz-dynamic' fluid system Fig. 7.25
151
Mn+1
45
40
35
30
30
35
40
45
Mn
match is sufficiently smooth to be treated as a map. By way of comparison, see the Rossler `maps' in section 7.11. Lorenz therefore established that, when r = 28, the trajectories converge into a region of phase space which has zero volume (based on the usual three-dimensional measure) and yet, within the approximation inherent in his match, appears to contain no stable fixed points nor stable limit cycles. Such a w-limit set can be reasonably called a strange attractor, because it also possesses the property of being sensitive to initial conditions (what Lorenz referred to as `the instability of nonperiodic solutions ...
in the sense that solutions temporarily approximating it do not continue to do so.') This was a major discovery, and would be sufficient to warrant the interest in the Lorenz system, but there are many other interesting features to be discovered. Before discussing these features, we will consider several other examples of physical systems which are associated with the Lorenz equations
7.5 A 'Lorenz-dynamic' fluid system In the derivation of the Lorenz equations for the case of Rayleigh-Benard convection, a number of severe approximations were used, whose nature and validity are hard to assess. Before proceeding to analyze further the dynamics of the Lorenz equations,
we will consider a simpler system, in which it is possible to specify clearly the approximations, because its dynamics is sufficiently restricted by boundary constraints (Yorke and Yorke, 1979). The system (Fig. 7.26) consists of a fluid in a circular tube, which stands vertically in a gravitational field. The tube has mean radius R, and a small cross-sectional area,
152
Models based on third order differential systems
Fig. 7.26
Cooled
8
Heated
A. The only approximation which we require is that the fluid velocity can taken to be v(r, 0, t) = v(r, t)0. This means that, in the convective aspects of the dynamics, the fluid is treated as incompressible. The wall of the tube is maintained at a constant temperature, Tw(0), which is an arbitrary function of 0. The fluid density depends on its temperature, T(0, t), p = po[l + a(T0(t) - T(0, t))], where a( > 0) is the coefficient of thermal expansion and To(t) is its mean temperature, f 2' T(0, t) dO = 2i To(t). The heat transfer between the wall and the fluid is given by 0
K(Tw(0) - T(0, t)]
where K is a constant. Finally, the fluid flow is opposed by a frictional force, f, proportional to its mean flow rate q(t) = Jv(r.
so that
F=9
-ppoq(t)d
where y is a constant. The equations of motion of the fluid now reduce to po
at = -VP(0,t)+ po(1 +a(T0 - T))g + f
where we have again ignored the variation of p on the left side, and P(0, t) is the pressure, g = - g sin 0 0. If this is integrated by 1
Zn
r
dA,
dO 27r
0
J
153
'Lorenz-dynamic' fluid system
we obtain dq
_ Aga f sin 0 T(0, t) dO-,uq(t).
(7.5.1)
2n
dt
The second equation we need is for the heat transfer. We will ignore the thermal conduction in the fluid (see Exercise 7.9), in which case at +
q(t) a8 = K [Tw(6) - T(9, t)].
(7.5.2)
1
Now, if we expand Tw(6) and T(6, t) in Fourier series T(0)= Wo + E Vn sin (n6) + Wn cos (n0) R=I m
T(6, t) = To(t) + Z Sn(t) sin (nO) + C,,(t) cos (n9)
(7.5.3)
n=1
the equation of motion, (7.5.1), only involves S1(t), and we obtain a closed system of only three equations from (7.5.1) and (7.5.2), aq
dtt
ddtt
= 2 AgaS1(t) - µq(t)
AR q(t)C(t) +
K[V1 - S1(t)]
(7.5.4)
q(t)S1(t) + K[W1 - C1(t)]
AR
the remaining equations from (7.5.2) are decoupled from these, dSn = n
dt
dd n
AR
q(t)Cn(t) + K[V,, - Sn(t)]
n q(t)Sn(t) + K[W, - Qt)]
(7.5.5)
and
at
o = K[W0 - To(t)]
The equations (7.5.4) can be put into the form of the Lorenz equations, (7.3.6), (or nearly so) by setting
i=Kt,
a=µ/K,
r=yW1,
r'=yV1
(7.5.6)
Models based on third order differential systems
154
where y = ga/2K1R, and identifying x(t) = q(t)/AKR,
y(t) = yS, (t),
z(t) = y [ W1
- C, (t)].
(7.5.7)
This yields
dx
To(y-x)
d
=
dy
=rx-y-zx+r'
d2 dz
dr
(7.5.8)
=xy-z
which, aside from the additional constant r', is (7.3.6) with b = 1. The constant r' comes from an asymmetric heating of the wall, V, in (7.5.3). If r' is too large, relative to r, then the flow is stable in the preferred direction (depending on the sign of V,). In the simplest situation, when Tw(9) = WO + W, cos 0, (7.5.3) yields decaying solutions
dt (S" +
Cn) _ - 2K(Sn + Cn )
and To(t) = WO + exp (- Kt)(To(O) - WO). Therefore, in this case, as t -> oo,
x-q(t),
y
[T( t) - T()]
fluid
Temperature
velocity
difference midway up the side
,
Z
CTw(O)-T(O,t)] Temperature difference at the bottom
Exercise 7.9 Add the thermal conductivity term .(82T/892)(. = x/R2) to the right side of (7.5.2), and show that we can obtain (7.5.8) with modified relationships, (7.5.6), and (7.5.7). In particular, determine the influence of a. on r, and hence on the stability of the flow.
An experiment involving a fluid arrangement of a similar nature has been carried out by Creveling, Paz, Baladi, and Schoenhal (1975). The experimental arrangement was slightly different from the above theory, as illustrated in Fig. 7.27(a). The bottom half of the convection loop was maintained with a uniform heat flux, whereas the upper half had a constant temperature wall. Temperature reversals were observed between the two points A and B, as illustrated in Fig. 7.27(b). This temperature reversal is also related to a reversal in the fluid flow in the loop. The qualitative
Dynamo dynamics
155
Fig. 7.27 (a) Free convection loop employed for experimental study (R = 38 cm, r = 1.5 cm). (b) Fluctuations in the temperature difference between sections A and B exhibiting reversion to original flow direction after a flow reversal.
(a)
4
5
10
15
20
125
1 Time (min) -2
(b)
behavior is clearly the same as the above simpler situation, and they obtained more detailed agreements with the experimental results, using a suitably modified theory.
7.6 Dynamo dynamics One of the most spectacular features of geophysical dynamics is the numerous erratic reversals of the Earth's magnetic field which have occurred over at least the last 150
156
Models based on third order differential systems
million years. The first observation of this temporal reversal of the magnetic field came from observations of the magnetization in lava formations of different ages. Such observations were made in Japan (1929) and later, more extensively in France and Iceland (1951-58). The magnetization observed later in igneous rocks is frequently
due to a magnetization acquired during cooling from above the Curie temperature
of certain magnetic minerals, in the presence of the Earth's magnetic field ('thermoremanent' magnetization). Depositional remanent magnetization, on the other hand, is acquired by sediments when magnetized grains settle out in water in the Earth's magnetic field. Until the introduction of K-Ar dating replaced the use of fossil dating around 1960, it was impossible to extend the history of such reversals Fig. 7.28 Reversal time-scale for the past 170 Ma (after Cox, 1982).
Dynamo dynamics
157
back more than 10 million years or so. Now, with a variety of dating methods, there are estimates of these reversals over a span of 165 million years. This history of the reversal is illustrated in Fig. 7.28 (after Cox, 1982), where the black indicates the periods with the present polarity. The methods by which such conclusions are drawn from world-wide data, and its relationship with plate tectonics, is a fascinating area of research which is discussed in many books and reviews (e.g., see Dynamo references).
A particularly pretty example of sequences of reversals can be found in the solidification of the magma on the ocean floor. An example of this is the East Pacific
Rise, which runs north-south off the west coast of Central America, and south-westward into the southern Pacific Ocean. Along this line there is a 75,000 kilometre rift in the ocean floor where plates of the oceanic crust are continually being formed by the upwelling of molten rock, causing the plates to separate at a rate of up to 18 centimetres per year. In this process, ferromagnetic minerals in the magma become magnetized by the geomagnetic field, and preserve a record of this orientation as it cools and solidifies. This leaves a series of magnetic strips running parallel to the rift, and a history of these reversals is recorded as a function of the distance from the rift. This history is rather limited (some 10 million years) but quite spectacular, as illustrated in Fig. 7.29 (after MacDonald and Luyendyk, 1981). The Fig. 7.29
Baja
California
Gulf of California
Mexico
158
Models based on third order differential systems
lines represent the boundaries of magnetic reversals detected by a magnetometer aboard a manned deep-diving submersible. The dates of several reversals (in millions of years) are indicated. While there has been considerable progress made toward obtaining a self-consistent theory of geomagnetic reversals, a complete theory has not yet been obtained (see Dynamo references). Here we will consider some simple dynamo systems, which also can exhibit such magnetic reversals, but which do not shed light on the geophysical situation, because they employ a topological structure which does not occur in the interior of the Earth (namely, the dynamo systems are not simply connected). Fig. 7.30
As an example, consider the disk dynamo, illustrated in Fig. 7.30. Here a conducting
disk rotates with an angular frequency w about a conducting axle. A brush contact with the disk and the axle completes a circuit through a coil (represented by only a single loop). An initial magnetic field in the disk produces a current through the coil, which in turn produces a magnetic field through the disk. Under suitable conditions this field-current situation can be self-sustaining, and indeed interesting dynamics can occur, as we will see. If (L, R) are the (inductance, resistance) of the coil, then the voltage across the coil yields the equation
Ldl dt
+ R1= Mwl
(7.6.1)
where 2iM is the mutual inductance of the coil and disk. The current increases with
159
Dynamo dynamics
time only if Mw > R, otherwise it decays. If co changes with time, then Jw will equal
the net torque on the disk where J is the moment of inertia of the disk. If there is an applied torque T, and a back-torque M12 due to the current in the disk flowing in the magnetic field, then (if IS = 0 in the above figure)
Jw=T-MI2. If T
(7.6.2)
0, there is a stationary state I = [i/M]112,
co = R/M
which is a center (elliptic point), and perturbations from this fixed point oscillate with
a period 2n(JL/2MT)1/2. Note that in this model the current (and hence B) cannot change sign, because (7.6.1) does not have a solution passing through 1= 0. Fig. 7.31
3
10
15
20
25
Time (JL/M)112
If a shunt s is added to the system, as shown in Fig. 7.30, equation (7.6.1) is unchanged, but (7.6.2) becomes
Jco=T-MI(I+I.)-µw
(7.6.3)
where, now, a bearing friction µw has been included. Moreover, if the shunt has a resistance R,, and inductance L, then LSI, + R,18 = MwI.
(7.6.4)
In this case the current still cannot change sign (because of (7.6.1)), but the oscillations will no longer be simply periodic. Instead they now grow and produce sharper `bursts' of current, as illustrated in Fig. 7.31, a numerical solution by Bullard (µ = 0). Clearly, any noise added to I when I 0 could produce a reversal of 1, and thereby a reversal of the magnetic field. The reversal of I can be produced in a deterministic fashion (without introducing
160
Models based on third order differential systems
noise) only if equation (7.6.1) is modified. This can be done by introducing a second coupled dynamo, as was proposed by Rititake (see index), or simply by introducing an impedance and resistance in a brush, as shown in Fig. 7.32. If the brush impedance Fig. 7.32
Lb, Rb
is Lb and its resistance is Rb, then (7.6.1) is replaced by the equation LI + RI + Lb(1 + Is) + Rb(I + Is) = M(9,
(7.6.5)
LSIS + RsI, + Lb(1 + Is) + Rb(1 + Is) = MwI.
(7.6.6)
and (7.6.4) becomes
Robbins (1977) has studied the solutions of (7.6.3), (7.6.5), and (7.6.6), and has found that there are four dynamically different regions, namely: (i) L/R > LS/RS and Lb/Rb > LS/RS: irregular reversals; (ii) Lb/Rb > LS/RS > L/R: no reversals after transients; (iii) LS/RS > L/R and LS/RS > Lb/Rb: periodic with reversals; (iv) L/R > Ls/Rs > Lb/Rb: no reversals after transients. The case (i) is of greatest interest since it has dynamic features similar to the reversal of the earth's magnetic field. The simplest case under (i) is if L, = 0, in which case (7.6.5) can be replaced by
LI+RI-RsIs=0
(7.6.7)
where (7.6.6) has been used. This situation has been studied extensively by Robbins (1977), and some of her results are illustrated in Fig. 7.33. In the top sample, the
number of oscillations between reversals range from 1 to 66. On the other hand, other solutions exist where, after 18 reversals, the solution decays to a steady state. Such results are quite reminiscent of Lorenz dynamics - for good reason, as we will see.
Dynamo dynamics
161
Fig. 7.33
II
ti
t U
M
2
U
I
t
00
Pseudo-random oscillations of the current in the coil of the dynamo (from Robbins).
t U
ti t U
Oscillations settling to a steady state (from Robbins).
Introducing the variable 0(t) = (T/P) - w(t) (7.6.3) becomes
JS2 = MI(1 + IS) - µS2.
(7.6.8)
Models based on third order differential systems
162
Also (7.6.6) becomes (with L. = 0) Lb(1 + IS) + (Rb + RS)(I + IS) - RS1= M(
µ-52) 1.
(7.6.9)
The equations (7.6.7)-(7.6.9) can be put into the form of Lorenz's equation by introducing the dimensionless time
2,=(Rb+R)t, Lb
and identifying the parameters as follows
b_
µLb , J(Rb + RS)
a _ (R + RS)Lb (Rb + RS)L'
_ r
RS
Lb(R + Rs)
(MT
µ +
RS/
Finally, if
x(t) = al(t),
y(t) = 13(1(t) + 1s(t)),
z(t) = yS2(t)
(7.6.10)
where (a, /3, y) are suitable constants, then (7.6.7)-(7.6.9), become (respectively) the Lorenz system
a(y - x)
i=xy - bz
-xz-y+rx. It is important to notice that the current 1(t) through the coil is proportional to x(t), by (7.6.10). Thus a change in sign of x(t) corresponds to a reversal of the current and hence a reversal in the polarity of the magnetic field. Therefore the erratic reversals
in the sign of x(t), which occur in the Lorenz system where r > r, = Q(a + b + 3)/ (a - b - 1) appear to be similar to the geomagnetic reversals. However, not only is the dynamo model much too simple to shed light on the geophysical situation, but even its `chaotic' dynamics may not be sufficiently `chaotic'. Thus, one of the observed features of geomagnetic reversals is the apparent stochastic behavior of the durations
of reversals. That is, it appears that the duration of the polarities is an independent (stochastic) variable (in contrast with the 11 year cycles of the sun). A detailed investigation of the statistical aspects of Lorenz chaos has not yet been carried out.
7.7 The Lorenz homoclinic and heteroclinic bifurcations We will now consider what happens to the global properties of the flow for values of r much below the critical value, r, considered by Lorenz. The discovery of a homoclinic bifurcation in this region is due to Kaplan and Yorke (1979). It is a global
Lorenz homoclinic and heteroclinic bifurcations
163
bifurcation, which cannot be detected by considering the flows only near the fixed points - because their stability properties do not change (recall the example of a global bifurcation in R2, discussed in Chapter 5). This bifurcation occurs at a value of r which is approximately ro 13.9656. Below this value of r the motion which starts out near the origin along its unstable manifold (W' ), that begins in the quadrant x > 0, y > 0, moves away from the origin and spirals into the fixed point C +. Similarly, the motion which moves away from the origin along the unstable manifold (WU ), that begins in the negative quadrant, x < 0, y < 0, then spirals into the fixed point C -. This is illustrated in Fig. 7.34(a). However, when r > ro, the motion which starts out along these unstable manifolds now spirals into the opposite fixed point than it did when r < ro. This is illustrated in Fig. 7.34(b). What has happened when r passes through the value ro, is that the unstable manifolds of the origin, W1 , have each moved from the basin of attraction of one fixed point, C + or C -, to the basin of the other fixed point. These two basins of attraction are separated by the two-dimensional
stable manifold of the origin, W°. States on one side of W° are attracted to C while those on the other side spiral into C -. This is illustrated only locally (near the origin) in Fig. 7.34(a). Fig. 7.34(b) is even more schematic, because it appears as if
nothing has happened to W° near the origin, whereas, in fact, a great change has occurred.
The transfer of a one-dimensional unstable manifold, W,,, from one basin of attraction to another basin is an interesting process in this three-dimensional phase space. It is important to recognize that no trajectory can pass through the stable manifold of the origin, W°, because every state on that two-dimensional surface must tend to the origin (and, of course, the solutions are unique). Moreover this manifold, W°, separates the basins of attraction of C + and C C. Therefore the only way that the unstable manifolds, W,,, can pass into the other basin of attraction, when r is
Local W°
r < r0
r > r0
Models based on third order differential systems
164
varied, is for the entire manifold, W,,, to make this transition through W° at the value ro. That means that the manifolds, W1 , must become embedded in the surface W°,
in other words become part of the stable manifold. Consequently these unstable manifolds return to the origin when r = ro, and therefore are homoclinic orbits. The nature of one of these homoclinic orbits is illustrated in Fig. 7.35 (recall that the other homoclinic orbit is similar because of the symmetry in the Lorenz system). Fig. 7.35
Now the essential, and interesting, feature to recognize is that these homoclinic orbits must be part of the stable manifold surface. In other words, the stable manifold
must be attached to both W' and W- at every point along these curves. On the other hand, we know that near the origin the stable manifold must be tangent to the plane containing the two stable characteristic eigenvectors (Exercise 7.3). The question therefore presents itself:
How can the unstable manifold, W,,, which near the origin is a curve that is essentially orthogonal to the `stable plane' defined by the two stable characteristic eigenvectors, at the same time be contained in the stable manifold?
The only possible answer is that the stable manifold near the origin is a singular surface, and not simply a plane subtended by the two stable eigenvectors. When
Lorenz homoclinic and heteroclinic bifurcations
165
r = ro, the stable manifold near the origin contains the z axis, and must be tangent to the plane containing this axis and the other stable eigenvector, and must contain the unstable eigenvector. If you try to draw such a surface, retaining its continuity, you will find that it must look something like Fig. 7.36. Fig. 7.36
The figure shows only selected pieces of W°, of course. The essential feature to note is that the surface must abut itself (intersect itself 'orthogonally') along an infinite
line which, at the origin, is tangent to the stable eigenvector with the largest (but negative) eigenvalue. This follows from the usual linear analysis at the origin, together with continuity considerations.
To understand what happens when r > ro, it is necessary to have a better appreciation of the global structure of W°. The z axis is contained in W°, and near this axis it is essentially a plane for z < 0, and a `double helix' structure when z > 0, as illustrated in Fig. 7.37. The `double' in this helix structure comes from the fact that one half of the helix contains trajectories which approach the origin with (x > 0, y < 0) and the other half with (x < 0, y > 0), which we denote as WS W. WS looks like Fig. 7.38(a). When this is joined with its inverted ((x, y) - (- x, y)) image, WS , the total W° looks something like Fig. 7.38(b) (in a rather limited region of the origin).
The basin of attraction of C+ is contained in those regions through which the illustrated orbit moves. We denote this as B. The other basin, B-, is on the other side of W. Further out along the positive x-axis, the `hanging curtain' portion of W° rotates clockwise into pleat formations, as illustrated schematically in Fig. 7.38(c). The innermost portion is WS , and WS alternate as we move away from the `x'-axis (which is parallel to the x-axis). These channels parallel to the x axis, about which
166
Models based on third order differential systems
Fig. 7.37
Fig. 7.38
(a)
Lorenz homoclinic and heteroclinic bifurcations Fig. 7.38 cont.
(b)
(c)
167
168
Models based on third order differential systems
the trajectories rotate are therefore not topological cylinders - this is necessary in order that this surface be able to connect with the double helix structure above. The `fan' of surfaces in the figure, amply illustrates the fact that all these surfaces retain their identity, even though they are very close together. Some trajectories in different regions of these basins of attraction are shown in Fig. 7.39. The subscripts on BK indicate the number of times the trajectories pass through the half-plane (x + y = 0, ± x > 0). Note that the rotations in parts (a) and Fig. 7.39
z
(a)
y 60
Bi
A
30
0 I
-30
-60 I
I
I
I
0
50
100
150
(b)
Lorenz homoclinic and heteroclinic bifurcations
169
(b) are orthogonal, so that the stable manifold must likewise change its structure, as illustrated. As r -, ro, where the homoclinic orbit exists, the portions of the stable manifold which `hangs down' nearest the z axis (but does not contain the z axis), becomes `tucked up' under the (x, y)-plane, as illustrated in Fig. 7.40(a). z
Fig. 7.40
170
Models based on third order differential systems
At r = ro, the stable manifold abuts itself, as was described above, and the three hanging portions of W. coalesce into the one portion attached to the z axis. This is illustrated, for a small portion of W, in Fig. 7.40(b). As r is increased above ro, the stable manifold of the origin acquires a convoluted, self-whorled structure. The first few layers of this structure are illustrated in Fig. 7.41. The remainder of this structure Fig. 7.41
is obtained by `backing up' the illustrated portion beside itself (throughout space), without limit (Fig. 7.42). The basins of attraction, Bt, are then densely interwoven in some regions off the phase space. It can take a point in these basins an arbitrarily long time to settle into a region near the stable fixed points, C', depending on where it starts in the basins. On their way toward C t, solutions cross the plane z = (r - 1) an infinite number of times. When they cross this plane near the z axis, with i < 0, they are either on the C+ or the C- side of a line which passes through x = y = 0, produced by the intersection of WS and this plane (call the line Y_ - see Fig. 7.43). When r < ro, all solutions intersect z = (r - 1) near x = y = 0 only on the C+ or C- side of Y-. However, when r > ro the solutions can intersect this plane on either side of E, and moreover they can do it in any arbitrary order. That is, once the above convoluted structure exists, it is possible to find solutions which cross z = (r - 1) on the positive (C+) or negative (C-) side of F,, in any order that we wish to prescribe. If we designate these crossings as P and N respectively, then a solution's `itinerary' can be denoted by some
Lorenz homoclinic and heteroclinic bifurcation Fig. 7.42
Fig. 7.43
171
172
Models based on third order differential systems
sequence of Ps and Ns. Except for unstable periodic solutions, all itineraries end with
either an infinite number of Ps (if it is in B+) or Ns (if it is in B-). Ignoring this `monotonous' part, the interesting part of the itinerary can be written (for example) as
PPNPNNNPPNP = P2NPN3P2NP. When r > ro, there exists an infinite number of solutions which have any (arbitrarily long) itinerary that we wish to prescribe,... P°NIP`N°.... When a system possesses such arbitrarily long, but finite, random sequences in the itinerary of some variable, we will say that it exhibits transient chaos. Of course the measure of such solutions (in some bounded region of the phase space) decreases as the itinerary gets longer, but all of these measures increase as r increases toward rt. Each of these itineraries belongs to solutions adjacent to some portion of Ws, as illustrated very schematically in the `ribbon' representation of WS (Fig. 7.44). All of the splits which are shown do not exist (the surface is connected everywhere), but are introduced so that the convoluted structure of W. can be represented. The actual width of this infinitely convoluted structure is illustrated in Fig. 7.45 (for r = 17),
showing where it intersects the plane z = (r - 1) = 16. The itineraries, and their relationship to this spatial structure can be described by a simple one-dimensional map, which is illustrated in Fig. 7.46. Indeed, the development of both the fixed point
N3
Lorenz homoclinic and heteroclinic bifurcation Fig. 7.45
173
Y
bifurcation and the homoclinic bifurcation can be partially represented by such a one-dimensional map, as shown. These maps have been discussed in some detail by Sparrow. This reduction of much of the information about Lorenz dynamics to these one-dimensional maps, depends strongly on the relatively strong rate of contraction in one direction in the (x, y)-plane at z = (r - 1), vis-d-vis the rate of expansion in the other direction. This produces the `one-dimensionality' of the features in that plane. Figure 7.46 shows the location of some intersections of WS with both the plane z = (r - 1) and the line x = y > 0 (for r = 17). This one-dimensional result is faithfully reproduced by the above map. The pairs of sequences which are listed (e.g., PNPN, PNPZ) designate the itineraries of solution which can be found adjacent (left or right)
to the indicated intersection. Between any two indicated intersections there is an infinite number of intersections that are not indicated. This type of a boundary `between' basins has been referred to as a fractal boundary
Models based on third order differential systems
174 Fig. 7.46 f(s)
f(s)
r<1
/
I
S
0
C-
I
I
0
C+
S
f(s)
I
I
.,..lIlI
C-
0
S
C+
Fig. 7.47
(McDonald, Grebogi, Ott and Yorke, 1985), and has the following property (Fig. 7.47): Consider a volume V enclosing a portion of this boundary region, and lets represent
some uncertainty range of initial condition (E3 << V). Some points P in V have the
property that, within the E neighborhood, solutions can tend to either C+ or C-.
The ratio of the volume of such `uncertain' points V(c) to V is called the
Lorenz homoclinic and heteroclinic bifurcation
175
Fig. 7.48 (Nk, Nk-1P)(k->oo)
(Pk-1N, Pk) (k - °°)
I (N4, N3 p)
(NPN2, NPNP)
(PNPN, PNP2)
(N2 PN, N2 P2) (NP' N, NP3 )
(P3 N, P4)
(PN3, PN2P)
(P2 N2, P2 NP)
(PN2, PNP)
(P2N, P3)
I
.(N3, N2')
(NPN, NP2)
(PN, p2)
(N2, NP)
(N, P) 4
0.10
4-- x=y
0.20
0.15
uncertainty fraction f (E) = V(c)/V.
In the case of regular boundaries f (e)
s'
(as c -, 0)
where a > 1 is called the uncertainty exponent. It is fairly clear that this uncertainty in the location of the basins is due to the dense character of the transverse cuts which the stable manifold makes through any a region. It is expected therefore that the capacity of this set of points (Ws) is directly related to a. The following theorem can be established without much difficulty (McDonald, Grebogi, Ott and Yorke, 1985). Theorem The uncertainty fraction, f (E) of a finite region of a D-dimensional phase space satisfies lim In f (e)/In e = a E-0
if and only if the basin boundary has a capacity dimension d,, = D - a.
Consider now a particular sequence of these intersections (N, P), (PN,
p2), (P2N, P3),.. . ,
(pk-1N, Pk).
The solutions on the right side of these intersections go to C- (since the `monotonous'
176
Models based on third order differential systems
part, which is omitted, is N°°), while those on the left go to C. If we consider the limit k -> oo in the above sequence, these intersections approach a limit point (indicated
by the arrowed top in Fig. 7.48). This limit point must correspond to a limit surface in the three-dimensional phase space, which is not part of Ws. To the right of this limit point all solutions tend to C+ (monotonously), but to the right of the points (Pk-'N, Pk), which approach the limit point from the left, the solutions tend to C-. F+, and it clearly is Hence on this limit suface there must be a periodic solution, unstable in one direction. For example all solutions along x = y, near this intersection, move away from the surface (note that the limit point in the figure need not be on F+, just on this limit surface). Due to the contracting direction of the periodic orbit,
the Lorenz flow (toward the curves W on WS at the homoclinic point, and their `ghosts' on WS when r > ro), the period orbit F+ has a two-dimensional stable manifold, Ws`, and a two-dimensional unstable manifold on one side of the stable manifold. On the other side of its stable manifold, however, there is a dense set of surfaces (parts of W°) `parallel' to its stable manifold. Therefore the limit cycle r+ is not a simple saddle-cycle, but has a more sophisticated structure. This is illustrated very schematically in Fig. 7.49. Note that inside the cylindrical region, formed by Ws`, is the basin B+, where the unstable manifold of the origin W- tends toward C+. (Clearly there is an identical structure around C-, with a corresponding limit cycle, F -.) Fig. 7.49
The limit cycle T+ is not the only one formed at the homoclinic bifurcation. Consider any `seed' sequence (e.g., NP), and the set which consists of longer and longer repetitions of this sequence, such as
NPNP, NPNPNP,... , (NP)".
Lorenz-Hopf bifurcation
177
Each of these sequences represent solutions which lie to the right of the intersection
of W° and the line x = y (z = (r - 1)) - that is, in B-. As k + oo, there must again be a limit surface which is a stable manifold of a limit cycle with the periodic behavior NP. Since this holds for any `seed' sequence (hence any periodic behavior of the limit cycle), there is an infinite number of unstable limit cycles formed at this homoclinic
bifurcation. In addition there are an uncountable number of aperiodic solutions (corresponding to the infinitely long aperiodic sequences). This profusion of various
types of solutions at the homoclinic bifurcation has been termed a `homoclinic explosion' (he) by Sparrow. It illustrates the richness of three-dimensional dynamics (e.g., contrasted with homoclinic bifurcations in two dimensions). The limit cycle T+ which simply circles C+ (in Fig. 7.49) is special because its stable manifold `encloses' the unstable manifold of the origin, WU . As r - rhe 24.06,
this unstable manifold tends closer to this limit cycle, and at r = rhe, WU tends asymptotically to T+ (that is, WU is contained in the stable manifold of T+). A solution y(t), such that Periodic orbit A
Periodic y
+x orbit B
is called a heteroclinic solution (in the present case A is a fixed point - a trivial periodic
orbit - and B is the unstable limit cycle). The fact that WU tends asymptotically to an unstable limit cycle is somewhat akin to landing an airplane on an icy mountain ridge. That would be a good analogy if the limit cycle were a saddle-cycle, but that is not the case here, as already noted. As r is increased through rhe 24.06, WU passes through the stable manifold of
F+ and enters the limit-set-of-surfaces region of W°, illustrated in Fig. 7.49. When r is increased toward the value rt 24.74 a complicated infinite sequence of homoclinic bifurcations occurs. These bifurcations involve the unstable manifolds, W , which return to the origin only after making some sequence of `loops' about the cylindrical basins of attraction of the stable fixed points Ct. These complicated
bifurcations are difficult to observe, and the topology of W° now defies the comprehension of this author. However, one essential point is that, in the parameter range r, > r > rhe, the phase space contains both an attracting chaotic region, as well
as two basins of attraction to the stable fixed points C. The chaotic region is no longer transient chaos, as it is when r < rhe, for the chaos now persists for all time. This multiple attractor situation changes to pure chaos as r-+r,, as we now will briefly consider.
7.8 The Lorenz-Hopf bifurcation As r - r, = 24.7 the limit cycle F+ contracts onto the fixed point C+ and the basin of attraction to C+ shrinks to zero. For r > r, C+ becomes a spiral-out saddle, with
178
Models based on third order differential systems
a two-dimensional unstable manifold, and the limit cycle F+ vanishes. Because the real part of the complex eigenvalues, A = A, ± ico, passes through zero at a nonzero rate (as a function of r) (c32,(r)/8r),t > 0
this is a Hopf bifurcation, discussed in Chapter 5.
Exercise 7.10 Determine under what conditions (involving a, b, r) there is a Hopf bifurcation at r = rt = a(a + b + 3)/(a - b - 1), by computing (8i using (7.3.10). Since, when r < r,, the limit cycle, F'+, has an unstable manifold, with C + as its co-limit set, this Hopf bifurcation is called subcritical. The standard representation of this bifurcation in the control-phase plane is illustrated in Fig. 7.50 (the phase plane Fig. 7.50
is the plane which contains the limit cycle). In the present case this `plane' is the unstable manifold of F+, at least when r < rt. A representation of all of the bifurcations which occur in the range 0 < r < 30 (say) is illustrated in Fig. 7.51. The last panel is obviously a futile effort.
7.9 Lorenz dynamics for various parameter values In this section we will consider, rather briefly, some of the forms of dynamics which can occur for other values of the control parameters (a, r, b). This is a very rich area
of dynamics, about which a great number of details are now known. The present discussion is limited to a very small portion of this wealth of information, in an attempt to convey some of the flavor of this dynamic variety. A detailed presentation of many of the known dynamical features can be found in Sparrow (1982).
The control parameter of primary historical interest is r, being related to the Rayleigh number. It might be expected that once r is increased above r, 25, where Lorenz chaos sets in, the system would remain chaotic for all larger values of r. That this is not the case can be discovered either by numerical methods or, perhaps more refreshingly, by analytic methods. We outline here an analysis due to Robbins (1979),
Fig. 7.51
/ /
I
/
/
One rn fixed
point
/
/
Models based on third order differential systems
180
which involves changing large-r equations into small-e equations, and then developing
a perturbation series in powers of e. Sparrow (1982) has presented details of an alternative perturbation theory, based on an averaging method. Unfortunately these perturbation results are too involved to discuss in any detail, but we can at least discuss the lowest order results. To this end we introduce the change in variables r = (ace)-',
x = x'/e,
y = y'/ace,
z = z'/ae2,
t = et'
so that the phase space is contracted and time is expanded. When this is substituted into the Lorenz equations
dx/dt = a(y - x);
dy/dt = rx - y - xz;
dz/dt = - bz + xy
and we drop the primes on the variables, we obtain
dx/dt = y - asx
dy/dt=x - xz - ey
(7.9.1)
dz/dt = xy - ebz.
The first interesting feature of these equations is that, in the limit e = 0 this system is 'conservative'- that is, the volume of a region of phase space is conserved as it moves according to (7.9.1), a (x) +
ax
a (Y) + a (z) = 0 OY
(e = 0).
OZ
Moreover the e = 0 equations have two integrals of the motion
K1=2x2-z;
K2=2y2-z+Zz2.
(7.9.2)
We can use these constants in several ways. Analytically they can be used to obtain a first order equation in a single variable, such as x. Thus
(x)2=y2=2K2+2z-Z2=2K2+(x2-2K1)-(2x2-K1)2 so that we obtain the differential equation
(dx/dt)2 = (2K2 - K;) + x2(1 +K,)-
1x4
which immediately leads to elliptic function expressions for x(t) (Robbins, 1979). The constants of motion can also be used to illustrate the limit cycles in the phase space, since they must lie on the intersection of the surfaces K1 and K2. There is one closed curve if K1 + 1 > (2K2 + 1)1/2 (Fig. 7.52), or two closed curves if (2K2 + 1)1/2 > K1 + 1, or a pair of homoclinic curves if (2K2 + 1)1/2 = K1 + 1.
While these solutions are interesting, conservative systems are not structurally stable, so it is important to determine which of these solutions exist and whether
Lorenz dynamics for various parameters
181
Fig. 7.52 Note scaling w.r.t. original (xo, Yo' zo) variables. z(= zo/r)
y(=y0/r)
if-Ki>1-(2K2+1)1/2 (otherwise two loops) x = [(ra)1 /2x0]
they are stable when E 0. Robbins showed by her perturbation analysis that the single symmetric limit cycle exists and is stable for certain parameter values. Sparrow (1982) gives many details of the existence and stability of these orbits for all parameter values, using averaged equations for the quantities K1 and K2, (7.9.2). This method begins with the equations for the time dependence of Kt and K2
dK1/dt=E(-6x2+bz) dK2/t = E( - y2 + bz - bz2).
(7.9.3)
The solutions (x(t), y(t), z(y)), when E = 0, are then substituted into the right side, and averaged over a period of the motion. This leaves two equations involving (K 1, K2 ),
and he discusses the existence and stability of the fixed points of these averaged equations. In particular, for the canonical parameters a = 10 and b = 8/3, there is only the single stable symmetric limit cycle. The (x, y) projection (left) and (x, z) projection (right) of the orbit (r = 350) are illustrated in Fig. 7.53 (the x-axis is horizontal). A topological rendition, including the stable manifold of the origin, is shown on the cover. As r is decreased, it is numerically found that the first bifurcation (near r = 313) produces an interesting new arrangement of limit cycles. The limit cycle, which is invariant under (x, y, z) - ( - x, - y, z), becomes unstable and two stable limit cycles are formed which do not possess this invariance. They have essentially the same period
as the unstable limit cycle, so that this is not a period-two bifurcation of the limit cycle. One limit cycle is illustrated in Fig. 7.54(a), for r = 250. The second cycle is obtained from this by using the invariance (x, y, z) --. (- x, - y, z) of (7.9.1), which is now a distinct orbit. This dynamic symmetry breaking thus produces a pair of attractors (stable limit cycles).
182
Models based on third order differential systems
Fig.7.54 cont.
z
Fig. 7.54
(a)
(b)
Lorenz dynamics for various parameters
183
Fig. 7.54 cont.
(c)
(d)
A particularly interesting feature of this dynamic symmetry-breaking bifurcation is that the resulting limit cycles are linked (like two links of a chain - Fig. 7.54(b)). The way in which such linked cycles are formed can be understood from the schematic figures. We begin with a stable limit cycle, which has a set of solutions that lie on a band-surface such that, as they move toward the limit cycle, the band twists around by 2n in one period of the limit cycle (Fig. 7.54(c)). When this limit cycle becomes unstable, the solutions on this band move away from it a small distance (due to nonlinear effects, as in the Hopf bifurcation), forming the two stable cycles on the edge of the band (Fig. 7.54(d)). It is clear that these cycles are now linked limit cycles. The unstable limit cycle is a saddle cycle, which is sometimes said to be `nonstable' (Sparrow, 1982), since it is also unstable as t - - oo. We also note that the above linkings of oriented curves possess a handedness (chirality). That is, the linked curves can not be deformed into their mirror-image linked cycles: the two sets of linked cycles are not TOE. Placing a thumb in the direction of the flow on the unstable cycle, we see that the stable cycles rotate about the unstable cycle in the sense of the right-hand fingers, so they form a right-handed link.
The actual linkage is sometimes clearer if we use other projections than the (x, z) plane. Thus, if we use the projection onto the (x cos 0 + y sin 0, z) plane the two stable limit cycles (one broken) appear as shown in Fig. 7.55 (for r = 290, 0 = 0.3 rad). The
portion of the curves with the downward arrows lie closest to the reader (above the crossing curves), as illustrated. A little survey shows that the broken curve has `come
184
Models based on third order differential systems
Fig. 7.55
through' the full-line-loop on the right, forming the link. We see that this Lorenz-linkage is right-handed, as in the last example. As r is decreased further, another quite different type of bifurcation takes place. This begins with one of the asymmetric (AS) limit cycles, and as r is decreased it bifurcates into a stable and unstable limit cycle. The stable limit cycle has twice the original period. In other words, this is a subharmonic, or period-doubling bifurcation. Fig. 7.56 Fig. 7.56
Fig. 7.57 AS
Lorenz dynamics for various parameters
185
illustrates such a stable orbit when r = 220. The nature of this bifurcation is clarified by again considering Poincare's first return map, illustrated schematically in Fig. 7.57.
When r is slightly above the bifurcation value, the first return map takes a point halfway around the fixed point (point 1-> 2). In other words, the characteristic multiplier is negative. As discussed in Section 7.2, this may mean that the solutions near the stable limit cycle flow toward that orbit in a spiral fashion. A particular family of these solutions can be viewed as moving inward along a surface illustrated in Fig. 7.57 (other possibilities were discussed in Section 7.2). When this limit cycle goes unstable in some transverse direction, a Mobius strip is `born', with a single edge that is the new stable cycle, which now has twice the period (since it must go around twice before closing). This stable limit cycle is illustrated in the bottom figure.
The associated unstable orbit is not shown, to simplify the figure. The unstable manifold of this orbit is the Mobius strip. Moreover this unstable orbit has a two-dimensional stable manifold which is infinite in extent, and attaches transversely to the Mobius strip (a thin strip is illustrated). These subharmonic bifurcations are also discussed in the study of Rossler's models. This example of a subharmonic bifurcation, where the new stable limit cycle is the edge of a Mobius strip, is not the only way that subharmonic bifurcations can occur. If the characteristic multiplier of solutions on the stable manifold of a limit cycle is
negative, then we saw in Section 7.2 that the solutions can rotate about the limit cycle any odd number of half-revolutions per cycle. In the above case they rotated one-half revolution. However, if the solutions rotate three or five half-revolutions, then locally there are families of solutions which can be viewed as lying on the strips which are illustrated in Fig. 7.58. Fig. 7.58
(a)
(b)
In these cases, if these stable manifolds bifurcate into unstable manifolds, the new stable limit cycle which is generated will form the (single) edge of such strips, and
186
Models based on third order differential systems
since it takes essentially twice the distance (and hence time) to complete a cycle, these bifurcations are both subharmonic. These limit cycles have the structure of so-called torus knots (Rolfsen, 1976), which in cylindrical coordinates is the set TP.e = jr = (1 + 1 cos (2ngt)), 0 = 2npt, z = i sin (2ngt); tE [0, 1] }.
In other words, it is a knot which wraps around a solid torus p times in the longitudinal
direction and q times in the meridional direction. The two cases illustrated in Fig. 7.59 are T2.3 (the trefoil knot) and T2.5 (the Solomon's seal knot). The first example above, involving the Mobius strip, produced the torus knot T2.1, which is a trivial knot, since it can be unfolded to a simple loop. Uezu and Aizawa (1982) established that the Lorenz equations produce the stable Solomon's seal knot limit cycle, T2.5, when b = 4, a = 16, and r = 340, which was studied further by Uezu (1983). A survey of some of the known occurrences of torus knots in various systems is given by Crawford and Omohundro (1984), who also give a number of nice illustrations of the global structure of the stable and unstable manifolds generated in such subharmonic bifurcations. Fig. 7.59 illustrates, through Fig. 7.59
T2,3
T2,5
Lorenz dynamics for various parameters
187
a series of continuous deformations, that a periodic solution with an itinerary N(NP)2 (Section 7.8) is topologically equivalent to a trefoil knot.
In the canonical Lorenz system, these period-two bifurcations continue as r is decreased, until r reaches about 214, at which point a chaotic region is found. The system has another large stable region, beginning near r = 166 and extending down to near r = 145. This `window' region begins with a symmetric mode, which is illustrated in Fig. 7.60, and this is followed by a series of period-two bifurcations. Fig 7.60
Yet another window has been found between r = 99.5 and r = 100.8, which contains only asymmetric orbits and period-two bifurcations. Note that, where two stable
orbits are `born' at a bifurcation, there is no period doubling. Period doubling is characterized by one stable orbit becoming unstable, accompanied by a stable neighboring orbit. This bifurcation scenerio is illustrated in Fig. 7.61 (adapted from Fig. 7.61
S: Symmetric AS: Asymmetric
Stable LC
- - - - Not stable LC Stable FP
*Not stable FP
0 a=10 b = 8/3
313
Sparrow (1982)), schematically representing the limit cycles (LC) and fixed points (FP) which occur. The hatched regions contain other narrow windows, not illustrated. Another interesting dynamical feature, discovered by May (1976) and Manneville and Pomeau (1979) involves an intermittent form of chaos which is reminiscent of the intermittency found in one-dimensional maps. As r is decreased toward the value
188
Models based on third order differential systems
r,, = 166.07, Manneville and Pomeau found that the chaotic motion is interrupted by
periods of `periodic' motion whose mean duration appears to increase at a rate proportional to (r - rj-1/2. Indeed they found that, from a Poincare map on the surface of section at x = 0, they could obtain a Lorenz `map' for the y variable, yn+ 1 = f (y.), which is fairly well defined (see Fig. 7.62). This figure shows the usual Fig. 7.62
42
40
yn 40
42
one-dimensional tangent bifurcation structure, which is consistent with the variation
(r - r,,)- 112. There is a similar intermittency effect near r = 100.8, which is also symmetric in character (x(t) and - x(t)). This may seem surprising, given the asymmetry of the periodic orbits below r = 100.8, but the intermittency now involves the `flip-flop' between two such asymmetric motions, yielding symmetry. A very extensive search of the bifurcation features of the Lorenz system has been made by Alfsen and Froyland (1984) in the (b, r) plane with a fixed at a = 10. Fig. 7.63 illustrates only a few of their results, together with other known features. It shows (a) the variation of the first homoclinic bifurcation of the origin's unstable manifold, W°; (b) the variation of the heteroclinic orbit, W°, to the unstable limit cycle nearest
Lorenz dynamics for various parameters Fig. 7.63
189
10
b
1.0
idows
\ Intermittencies bistabilities
Linked knots 0.1
r
C}, (c) this separates the region of convoluted W° from the `preturbulence' region, which is bordered on the other side by the Hopf bifurcation at Ct, (d) this gives rise to the Lorenz chaos, and then regions of periodic windows (not shown) intermittencies etc.; (e) finally, for sufficiently large r, the system settles into the stable symmetric limit cycle.
Another aspect of Lorenz dynamics which has been found for very different range of a (e.g., a = 300), is bistability - the coexistence of two stable limit cycles. Associated
with this situation is the phenomena of hysteresis, when r is increased and then decreased (Fowler and McGuinness, 1982). Specifically, if b = 1, and r = 220 there is a stable symmetric limit cycle, Tt, and the stable fixed points, C±. However, if r = 240, there are two stable limit cycles, T1, T2, and the stable fixed points. If r = 250, only the asymmetric stable cycle T2 remains, together with the stable fixed points. This `bistability' may also involve, for some 220 < r < 250, a stable limit cycle and a chaotic
region, as occurs in the parameter region (a = 10, r = 15, b = 0.1) (Froyland and Alfsen, 1984), to be discussed in the following section on Lyapunov exponents. This very brief look at some of the dynamics in the 'chaotic-plus-windows-plus-etc.'
region does not do justice to the variety of bifurcation phenomena which exist in this system. The phenomena of bistability, hysteresis, and the coexistence of stable periodic orbits and chaotic regions, will be examined further in the next section, after
190
Models based on third order differential systems
we have discussed Lyapunov exponents. Other discussions and references concerning such effects and multiple basins can be found in Schmutz and Rueff (1984). As they
noted, and perhaps should be emphasized, the concept of basins of attraction may not be appropriate when the basins are intertwined in an extremely complicated fashion. We have already seen in Section 7.6, this convolution of two basins, even in the nonchaotic region, 14 < r < 24, produces fractal basin boundaries. What is needed in the future is to sift out from this rich variety of dynamics those features which are important for physical processes. That program has hardly begun.
7.10 The Lyapunov exponents Having now discovered a wealth of `continuous chaos', as contrasted with the discrete
chaos already found in maps (and, of course, the two are connected by Poincare maps, and Lorenz `maps'), we now turn to one of the methods of characterizing such
chaos, as well as nonchaotic motion. This goes by various names: characteristic multipliers and exponents, or Lyapunov multipliers and exponents. We will arbitrarily use the latter terminology. The Lyapunov exponents discussed here are simply the generalization (to continuous motion and higher dimensions) of the ones introduced for maps.
Chaotic motion can be `measured' (characterized) in a variety of fashions, but it is now widely agreed that before the dynamics qualifies to be called `chaotic' (in any sense), it must at least exhibit the property of `sensitivity to initial conditions'. More specifically, if xl (t; x°) and x2(t; x2) are two solutions of dx/dt = F(x, t)
(xER"),
where xk are the initial conditions, then the dynamics is sensitive to the initial conditions if these solutions tend to separate on the average at an exponential rate (in time), when their initial conditions are infinitesimally close to each other (Fig. 7.64). It was this type of sensitivity which was first discovered by Cartwright and Fig. 7.64
x2 (t)
d (t) - do exp At
Littlewood (1945) and Levinson (1949), in the case of the periodically forced van der Pol oscillator. We now, however, restrict our attention to autonomous systems
dx/dt = F(x)
(xeR").
(7.10.1)
191
Lyapunov exponents
Recall that, if F(x) satisfies the Lipschitz condition
IF(x)-F(y)I
Ix(t)-y(t)I < Ix(0)-y(0)Iexp(Kt) (e.g., see Ince, 1927, Section 3.22). This is the reason why the exponential separation
is usually the `worst possible' (rather than, for example, exp(Kt2)). Obviously, if solutions remain in some bounded region of phase space, they cannot separate exponentially for all time, so we generally must view this temporal separation after we take the limit of an infinitesimal separation of their initial conditions. This limiting procedure is most readily carried out by considering some solution x(t) of (7.10.1),
and then a second solution which is infinitesimally close, x(t) + 6x(t). Substituting the latter solution in (7.10.1), and making use of the fact that 8x(t) is an infinitesimal to expand the right side only to first order (which is equivalent to carrying out the limiting procedure), we obtain dx + d aF 6x = F(x) + 8x dt dt
8x X(t)
or
d Sx, =
dt
"
i=1
axi OF
Sxi
(1= 1, ... , n)
X(f)
where we have used both the fact that dx/dt = F, and have written out the resulting n equations in detail, so that there is no misunderstanding about the notation. Having done that, we now return to the abbreviated notation dz/dt = (aF/8x)X()z(t)
(zER").
(7.10.2)
This equation is known as the variational equation of (7.10.1), and in the Hamiltonian context, was frequently used by Poincare. This will be discussed further in the next chapter.
While the variational equation is linear, the coefficients are functions of the particular solution, x(t; x0) - depending on the initial condition x0 - and so we cannot usually obtain an analytic solution, z(t). In any case, the dynamics is considered to be chaotic only if, for most initial conditions z(0) (all but a set of measure zero), the magnitude of z grows exponentially for large t I z(t) I -> A exp (At).
This can be expressed in the form 2(x) = lim 1 In I z(t) I .
too t
(7.10.3)
192
Models based on third order differential systems
Here the magnitude of the vector can be taken to be the usual zI =(Yzz), " I
A necessary condition for the dynamics in the region of the solution, x(t), to be `chaotic' is that J.(x) > 0. This condition, however, is not sufficient (see the following exercise). The quantity 2, defined by (7.10.3), is called a Lyapunov characteristic
exponent (there are others, as we will see). It is a generalization of the usual characteristic exponents which are obtained to assess the stability of fixed points. In that case the special solution, x(t), is simply a constant, x0, and the solutions of the variational equations are easily obtained. It should be noted again that, in general, the Lyapunov exponent depends on the particular solution of the nonlinear equation which is used in the variational equation. Put another way, the `quality' (or existence) of the chaos may depend on the region of the phase space through which x(t) passes. If the dynamics is ergodic, then the exponent is independent of x(t), since it approaches arbitrarily close to all points in the (relevant) phase space. Alas, ergodic systems are not always as frequently found as theoreticians used to believe, and the dynamics in
phase space often has much more structure than previously expected. One of the useful measures of this `structure' is the dependence of the Lyapunov exponents on the solutions, x(t). More details about this can be found in the next chapter. Exercise 7.11 Obtain the solution of dx/dt = ax + b satisfying x(0) = c, and determine the Lyapunov exponent, ), of this solution, both when a = 0 and a 0. Does 1. depend on c? This solution is clearly not chaotic, even if ), > 0. What additional property of the dynamics generally insures chaos?
Because we can rarely obtain the solution of the variational equation, the Lyapunov exponent must usually be obtained numerically. One of the numerical problems which
is encountered is that any exponential solution of the variational equation soon becomes too large to treat computationally. This difficulty can be overcome (Benettin
et al., 1976, 1980) by making use of the linearity of the variational equation. Let x(t) = T`xo be the solution of (7.10.1), with the initial condition x0, and represent the solution of the variational equation by z(t) = dTxozo. The symbol dTXO is just an exotic way of representing all of the facts about the map from zo to z(t), in particular Fig. 7.65
x(t) = Tzo
Lyapunov exponents
193
showing that it is related to the nonlinear solution through the point x0. It is called the tangent map of Ti. This is illustrated in Fig. 7.65. This mapping clearly has the composition property dTXOSzo = dTx=(dTzpzo),
(7.10.4)
where s is any arbitrary time. That is, the solution after time t + s is the same as propagating the initial condition dTxozo forward for a time t, using the new initial condition xs = x(s) for the nonlinear solution x(t). Now assume that, beginning with (x(0)) = 1, the magnitude of the vector z(t) gets larger than we want to use in a computer, when t = s. Let us designate its magnitude by a1 = Iz(s)I.
Instead of continuing to use this large solution, we consider a new solution of (7.10.2),
which is vectorially the same as z(s), but of unit magnitude. Thus we take a new initial condition z° = Z(S)/al
and, the nonlinear solution now has the value xs = x(s; xo). After a time s this new solution will have some magnitude a2 = I z1(s) I. Because of the composition (7.10.4) and the linearity of (7.10.2), it is clear that the magnitude of z(2s) is the product of a1 and a2 Iz(2s)I = a1 a2.
(Fig. 7.66.) We can proceed in this fashion, taking new initial conditions every s time Fig. 7.66
intervals Zk = Zk - 1(S)/ak;
«k = I Zk - 1(s)
and computing the new solutions, zk(t). Then the original variational solution after n steps has the magnitude Iz(ns)I=a1a2...a..
194
Models based on third order differential systems
Substituting this into the definition for the Lyapunov exponent (7.10.3), yields n
21(xo)= llm
E Inak.
(7.10.5)
n-.onSk=1
We have now appended a subscript `1' to this exponent, because it is only one of n exponents (in Rn). It is, however, one of the most important exponents, because none of the others are larger in value. Hence it is necessary for 21(x0) > 0, if the system is sensitive to initial conditions in the region x0.
Note also that obtaining the numerical solution of z(t) usually requires the simultaneous numerical solution of the nonlinear equation for x(t). Thus the iterates x(tk), must be substituted into the coefficients of the variational equations, (aF/ax)x(t,,), before it can be iterated to give the values z(tk+ 1 ). It might also be noted that s is
frequently taken to be between one and five times the step used in the numerical computations (see below). The other Lyapunov characteristic exponents give a more complete indication of
the nature of the flow in the neighborhood of a solution x(t). This is illustrated in Fig. 7.67, showing an initially spherical region around x0 becoming distorted into
some ellipsoidal shape. The Lyapunov exponents 1k (k = 1, 2, 3) measure the exponential growth of the principal axes of this ellipsoid. The exponent Al measures the largest of these exponential growths, because most initial vectors, z0, will tend to orient along this direction as t increases. Therefore we cannot determine the exponent associated with another axis of the ellipsoid simply by considering another initial vector, since it will almost certainly become `captured' by the direction of most rapid growth. To measure the next most rapid growth rate, we can use the fact that most initial vectors will move into the plane subtended by the two principal axes with the largest growth rates. If we can measure how the area in this plane changes with time then, since
Area ' a 1 exp (A 1 t)a2 exp (A2 t),
we have 1
lim In (Area) = A 1 + 22. t-. t
195
Lyapunov exponents
If we know Al by our previous calculation, then we can obtain 22 from this result. The difficulty which now occurs is that it takes two vectors to obtain the behavior of the area, but these two vectors tend to become parallel (along the `most' principal axis). Thus there is a second computational difficulty, caused by the angle between two vectors becoming too small to compute. A nice way to overcome this difficulty
(Benettin Galgani and Strelcyn, 1980) is to orthogonalize the vectors, using a Gram-Schmidt method, at the same time as we renormalize the vectors to overcome the magnitude problem. The procedure is not difficult to visualize, although it is a little laborious to write down in detail. Consider two vector solutions zG)(t) and z(2)(t) which tend to become nearly parallel at t = s, as illustrated in Fig. 7.68. Let the magnitude of z" (s) again Fig. 7.68
Zj"(s)
'21
Z
(S)
-
-
1 /
r 77
ZO)(t)
be designated by a( l) I
= Iz")(s)I,
so that zi')(s) = z(')(s)/a(l 1)
is a unit vector. The area associated with the two vectors z") and z(2) at t = s equals the base times the height of the parallelogram, Area = Iz'1)(s)I IZ(l)(s) _ (Z(l),Zil))Zil)I
where (z(2)(s), z(')) is the scalar product of the two vectors. If we now define a second new vector which is orthogonal to zi' ), l) zi'))zi1))/ai \ 1 - (z(2), 1 1 1
Z(i2) = (Z(2)(s)
l
and which is a unit vector if ail) =
I Z(2 )(s) - (z(2), zil))zi1) I,
then the above area can be expressed in the form Area = ai1)a(2)
Using 4 and zit) as new initial conditions in the variational equations, as before, we obtain new magnitudes at time s later a2') = zl')(s)I,
a22)
= jZ1 (s)
-
(Zi2),
Z21))Z21)I
Models based on third order differential systems
196
where zz11= zl' 1/x211 and the area associated with the original solutions, z111(t) and z(2 )(t), now equals Area = (a j1)a(2))(a(1)x2)).
Carrying out the limiting process, we rind for the sum of the two Lyapunov exponents, 1
n
x fS
n
n
lnak'1+ E k= 1
inak211.
k= 1
l
The first sum on the right is simply 11, so we obtain 1
n
I In n-.D ilSk=1
12(x0) = lim
ak2).
The third Lyapunov exponent can be similarly obtained by determining the volume
subtended by three vectors, and orthonormalizing all three vectors at each time tk = ks. Nothing new is involved except extra labor! For more details and a discussion
of more fundamental issues, see Benettin, Galgani, Giorgilli, and Strelcyn (1980). Also see Shimada and Nagashima (1979) and Wolf, Swift, Swinney and Vastano (1985) for issues related to experimental applications. An alternative method of computing the Lyapunov exponents has been described by Froyland (1983), and applied to the Lorenz equations. As an example of the behavior of three Lyapunov exponents when a control parameter is varied, Fig. 7.69 Fig. 7.69
a;
200
300
400
500
r
shows the results obtained by Froyland and Alfsen (1984) for the Lorenz system with
a = 10, b = 3. They varied r in steps Ar = 0.1, connecting the resulting values by straight lines for purposes of visualization. There is, therefore, additional structure
197
Lyapunov exponents
which is not represented, but the major as well as many minor `windows' can clearly be seen where no exponents are positive. Many of the stable bifurcations can also be seen, where two of the exponents become zero. To demonstrate how the Lyapunov exponents can depend on the orbit, we illustrate (Fig. 7.70) here an interesting example found by Alfsen and Froyland (1984) for the Fig. 7.70 0.2
0.0
-0.2
0.2
0.0
-0.2 0.15
0.1
b
Lorenz system, with o = 10, but now r = 15. Beginning with b = 0.15 they found a stable symmetric limit cycle whose evolution they followed as b was slowly decreased. The two largest Lyapunov exponents are shown in the upper part. While a topological
change takes place in the limit cycle near b = 0.09, it remains stable down to about b = 0.06. Reversing the trend in b produces the Lyapunov exponents of the lower part. For b < 0.9 the upper and lower figures agree as they should, since they are on the same limit cycle. When b = 0.09 this goes through a period-two bifurcation and becomes chaotic around b = 0.1, where one exponent becomes positive. It is not until b = 0.12 that it enters the basin of attraction of the stable periodic limit cycle which was traced down from b = 0.15 (note again that the upper and lower figures agree for b > 0.12). These results imply that there are two topologically distinct stable limit cycles in the range 0.1 > b > 0.09 (bistability) and that there is both a chaotic region and a stable periodic region of the phase space in the parameter range 0.12 > b > 0.10. The coexistence of a stable limit cycle and a chaotic region is obviously of considerable
interest, and this use of Lyapunov exponents to document this fact is quite nice, because the chaotic regions or basins of attraction may otherwise be quite difficult to discover.
Models based on third order differential systems
198
7.11 Rossler's models The chaotic motion which was discovered by Cartwright and Littlewood and Levinson, involving a periodically forced van der Pol oscillator, and Lorenz's autonomous three-variable system are very important both historically and because
they are related to real physical systems. Prior to Cartwright and Littlewood's discovery, there was only Poincare's analysis, based on the conjectured homoclinic dynamics and Birkhoff's abstract analysis of area-preserving maps, to indicate the possible nature of chaotic motion. Following the discovery of Lorenz chaos, Rossler developed a number of simpler mathematical models which shed light on the minimal dynamic properties which will yield chaos in various forms. These models are largely
pedagogical in character, and not encumbered by any attempt to (immediately) associate them with physical systems. The ultimate aim is to develop an understanding, or feel, for different types of chaos by constructing models which exhibit a `hierarchy'
of chaotic forms (e.g., in increasingly higher dimensions). We will limit our considerations here to three dimensions (the minimum number). A rough description of Rossler's suggestion is that perhaps we can understand complicated motion in higher dimensions by using a `mental bootstrap' technique (i.e.,
`lift oneself by your bootstraps' - Fig. 7.71), which helps us `lift' the lower
Fig. 7.71
dimensional motion into a higher dimension and then 'reinjects' it back. This program
is obviously only suggestive, and requires considerable cleverness to make it enlightening.
The general idea can already be illustrated by the nonchaotic motion in low dimensions (Fig. 7.72). In one dimension, the elementary dynamics is either a simple
flow along a line, or a flow to/from a fixed point. From these we can, of course, construct composite flows, which require that nonlinearities be introduced. We can next move up to two dimensions in several ways. We can `lift' a simple one-dimensional
flow and reconnect it to form a periodic orbit, or spiral it around a fixed point to form a focus. We could also lift off from a fixed point flow, to obtain saddle points or nodes. All of these lifts are relatively trivial, because they involve only two linear
Rossler's models
199
Fig. 7.72 Dimension
Linear components
Nonlinear compositions
Il
(D 9 (etc.)
flows - that is, the lifting operation is linear. We can, however, introduce nonlinearities, and generate a variety of composite flows such as limit cycles, heteroclinic orbits (e.g., separatricies), and homoclinic orbits. For reasons which will soon become apparent, we wish to focus on the homoclinic, or near-homoclinic case, because the
motion is essentially reinjected back into the one dimension from which it came. However, because of the uniqueness of the solutions, the two-dimensional motion can never exhibit sensitivity to initial conditions (a positive Lyapunov characteristic
exponent) if the motion is bounded. To find chaos we must move up to three dimensions. We can move up to three dimensions by first using linear lifts, which produces the
nondegenerate flows discussed in Section 7.1, and the degenerate cases which may be controlled by nonlinear effects. However, now because of the third dimension, we
can visualize a nonlinear lifting and reinjection which is not simply homoclinic (although that is possible, of course), but which lifts off from, and drops back onto, a spiral motion. If the reinjection develops a random character, we can have a simple form of chaotic motion. This is the basis of Rossler's first model of chaos, which we will now reconstruct.
200
Models based on third order differential systems
Fig. 7.73 ac
x
-c
@1e
c( t)
We begin with a simple unstable spiral motion in the (x, y) plane, but displaced from the origin, as illustrated in Fig. 7.73
z= -Y-C;
y=x+ay
(a<2)
so that x = ac + Re [A exp (iwt) ] exp (at/2)
y = - c - Re [(a/2 + iw) A exp (i&t)] exp (at/2) where w = (1 - (a/2)2)172. We then visualize lifting this off the (x, y) plane by taking
c to be a variable along a perpendicular axis, which we will initially specify as a function of time (in other words, the system is no longer autonomous - and the new axis is related to the time). We note that, if c increases, the `center' of the spiral motion tends to move down and to the right in the (x, y) plane. If c(t) subsequently returns to its original value, the center of the spiral also returns, and the motion is reinjected into the (x, y) plane. However, where it is reinjected is a sensitive function of the history of c(t), so its subsequent motion in the plane may not be simply related to its previous motion in the plane. Now to return to an autonomous system, we make c(t) a dynamic variable, z(t), and trigger the behavior of z(t) by the value of one of the variables x(t) or y(t). This interdependence of z(t) and (say) x(t) requires a nonlinear interaction. Moreover, in Rossler's first model he took z(t) to remain small until x(t) reaches some prescribed value, where it rapidly increases, and then decrease when x(t) returns (by the spiral
motion) below the prescribed value. The rapidness of the increase and decrease, together with the triggering specification, may (but need not) produce a sensitivity in the flow, which then generates chaos.
Rossler's models
201
Rossler's first model, based on this idea, is
z= -y-z;
y=x+ay; (a=h='-s,c=5.7).
b+z(x-c) (R1)
We see that, if x < c, z(t) settles down nearly to the value b/(c-x) (and rapidly, if c - x is large), whereas if x > c, z(t) increases exponentially. Thus the only nonlinearity in this dynamics involves this lifting and reinjection of the z(t) dynamics relative to the `linear' (x, y) motion. Put another way, the motion in the (x, y) plane provides a stretching operation, whereas the z(t) motion produces a folding of the dynamics back
to the region near the center of the spiral motion and a z compression ('squeezing') when x < c. Fig. 7.74
Rossler (1979) has illustrated some of the mixing aspects of his attractor by paper (origami) models, from which Fig. 7.74 is derived (also see Abraham and Shaw (1983)
for many instructive illustrations). This figure shows two circulations of the origin by one trajectory, illustrating the initial repulsion of the trajectory in the (x, y)-plane from the origin, followed by its 'reinjection' near the origin through its z(t) motion.
It will be noted that the `folding over' of this surface by the z(t) motion is very
202
Models based on third order differential systems
similar to the orientation inversion one obtains in a Mobius strip, except that now it joins another orientation-preserving portion of the same surface. The surface cannot, of course, be accurately represented by a paper model, since such a surface cannot have unique trajectories on it (because of the merging trajectories along the reattachment line of the fold). In other words `surface' must have some thickness (a cardboard model is more accurate, but more difficult to construct). The fact that it is a strange attractor can be established by determining the Lyapunov characteristic exponents. It has been found (Froehling, Gutchfield, Farmer, Packard and Shaw 1981) that they are (0.075, 0, - 5.372). To understand the nature of such an attractor better, we consider first the Poincare surface of section (P1) illustrated in Fig. 7.74; that is the surface
x<0.
y=0,
(P1)
When trajectories pass through this surface many times, they settle toward the Rossler attractor which is contained in a very thin region when it passes through (P1). This is illustrated schematically in Fig. 7.75. Because the variation in z is very small the Fig. 7.75
Ln+1
0.15
6.5
x
Ln
value of x is essentially determined by its value at the last passage through the surface. This means that the Poincare map P: (xn, zn) -. (xn+ 1, zn+ 1) can be reduced to the
Lorenz `map' L:xn - xn+ 1. This is also illustrated in the figure on the right, where the Lorenz variable has been taken to be Ln = - xn, for convenience. Note that, in this model, we can even define the Lorenz variable as the local minimum value of x, similar to Lorenz's original `map'. We will discuss this map in more detail shortly. First, however, it is instructive to consider another Poincare surface, such as X= y > 0.
(P2)
The set of points generated by a trajectory passing through this surface for a long time looks something like Fig. 7.76 which clearly illustrates the folding action of the dynamics. The x variable can no longer be used for a simple map because of the double-valued nature of the limit set as a function of x. We might attempt to use the Lorenz variable L. = z,,. However, even using z we do not obtain a very well defined map, because the attracting set has considerable thickness on this Poincare surface. This is illustrated schematically in Fig. 7.76 on the right. This demonstrates
Rossler's models Fig. 7.76
z
203
Zn+1
Z
Zn
X
Fig. 7.77
Ln+ i 10 r-
5
5
10
that a Lorenz map is generally a useful quantitative construct only on special Poincare surfaces.
Now let us examinine the Lorenz `map' (Fig. 7.77), obtained from the Poincare surface (P1). This is illustrated in greater detail below. This map contains portions which are outside the attractor region, obtained, for example, by taking initial
204
Models based on third order differential systems
conditions with small values of (x, y, z) and determining their history. The `map' has the same simple structure as the logistic map, and we can determine the attracting
region (or periodic orbits) in the same way as was done there. Below the map is illustrated a sequence of regions which show in successive refinements where the Rossler attractor is located. All points which are attracted come within the largest region shown in the figure. These points, in turn, are mapped into the next two smaller regions. These two regions map into the next four smaller regions, and so on (for a further discussion, see the logistic map). The Rossler attractor lies inside these (closed) regions, and what we are constructing along the x axis is simply a classic Cantor set. Therefore the above paper (or cardboard) `surface', while continuous
along the direction of the flow, has a Cantor structure in one of the transverse directions (the other direction will be considered shortly). It is the dynamics associated
with the approach to this Cantor structure which produces the observable (macroscopic) chaotic properties (note that this Cantor structure extends over a distance Ax ^-, 6.5, which is macroscopic on any graphics!) This Cantor structure is so large that part of the structure is clearly visible from an (x, y) projection of the dynamics (Fig. 7.78). This macrochaotic structure is analogus to the `butterfly wings' Fig. 7.78
structure of the Lorenz attractor. There is also, however, a microscopic form of chaos which is difficult to detect at the surface (P1), but which is responsible for the lack of a `good' Lorenz map on many other Poincare surfaces (e.g., (P2)). To understand the other structure of the Rossler attractor, it is useful to consider what happens to a `block' of phase space as it passes successively through the Poincare
surface (P1). We take the x extension of the block to be the largest region of the previous map. Fig. 7.79 illustrates three modifications of the original phase space block, each of which involves a stretching, folding, and compression. The first circulation of the z axis produces a `folded towel' structure, except that the ends of
Rossler's models
205
Fig. 7.79
the towel do not coincide. This can be understood by considering the Lorenz map of the smallest and largest value of the Lorenz variable, L = - x, in the Rossler interval. Rossler called the noncoinciding ends structure, which is produced by the map of the entire interval, a `walking stick'. As the above figure indicates, this walking
stick maps into a more curious structure, which can again be understood with the help of the Lorenz map. Now both ends do not coincide with the boundary of the region, and there are four, three, or two `layers' in the vertical (z) direction, depending
on the Lorenz variable (- x). This complicated folding procedure is illustrated for one more circulation about the z axis. We see that this forms a complicated structure in both the x and z directions, the former being given by the Lorenz `map' (which ignores the z complications). If we could ignore the x structure, the z structure would be a relatively trivial form of towel folding, which produces 2" layers at the nth fold (Fig. 7.80). In that case we would simply be constructing a classic Cantor Fig. 7.80 0)
set, similar to the one we just found in the x direction. They differ in details, of course,
(e.g., possibly in dimensions), but they are conceptually the same as the Cantor middle-third set. If we return to the Rossler attractor, with its Cantor set structure in the x direction, it is more difficult to decide precisely what is the limiting z structure of this attractor. While it may have varying detailed properties as a function of x, it seems clear that the z structure is likewise a Cantor set. Thus the Rossler attractor has a structure of two transverse Cantor sets `carried along' by a continuous flow. The motion toward the latter Cantor set, however, does not produce a noticeable (macroscopic) chaotic
Models based on third order differential systems
206
behavior - that property is described by the Lorenz map. However, the Cantor structure in the z direction on (P1) is associated with the `transverse fuzziness' of the set of points observed on other Poincare surfaces (such as (P2) above), and for the corresponding lack of a useful Lorenz map. The behavior of the dynamics produced by the model (R1) is strongly dependent on the values of the parameters (a, b, c). In particular there are many parameter values which give periodic, rather than chaotic motion, and varying the parameters produces various bifurcations. One study of the bifurcation sequence produced by increasing
c (Crutchfield, Farmer, Packard, and Shaw, 1980) shows that this dynamics goes through a series of period-two (subharmonic) bifurcations before it exhibits `chaotic' characteristics. Several bifurcations are illustrated in Fig. 7.81. For several reasons Fig. 7.81
A
0
4 0
O .o o
o
-4 0
10 20
30 40 50
Frequency (Hz)
it is difficult to determine whether higher order stable periodic solutions exist. Part of the difficulty involves numerical accuracy, but there is also the `real' phenomena associated with the time required for solutions to be attracted to neighborhood of the stable periodic orbit (a `relaxation time'). In the neighborhood of the periodic orbit this is measured by the value of the negative Lyapunov exponents, but there is no simple global measure for this time. In any case, when the power spectral density
Rossler's models
207
(PSD) of dynamics are obtained (always over a finite time, of course), this relaxation
effect tends to broaden the determined spectra. Nonetheless the calculated power spectral density, shown in the lower half of Fig. 7.81 clearly shows the discrete fundamental and harmonic frequencies above the `noise' level. In addition to the power
spectral density, the largest Lyapunov exponent, A,, has also been determined for this range of c (Fig. 7.82, Crutchfield et al. 1980). The letters in this figure correspond Fig. 7.82
0.1
Periodic
C
Chaotic
0
1
CA D
-0.1 3.0
5.0
4.0
C
6.0
11
to those in the last figure. It was found that this Lyapunov exponent becomes positive when c > 4.2, as illustrated, which establishes a lower bound for chaotic behavior. It
is worth noting that this Lyapunov exponent criterion appears to be a clearer indication for the onset of chaos than the use of the power spectrum. However, the fact that this simple Rossler case retains sharp frequencies in the presence of the broad spectrum gives additional insight into the nature of the chaos, which is not indicated by the Lyapunov exponents (or the Kolmogorov entropy). Fig. 7.83
208
Models based on third order differential systems
The fact is illustrated by noting that the Rossler model exhibits other bifurcations of a more sophisticated nature. Thus, if the parameter a is increased slightly (a = 0.3,
b = 0.2, c = 5.7), the Rossler attractor acquires a funnel structure, as is illustrated schematically in Fig. 7.83. Associated with this topological change in the structure, there is a significant alteration in the Lorenz map (Fig. 7.84) at the Poincare surface (P1). Fig. 7.84
-Xn+]
4k 3
- I
2
1
-xn 1
2
3
4
Rossler called the first attractor (a = 0.1) a 'spiral-type chaos' (it spirals each time around the center in the (x, y) plane), and the latter attractor (e.g., a = 0.3) a `skrew-type
chaos' (there are a number of `visits' near the (x, y) plane which do not circle the center). A third possibility (torus-type) was also suggested. A difference in these forms of chaos can be exhibited by examining their power spectral density (Farmer, Crutchfield, Froehling, Packard, and Shaw, 1980). Fig. 7.85 shows the power spectra for various values of a, all with b = 0.4, c = 8.5. Specifically,
in (A) a=0.15, (B) a=0.17, (C) a=0.18, (D) a=0.19, (E) a=0.2, and (F) a=0.3. It can be clearly seen that the sharp frequency lines are lost as a is increased successively over the values A, a = 0.15; B, 0.17; C, 0.18; D, 0.19 (where the funnel structure beings); E, 0.2; F, 0.3. Notice that, not only do the discrete lines vanish as the funnel is fully formed, but the spectrum also no longer goes to zero at zero frequency. The occurrence
of sharp peaks in broad power spectrum, and a nondecaying component of the autocorrelation function, has been termed `phase coherence' (Farmer et al., 1980). This feature is similar to many experimental power spectra in turbulent fluid situations which retain a large-scale order in the presence of small-scale chaos (e.g., `chaotic'
Taylor vortices, Figs. 7.86, 8.68; Fenstermacher, Swinney, and Gollub, 1979). Rossler's model illustrates how this phase coherence can occur in a simple flow, and how it can disappear with a topological change in the structure of the attractor (here from spiral to skrew or funnel structures). The disappearance of the sharp peaks in the power spectrum signifies a dispersal of an ensemble of initially nearby solutions
Rossler's models
209
Fig. 7.85 A
B
C
D /._
E
F
7
i
/1 p
0
0.1
0.2
0.3
Frequency
Fig. 7.86
0.4
0.5
Models based on third order differential systems
210
along the direction of the flow, but not (necessarily) in an exponential fashion. Thus
the associated Lyapunov exponent can be zero, and yet there can be several distinguishable situations in this chaotic attractor (the other two Lyapunov exponents are + and - respectively). The distinguishing cases depend on whether there is rapid loss of phase coherence or not, which can be detected by the power spectrum. Fig. 7.87 The mixing process of the simple Rossler attractor; xy projections of the evaluation of an ensemble of initially neighbouring points. (a) 11 strobes, 64 points, beginning at t = 6.8. (b) 11 strobes, 128 points, beginning at t = 12.30. (c) 11 strobes, 256 points, beginning at t = 13.67. (d) 6 strobes, 256 points, t = 20. (e) 11 strobes, 256 points, t = 159. (f) 2 strobes, 64 points, t = 4723. (a)
(b)
(c)
(d) .J
(e)
An example of a slow loss of coherence is illustrated in Fig. 7.87 (Farmer et al., 1980) for 256 initial conditions in the simple (spiral) Rossler attractor. The figure shows a sequence of `stroboscopic' views of the points (6, 11 or 2 strobes in various
Rossler's models
211
frames). The last frame shows two strobes after nearly 5000 revolutions about the origin. It can be seen that this group of initial conditions is still not spread out along the flow over the entire attractor. It is this slow loss of initial information which is the cause of sharp peaks in the power spectrum. Similar figures in the case of the funnel Rossler attractor show a much more rapid loss in phase coherence, induced by the additional spiral return to the (x, y) plane. A similar analysis of the Lorenz attractor shows a power spectrum with little indication of phase coherence, particularly in the x variable (Fig. 7.88a). An examination
6
4
0
-2 0.1
0.2
0.3
0.4
0.5
Frequency
of the behavior of an ensemble of nearby initial states shows that their mixing in the direction of the flow is produced by the fixed point at the origin. Here the passage of time past this region is a sensitive function of the distance the solution passes from the origin, and this apparently produces the fairly rapid mixing of the states along the flow direction, helping to spread them uniformly over the Lorenz strange attractor. While Rossler's model (RI) is very instructive, many physical systems do not have
its feature of a single `triggering' value (i.e., x = c), but instead are based on the phenomena of bistability. It will be recalled (see the index) that there are systems
212
Models based on third order differential systems
which have equilibrium states on a Z-shaped curve in an appropriate state space (Fig. 7.89). Typically the top and bottom lines of the `Z' are stable (hence `bistability'), Fig. 7.89
whereas the connecting line is unstable. We have seen such a Z-shaped curve also occur in the analysis of relaxation oscillations, where the curve is associated with the `slow' dynamics, whereas the fast' dynamics approaches (or leaves) the stable (unstable)
portions of the curve. The canonical examples of relaxation oscillators are the van der Pol oscillator (Fig. 7.90(a))
Eu+(D(u)U+u=0
(E<< 1)
and the Rayleigh oscillator (Fig. 7.90(b))
Ez+F(z)+x=0, which are related by u = z, (D(u) = dF/du. For our present purposes it is useful to rotate Fig. 7.90(b) and to introduce Lienard's representation of the dynamics, by setting z = dx/dt (Fig. 7.90(c)). In that case the equations for the relaxation oscillator become
Ei= - F(z) - x
x= Z,
(E« 1).
The small value of e ensures that dz/dt is large (fast' dynamics), unless the state is near the curve x = - F(z). A simple example of such a curve is
x= -z3-Cz
(C>0).
We are now interested in taking this phenomenon of relaxation oscillations `out of plane' (or `lifting' it) into three dimensions, by introducing a new variable y (t), with the resulting dynamical equations
z= -Y-Z,
y=x+ay,
Ei=f(x,z,E).
(7.11.1)
Again, if E is very small, z tends rapidly toward the stable branches of the surface f (x, z, 0) = 0
or z = gk(x)
and, if z is reasonably constant on these branches (as in a Z surface), the (x, y) motion
Rossler's models
213
4(u)
Fig. 7.90
u
(a)
F(u)
(b)
U
(C)
is again a simple unstable focus (a > 0) about the fixed points (a, gk(z), - gk(z)) depending on which branch the motion (temporarily) takes place. The motion in the (x, y) plane in this `lifting' is linear, which is obviously the simplest possible mechanism. In fact, historically, interest focused on limit cycle motion in the (x, y) plane (also), but not necessarily in the relaxation limit. A particularly simple example is
z=- y+(ra-x2- y2)x,
Y=x+(ra-x2- y2),
Ei=f(x,z,E)
where the (x, y) motion is not influenced by the z motion, and has a circular limit cycle x2 + y2 = ro. If the surface f (x, z, 0) = 0 has a single root, z = g(z), then the limit cycle only becomes mildly distorted in the z direction (Fig. 7.91). Rossler noted the similarity of this limit cycle with Salvador Dali's Soft Watch painting of 1933, so it
is labeled accordingly. However, if f (x, z, 0) has three roots, and dgk/dx = oo for 0 < x < ro, the limit cycle dynamics acquires a relaxation characteristic (Fig. 7.92),
214
Models based on third order differential systems
Fig. 7.91
Fig. 7.92
the study of which dates back at least to Khaikin's (1930) electronic `universal circuit'. This is discussed in some detail in the book by Andronov, Khaikin, and Vitt (1966).
The linear behavior in the (x, y) plane, described by (7.11.1), is recovered in the limit of a large limit cycle (i.e., ro >>x2 + y2). Rossler's second model is of this type, where moreover the function f (x, z) is taken to be f (x, z, e) = (1 - z2)(x - l + z) - ez
(R2)
which vanishes on nearly-perfect Z-curve, as illustrated in Fig. 7.93. The signs in the Fig. 7.93
figure indicate the sign of f (x, y, e), and hence dz/dt, and the dynamics of interest is confined to the range near 0 < x < 2. For small s, f (x, z, e) = 0 near z = ± 1, in which
Rossler's models
215
case the spiral motion is centered about y = T 1, x = ± a. Since the fast z motion occurs near x = 0 and x = 2, the lower spiral is centered far from the fast upward motion at x 2, whereas the upper spiral can be centered near the fast downward motion near x = 0, if a is not very large. Rossler took the value a= 0.15 for his illustration of this bistable form of chaos. The spiral and skrew-type bistable forms of chaos are illustrated in Fig. 7.94 (after Rossler). They are upside-down from Rossler's
example (change z into - z). Fig. 7.94
(a)
(b)
It might be remarked that the numerical solution of relaxation oscillations takes some care if numerical instabilities are to be avoided. Note that the time scale At is effectively changed to At/E for the z motion, and that if (z2 - 1) becomes appreciable, then this will tend to change sign and grow at each iteration, producing the instability. An adjustment of At can, of course, avoid this problem. These models of Rossler (and others not mentioned) give a better insight into some of the forms of chaos, but clearly much more needs to be learned, even in three dimensions. What produces chaos is the exponential divergence of nearby solutions in a bounded region of phase space. This second condition means that solutions must continually diverge from new neighboring solutions, which they temporarily approach
in the course of staying in a bounded region of phase space. This can most simply be viewed in terms of various stretching-and-folding scenarios, such as kneading dough or `pulling' taffey. A sophisticated way of doing this is with a taffey pulling machine, illustrated schematically in Fig. 7.95 (Rossler, 1983).
Notice that these `mixing' operations take place in physical (configuration) space. When a piece of taffey passes through a point in this space its future behavior is not unique. This can be seen clearly in the taffey machine, whose mixing operation is
216
Models based on third order differential systems
Fig. 7.95
Fig. 7.96
ION
ti
illustrated in Fig. 7.96 (Rossler, 1983). The figure shows `dot' and `dash' taffey being mixed over nearly one cycle. Note that this mixing even has some of the `walking
stick' features of Rossler's first model. This space is not a phase space, which has unique solutions through each point. The reason for this, of course, is that there is
Lyapunov exponents and a strange attractor
217
a machine which makes the taffey dynamics nonautonomous. However the taffey machine is periodic, so that we can construct a finite phase space with this dynamics by using the extended phase space (one axis being the periodic time axis - that is, a one-torus). This can be nicely visualized (Rossler, 1983) by putting the taffey machine on a rotating turntable (`lazy susan'), where the turntable's period is the same as the taffey machine. The synchronous motion is required in order for the dynamics in the
taffey torus to be unique (as in Poincare's first return map).
7.12 Lyapunov exponents and the dimension of a strange attractor It has been conjectured (see below) that, when there is a strange attractor, there might
exist a relationship between the Lyapunov exponents of the trajectories which are attracted and the dimension (e.g., capacity) of the attractor. To see why we might
expect some relationship between these quantities, we consider first a twodimensional mapping of a set of points which are attracted to such an attractor (recall that in two dimensions only a mapping, not a flow, can give a strange attractor). For a more extensive discussion of these concepts see Farmer, Ott and Yorke (1983), and Ledrappier and Young (1985). Assume that the attractor is inside a square, whose sides are normalized to unity. The mapping stretches one side by a fraction L1 > 1, and compresses the other side by fraction L2 < 1, as illustrated in Fig. 7.98. We assume that 1 > L1L2, so that the
area is decreasing and fits back inside the unit square in the shape of a horseshoe (this is not the famous horseshoe map, and indeed this shape could just as well be an `S' or `Z', etc. - people and dynamics seem to like horseshoes!). The strange attractor
218
Models based on third order differential systems
Fig. 7.98
L2
must lie inside this horseshoe, so we can cover this set with squares that have sides L2, and the smallest number of such squares is essentially N(L2)
L1/L2
(or the next largest integer). If we map this horseshoe, it folds into itself in a similar fashion, as shown. The length of this region is now L2 and its width is L2, and the strange attractor is somewhere inside. It is our second estimate of the attractor. We can now cover the attractor with smaller squares, with sides L2, and the smallest number required is N(L2) = (L1/L2)2.
Clearly, if we proceed with k maps and obtain the kth estimate of the attractor, we will be able to cover the attractor with squares of size L2, and the minimum number is N(LZ) = (L1/L2)k.
These squares are becoming smaller and smaller, so we can use them to determine
Lyapunov exponents and a strange attractor
219
the capacity dimension, D, defined by
lim N(E) - C'D
(7.12.1)
E-0
where N(s) is the minimum number of E squares which covers the set. For large k we therefore have N(LZ)
(L1/L2)k
(L k2)-D.
If we solve this for D, we obtain
D=1+
lnL1 I In L2 I
In this discussion we have implicitly taken the two directions to lie along the principal axes, and the numbers Lk (Lyapunov numbers) are related to the Lyapunov exponents by Lk = exp (1k),
so we have the relationship
D=1+A I'I. The question is whether this type of relationship can be generalized to higher dimensions (in R"). It was conjectured by Kaplan and Yorke (1979) that such a generalization might indeed hold. To specify this relationship, we first order the exponents according to A l 1> 12 >,'
,
Next let j be the largest integer such that Ak1> 0-
k=1
In terms of the Lyapunov numbers, this means that j is the largest integer such that L1L2 Lj > 1. In other words, j is the `expansion' dimension of the dynamics - there are volumes of this dimension which expand - and it is certainly reasonable to expect
that the dimension of the attractor will be at least this large. Kaplan and Yorke conjectured that the dimension of the attracting set is given by
DKY=J+
k=IAk. I1j+1I
We note that, in the above two-dimensional case where Al + 12 < 0, we have j = 1, and so DKY = D. To see why DKY may equal D in many circumstances, but not necessarily in all (Grassberger and Procaccia, 1983), we will consider some examples in three dimensions. These examples illustrate the fact that, while the Lyapunov
220
Models based on third order differential systems
exponents give information concerning the stretching and squeezing of a set of solutions, they do not give information on how the solutions are `folded back', which is necessary to produce the strange attractor - and that folding process has an effect on the dimension of the attractor.
Let L, > L2 > L3 be the three Lyapunov multipliers, and L,L2L3 < 1, so that L3 < 1. Consider first the case when L, > L2 > 1, so that there is expansion in two directions. In this case there must be folding along both the 1 and 2 axes, for the attractor to be in a bounded region (a possibility is illustrated in Fig. 7.99). We can Fig. 7.99
3
1 cover the attractor, which is inside this region, with cubes having sides L3, and this requires N(L3) = (L,/L3)(L2/L3)
cubes. Note, again, N(L3) is the minimum number of L3 cubes. Repeating this k times,
we find that to cover the kth estimate requires N(L3) = (LIL2/L3)k,
and taking the limit of large k, we obtain for the dimension (L,L2/L3)k = (L3)-D
Solving for D, and introducing the Lyapunov exponents, yields
D=2+In(L_L2)2+A1 IInL31
+22 1231
Since A,, + 22 + 13 < 0, we have j = 2, and this agrees with the Kaplan-Yorke conjecture, DKY = D.
Now, more generally, a possible simple `rationalization' of the Kaplan-Yorke conjecture goes as follows: There is a j-dimensional subspace of R" which is expanding,
and toward which all trajectories are tending. The direction of slowest approach to this subspace is along the j + 1 axis, so if we take (hyper)cubes with sides LJ +, we
Lyapunov exponents and a strange attractor
221
can certainly cover the attractor by using (L1L2...LJ/LJ+1)k such cubes (note that, since LS/LJ + 1 < 1 for all s = j + 1, ... , n we only need one cube `thickness' in these directions). If we use this as an estimate for the (minimum!) number of cubes with sides L;+ required to cover the attractor, then we obtain the Kaplan-Yorke estimate: 1
+1)k=(L;+1)-DKY
The possible error in this estimate is due to the fact that we may be able to cover the attractor with fewer than NKY(L;+ 1) cubes with sides L;+ 1. Whether this is true
or not depends on how the set is folded by the dynamics. A simple example (Grassberger and Procaccia, 1983) is the case L1 > 1 > L2 > L3, and the folding is along the faster contracting direction (3 axis, as illustrated in Fig. 7.100). It is clear Fig. 7.100
that LZ cubes may enclose many layers of the folding in the 3 direction, so that the minimum number of LZ cubes is less than the Kaplan-Yorke estimate N(Lk2) < NKY(LZ)
which in turn implies that DKY > D. For their example, (21 + 22 < 0) Grassberger and
Procaccia obtained D = 1 + 21/1231 (see the exercise) as compared with the Kaplan-Yorke value of either DKY = 2 + (21 + 22)/ 1231 (if Al + 22 > 0) or DKY = 1 + 21/1221 (if 21 + 22 < 0), both of which are larger than D. If the dynamic folding is in the `slower' contracting direction, as illustrated in Fig. 7.101, then the Kaplan-Yorke estimate using LZ cubes is accurate, because no Fig. 7.101
multiple L3 layers are contained in the LZ cubes. In this case DKY = D. Clearly, in higher dimensions the folding possibilities are much more varied and these can be expected to yield dimensions less than or equal to DKY.
Exercise 7.12 There is an interesting question which arises here, namely why use `large' cubes (with sides L;+ 1) to cover the set rather than the `smallest' cube with
Models based on third order differential systems
222
sides L'? If we use the latter cubes we know that the minimum number required to cover the kth estimate of the attractor set is simply (L1L2 . . . Ln - 1/L:
-1)k. This is true,
no matter how the set is folded, provided only that we can neglect effects due to the `bends' of the folding - which we also did above. This then gives the estimate for D (L1 ... Ln - 1
/L- 1)k = (Ln)-D* n
or
D*=(n- 1)+Lk-1Ak 1;n1
Note that there are at least (n - j - 1) terms in last ratios which are less than - 1, Ak/I A. I < - 1, so that D * is near j. Why isn't D * = D?
For the Lorenz system (a = 10, b = 3, r Ak = 1, 0, - 14.5. This gives an estimate
30), the Lyapunov exponents are roughly
D*=DKY^'2.07, which is a remarkably good estimate (D = 2.073, Lorenz, 1984). The following table shows some comparisons of DKY and D, defined in (7.12.1), for several maps (Russel, Hanson and Ott, 1980). Further comparison can be found in the literature, but these
DKY
D
1.200 ± 0.001 1.264 ± 0.002
1.202 ± 0.003 1.261 ± 0.003
Henon
(xn+1 =yn+ 1 -axn,yn+1 -bxn) a = 1.2, a = 1.4,
b = 0.3 b = 0.3
Kaplan- Yorke (xn + 1 = 2xn(mod 1 ) ,y , ,+1 =
1.4306766
1.4316 ± 0.0016
examples appear to be representative of the accuracy of the Kaplan-Yorke estimate. Under suitable restrictions, much more definitive conclusions can be drawn, but these are beyond the scope of this introductory discussion (e.g., see Ledrappier and Young, 1985).
7.13 Open systems - chemical oscillations In this section we will consider some fairly familiar forms of nonlinear dynamics, but
in an entirely new physical context - one which involves open systems. An open system is one whose dynamics is influenced by the input of some (exogenous) agents
Open systems - chemical oscillations
223
to the system, and the output of other agents ('agents' = matter, energy). These are represented by (A, B) and (C, D) respectively in Fig. 7.102. The input agents directly Fig. 7.102
influence the system's (endogenous) variables, and will here be treated as constant source terms, whereas the output agents have no time dependent influence on the system. They are treated as a non-dynamic by-product of the system's dynamics. This is the simplest example of open systems, which more generally involve variable inputs and adaptive responses of the system. The most complex example of such systems relates to living systems. As Bertalanffy (see Davidson, 1983) emphasized since 1928, one of the essential features of living systems (at all levels) is that they are open systems. This is now more widely appreciated and discussed (e.g., Blumenfeld (1981), Prigogine and Stengers (1984)), but there are only rather limited explicit models of such processes. Some higher order models of such systems will be discussed in
Chapter 8, but here we will consider one class of models concerned with the phenomena of chemical oscillations. While there were speculations about the possibility of chemical oscillations in the 1920s, the subject was widely discounted until Belousov's discovery in 1959 of chemical oscillations which could be observed as changes in color (e.g., colorless to yellow to colorless, twice a minute). This discovery was followed by Zhabotinskii's observation in 1967 of spatial structures. Since then there have been several models proposed to
explain various aspects of chemical oscillations, and specifically the BelousovZhabotinskii reactions. In the present section we will not consider any of the spatial aspects of these dynamics, but only some of the simpler models describing the temporal oscillations. Some of the surprising spatial influence of diffusion will be discussed in
the next chapter. One of the earliest models of chemical oscillations was due to Prigogine and Lefever (1968), and has subsequently become known as the `Brusselator' model, after the city of its birth. This model involves only two-system chemicals (X, Y), and is sufficiently simple that it can be usefully applied to the spatially inhomogeneous situation. We will therefore delay discussing this model until the next chapter. The simplest model which specifically concerns the Belousov-Zhabotinskii (BZ) reaction is due to Field and Noyes (1974), which they named the `Oregonator' (not the `Eugenetor'!), after the
Models based on third order differential systems
224
state of its birth. In its simplest form, let the concentrations of the exogenous agents be A = [BrO 3 ], B = [BrMA], and P, Q = P + A the by-products. The concentrations
of the dynamic variables are represented by X = [HBr02], Y= [Br-], and Z = [Ce+4], and the reactions are A + Y -+ X;
X + Y--+ P;
B + X --> 2X + Z;
2X
Q;
Z - f Y.
When the reactants are well-stirred, so that no spatial inhomogeneities occur, the dynamics is governed by the system of ordinary differential equations dX/dt = k, AY - k2XY + k3BX - 2k4X 2
dY/dt= -k,AY-k2XY+ fk5Z dZ/dt = k3BX - k5Z
where the parameters, ki, range in value from 1 to 109. Here f is a suitable stoichiometric coefficient for the overall reaction (Noyes, 1976)
fA+2B-+fP+Q (e.g., if f = 1, B - P). These equations can be put into various dimensionless forms (Tyson, 1979), some of which make the phase portraits considerably simpler. The variables used by Field and Noyes yielded the following system of dimensionless equations
dx/dt=s(x+y-xy-qx2) dy/dt=s-1(-y-xy+z) dz/dt = w(x - z) where they set s = 77.27,
q = 8.375 x 10-6,
and took f and w as the control parameters. The fixed point of that equation is independent of w, xo = zo
q + [(1 - f - q)2 + 4q(l +.f )] 1/2}/2q
Yo = .fx0/(1 + xo).
Field and Noyes determined the stability of this fixed point as a function of (f, w). Their result is illustrated in Fig. 7.103. They moreover found that the numerical solutions of these equations tended toward the desired limit cycle motion, representing
the stable Belousov-Zhabotinskii oscillations. Later, Hastings and Murray (1975) were able to prove, using the Brouwer fixed point theorem, that these equations have at least one periodic solution.
Open systems - chemical oscillations Fig. 7.103
225
480 (0.9935, 450.05)
3
Stable
N O O
B 2.414
F
It should be noted that, although the physical origin of these equations relates to an open system, the exogenous variables are taken to be constants. Hence there is no dynamic coupling, such as a feedback effect, nor even a statistical influence of the `outside world' on the system. Moreover, there is no threshold of exogenous agents necessary to produce a nontrivial ('live') system. Such a model of an `open' system is therefore quite rudimentary, and not adequate to address many questions about living systems.
Experimentally the concentration of specific ions are determined by a probe, as a function of time. Fig. 7.104(a) illustrates the possible character of log [Br-] vs time, which shows a strong relaxation characteristic. This same relaxation characteristic was found in the Field-Noyes calculations, illustrated schematically by the full curve in Fig. 104(b). They noted that, if we take into account the large value of s, the first equation approximately requires that
x+y-xy-qx2^-0. This quasi-equilibrium curve, which is sometimes referred to as a nullcline, is illustrated by the broken curve in Fig. 7.104(b). Thus in this situation, (x, y) are respectively the (fast, slow) dynamic variables. Tyson (1979) reformulated this result of Field and Noyes by rescaling their equations
226
Models based on third order differential systems
Fig. 7.104
(a)
log x
5
4 3
2
1
0
-3
-2
-1
0
3
2
1
log y
(b)
into the form
EJC=µy-xy+x-x2 Sy=µy-xy+ fz =x - z
where e 4 x 10- 5, S 2 x 10- 6, p 10- 5, f 1. With this scaling the quasiequilibrium condition is due to the second equation rather than the first, whose nullcline is given by
Y=fz/(µ+x). That is, (x, y) are now the (slow, fast) variables. Eliminating y from the first equation yields
Eat=x(1-x)-fzX +µ x
µ
Open systems - chemical oscillations
227
These equations for x and z are similar to a model of the BZ reaction suggested by Rossler (1972). The nullclines of these two equations, fz-x(1-x)(x+µ)
x - z,
x-µ
are illustrated in Fig. 7.105. Their intersection is the fixed point of these equations, Fig. 7.105
1,
z
I-D
I -D
2
x
which is unstable if 1 + 21/2 >f> Z. It will be seen in the next chapter that the model
of Tyson's is quite useful to understand an `echo wave' (Krinskii, Pertsov and Reschetilov, 1972) which is induced by Turing's diffusive spatial coupling of two such oscillators. This will be discussed in the next chapter.
Comments on the exercises (7.1) If we take y1 = x1, Y2 =
x1
a12
a13
x2
a22
a23
x3
a32
a33
and
y3 =
x1
a12
x2
a22
+
x1
a13
x3
a33
then the differential equation for yl, Y21 and y3 are .Y1 = TY1 - Y3;
.Y2 = DY1;
.Y3 = MY1 -Y2
These only involve the three parameters (T, D, M), as desired. Other sets of
variables can also have this property (see Chapter 8 for a more general discussion).
(7.2) x1 =(R+rsinO2)sinO1; e1=(w1t+4'1) x2 = (R + r sin 02) cos 91;
x3=rcos02.
02 =(w2 t + 42)
Models based on third order differential systems
228
For a nonintersecting surface, R > r. The motion is periodic if mw, - ncu2 = 0 for some integers m, n which are not zero (the (Oks are `rationally dependent'), otherwise the motion covers the torus ergodically. The solution of all linear equations (7.1.1), which are not confined to a plane,
either tend to a fixed point or are unbounded (there is always one real characteristic exponent). (7.3) Fig. 7.9(a): I Mk I < 1 (F is an attractor), and the Mk are not real (there is
rotation). Correspondingly, Re(; 0 < 0, and Im ()k) 0 0, n. Fig. 7.9(b): I M, I > 1 and I M21 < 1, so these cannot be complex conjugates.
Hence the Mk are real, but may be either positive or negative (but M, M2 > 0 - why?). Negative Mk correspond to reflections through the origin, which will be discussed further after this exercise. Correspondingly, Re().,) > 0 and Re (.12) < 0, and Im (Ak) = 0 or it. Fig. 7.10: Since there is only rotation, I Mk I = 1, or Re (Ak) = 0. More generally
the relationship between the characteristic multipliers and exponents is illustrated in Fig. 7.106.
Fig. 7.106
Im(M)
E
Attractive
Repulsive
(7.4) Since Ws and W' are both two-dimensional, they must `typically' (but not necessarily) intersect in R3, and since the fixed points are saddles one branch of W- must be the same as WS (there is a heteroclinic orbit joining the fixed points). To picture the flow, put the fixed points at the poles of a spherical surface, which is both WS and W,; , and take the axis to be W- and WS . Inside the sphere there is a periodic orbit (e.g., parallel to the equator). The equator can, of course, be taken out to infinity, in which case W. and W do not intersect. (7.5) What we want is a function L(x, y, z) > 0 such that dL/dt <, 0 everywhere, and
dL/dt = 0 only at the origin. We note that a(yy + zz) = oc(rxy - y2 - bz2) eliminates the cross terms xyz. If we add this to x.x = - ax2 + vxy, we cannot
Open systems - chemical oscillations
229
eliminate xy. But if we take a = a we obtain
2dt(x2+ay2+az2)=a{-x2+xy(1 +r)-y2-bz2}
-a{[x-Z(1+r)y]2+[1-4(1+r)2]y2+bz2}. Therefore, if r < 1, L = x2 + aye + az2 is the desired Lyapunov function. (7.6) The eigenvectors of (7.3.6) satisfy
-v
a
0
x
r
-1
0
y
0
0
-b
z
so that ay = (a + ).)x, and x = y = 0 if 2 the x axis is
tan0= 1 +(A/a)=
x
=A
y z
b. The tangent of the angle with
{(a - 1)±[(1 -6)2+4ar)]112} 1 2a
= Zo{1 ± [1 +(40r/81)]1/2},
if u=10. The variation
is illustrated in Fig. 7.107. ii = 2+u - 6(s + u)z; s = 2_s + Q(s + u)z where Cr = a/(1 + - A-) and i = - bz + (s + u)(sc _ + uc+ )
where
c± =(a+2t)/a(1+ _
Fig. 7.107
A_)2.
0 8+
-0_ /1010
100
r
(7.7) The linearized equations about C+ are 9 = a(y - x), y = x - y - az, 1= a(x + y) - bz where a = [b(r -1)] 1/2, from which we obtain (7.3.8). The real roots are found to be: E=
A1=
1.05
1.1
-0.0946
-0.19 8 -2.44 3 -11.02 6
22= -2.56 20 = -11.03
1.2
-0.445 -2.171 -11.052
1. 3
-0. 813 -1. 777 - 11. 077
1.34560
- 1.280 - 1.299 -1 1.088
230
Models based on third order differential systems
(7.8) )o = - (a + b + 1); and eigenvector is [- aoc, (b + 1)a, b(a + b + 1)] where a = [b(rt - 1)] 12. (7.9) The relationships (7.5.6) become 2 = (K + 2)t, a =,u/(K + A), and y = gaK/24R(K + A)2 whereas (7.5.7) become
x(t) = q(t)/AR(K + A), y(t) = ys1, z(t) = ( KK 2 W1 - C1
y(K + )2)/K. The parameter r = yW1 decreases as (K + 2)- 2 for where increasing heat conductivity, stabilizing the system. (7.10) From (7.3.10), (a + b + 1) = - 2 - 22r, so that a20/ar = - 2(a2,/ar). Also, from (7.3.10), 2ab(r - 1) + 2ob(r + a) = 2202r. Differentiating with respect to r, and using the previous result, yields
2ab + Job = [2b(r + a) - Go),,. +1] 20 ar
At r=r1, 2r=0 and 2o= -(a+b+ 1), so a2r
b(a - b - 1)
ar
2b(r, + a) + 2(a + b + l)2
The condition for a Hopf bifurcation is therefore a > b + 1. (7.11) A= a for all a. For chaos there must also be a contracting direction, producing bounded motion (see Section 7.11 for illustrations). (7.12) The fallacy in this reasoning involves the use of cubes with sides L to cover the kth estimate of the strange attractor. The quantity is the minimum number of L cubes which cover the attractor, not our kth estimate of the attractor. In the example of this section, where L1 > L2 > 1, these two directions are expanding, and hence LT and LT are always larger than L3 for any m, k. However, if 1 > L2, then this is no longer true if m is much larger than k. This means that, for some m > k, the mth estimate (and hence the attractor itself) can be covered by fewer L
cubes than estimated in the exercise, so D* > D. In the Grassberger-Procaccia example, we can take such small L3 cubes, and then consider the mth estimate, where L3 > LT . If there is not more than one `layer' of the mth estimate inside a L3 cube, then their result follows, D = 1 + Al / 123
'Moderate-order' systems
This chapter will contain a very loosely structured collection of concepts and results related to 'moderate-order' systems, beginning with fourth-order systems, and
some of the problems of representing 'very-high-order' systems (e.g., fluids) by moderate-order models. Obviously this latter objective cannot always be met, but there are some experimental tests which give indications if such a representation is possible, as will be discussed in Section 8.13.
The approach will be to first consider a few general aspects of linear equations, including Turing's instability, leading to chemical morphogenesis. Then we will consider integrable Hamiltonian systems, the KAM theorem and near-integrable Hamiltonian systems, and their relationship with earlier theorems of Poincare and Fermi. Next we will turn to more specific nonlinear models, beginning with the seminal computer study by Fermi, Pasta, and Ulam (FPU), and the surprising light it shed on some treasured concepts of irreversibility. Fermi found these results sufficiently surprising and important to call them `a little discovery'. Indeed these results stimulated research which ultimately led to the discovery of solitons and the inverse scattering transformation method, discussed in Chapter 9. The FPU results, and later computer studies inspired Toda to search for, and find, a nonlinear lattice with exact `soliton' solutions (now in a discrete lattice). The breakdown of this integrability, produced by introducing diatomic lattice particles, and some aspects of `irreversibility' are then discussed. This is followed by quite a different and mysterious perspective of 'integrable systems', suggested by their Painleve property. Next we explore the development of spatial structures in some coupled dissipative systems, and then examine Smale's interesting, but rather abstract, nonlinear example
(in R8), based on Turing's morphogenic equations. These studies initiate our considerations of nonlinear space-time patterns, which become more prominent in Chapters 9 and 10. Following this we will review a nice method of using empirical data to embed (faithfully represent) the asymptotic behavior of 'large-order' dissipative
systems in 'moderate-order' phase spaces. The empirical data may be either
'Moderate-order' systems
232
computational or experimental, and several examples of the latter use of this embedding method will be discussed. Finally, we will explore some elementary aspects of the dynamics of `living systems'. This area of dynamics attempts to define and analyze those properties and interactions
which are basic to `living' units at various levels of complexity. It is a vast, controversial, and very active area of research, and surely presents some of the greatest conceptual and analytic challenges in the field of nonlinear dynamics.
While the first nine sections of this chapter maintain a certain logical ordering, the topics in the following sections are largely autonomous in character, and can be read as your curiousity dictates.
8.1 Linear systems In this section we will consider a few simple features of equations obtained from a linear expansion about some known solution of (8.1.1)
(yeR").
Y = F(y, t)
From such linear equations we can examine the behavior of solutions which start near to the known solution, and thereby determine the stability (in the sense of Lyapunov) of the original solution. This method is most commonly applied to the study of the stability a fixed point, F(y) = 0, both because the stability is important to know, and because such solutions are easy to obtain. More generally, however, we can consider any known (exact) solution y(t) = y°(t) of (8.1.1), and consider another solution (xeR").
Y(t) = y°(t) + x(t)
(8.1.2)
Substituting this into (8.1.1), and expanding F(y°(t) + x(t), t) in a formal power series in x yields Y°(t) + x(t) = F(y°, t) +
(8.1.3)
xj(aFlay°) + ...
where
aF/ay° - aF(y°, t)/8y°
(8.1.4)
is now a known function of t. Since y°(t) is a solution of (8.1.1), the series (8.1.3) reduces to n
n
(a2F/8y°ay°)x;xj +
(aF/8y°)x; +
z(t)
i=i
.
(8.1.5)
i,;=1
The linear part of the series (8.1.5) is known as Poincare's variational equations,
x(t) = E [aF(y°(t),t)lay°lxj,
(8.1.6)
Linear systems
233
which can clearly bear a relationship to the solutions (8.1.2) of (8.1.1) only if x(t) is sufficiently small.
In particular, a necessary condition for the solution y°(t) of (8.1.1) to be stable (in the sense of Lyapunov) is that all solutions of (8.1.6), with initial conditions I x(to) I < E,
satisfy Ix(t)I < S(s) for all t > to and some 5(s), provided that s is sufficiently small.
Unfortunately it is generally very difficult to obtain solutions of the variational equations (8.1.6), even if we can obtain some exact solution, y°(t), which is needed for their construction. Indeed the two solutions, x(t) and y°(t), are closely related, as is
shown by Poincare's derivation of the variational equations. Poincare established that there is a relationship between a family of exact solutions of the nonlinear equation (8.1.1), and a solution of the variational equation (8.1.6). This result shows that it is as difficult to obtain solutions of the linear variational equations as it is to obtain some special family of solutions of the original nonlinear equations. To see this, consider an exact solution of (8.1.1), (8.1.7)
Y°(t) = 9(t; Yo),
where yo are the initial conditions 9(to;Yo) = Yo.
Assume now that we know a family of such solutions, characterized by some parametric relationship between the initial conditions; that is, all yo are functions of
some parameter, ,, so yo = yo(),). If we substitute (8.1.7) into (8.1.1), and then differentiate with respect to ,l, we obtain (aY°layoj)Yo j = E (aFlaY°)(aY°layoj)Yoj
dt
where yo j = dyo j/d,? . Comparing this with (8.1.6), we see that an exact solution of the
variational equations is x(t) _
(aY°/ayoj)Yoj j=1
where the y'0 are entirely arbitrary. If we know several such families of solutions, we can of course form linear combinations to obtain other solutions. In particular, if we can vary the initial conditions independently, then we can take the yoj to be arbitrary constants, aj, and obtain the solution
x(t)_
(aY°lay0j)aj j=1
and, if the Hessian does not vanish ayio l 960, aYok
(x,yeR"),
(8.1.8)
'Moderate-order' systems
234
then (8.1.8) is the general solution of the variational equation, (8.1.6). This result will be useful when we consider integrable Hamiltonian systems, later in this chapter.
Because of this difficulty in obtaining solutions of the variational equations, investigations usually concentrate on one of the following important special cases: (A) y°(t) is a fixed point (constant solution), F(y°) = 0; (B) Asymptotic properties of x(t) (the Lyapunov characteristic exponents) are obtained, usually numerically, provided that they exist; (C) y°(t) is a periodic function of time, y°(t + T) = y°(t) which leads to Floquet theory; (D) y°(t) is a quasi-periodic function of time (a particular type of sum of trigonometric functions, with a finite number of incommensurate periods). We will discuss (very) selected aspects of these cases.
Case A We first consider the case where y°(t) is a fixed point, which we can take to be yo = 0, so that (8.1.6) reduces to the form n
zi = > aijxj
(aid: constants; i = 1, ... , n).
(8.1.9)
Exercise 8.1 Show that, if x = x * is also a fixed point of (8.1.9), then the fixed points x = 0 and x = x* are connected by a continuum of fixed points. In other words x = 0 is the only possible isolated fixed point of (8.1.9). This, of course, is a major distinction between linear and nonlinear equations.
Such a system can always be transformed to one which involves only n parameters, rather than the n2 parameters, aid, which occur in (8.1.9). This can be accomplished in a variety of ways, one of which we now illustrate for n = 4 (and then follow by a general prescription). The point of this exercise is simply to show that an nth order linear system (8.1.7) can have at most n independent control parameters. The approach used here is exploratory and heuristic, rather than elegant. Consider the case n = 4. If we set y1 = x1 and then take
Y2 =
x1
a12
a13
x2
a22
x3
a32
... ...
x4
a42
...
a14
(8.1.10) a44
we obtain the sim pl e rel a ti onshi p Y2 = DoYI
where Do = det I aid 1.
(8.1.11)
Linear systems
235
We next search for a variable, y3, whose time derivative can be expressed in terms of only yl and Y2. The form of Y2 suggests that it might be useful to look at the time derivative of other determinants, such as d
dt
x1
a12
a13
x2
a22
a23
x3
a32
all
...
a13
a31
...
a33
...
a12
a14
+ x4
= x1
a33
a32
The last term is part of the expansion of the determinant which defines Y21 (8.1.10), and clearly we can get the other two parts (proportional to x2 and x3) by taking the time derivatives of xl
a14
a12
x2
a22
a24
x4
a42
a44
and
xl
a13
a14
x3
a33
a34
x4
a43
a44
The terms proportional to xl are, of course, not simply the ones obtained by expanding
(8.1.10), so the time derivatives of all of these do not just add up to Y2, but also involve yl. Specifically, if we take Y3=
xl
a12
a13
xl
a12
a14
xl
a13
a14
x2
a22
a23
x2
a22
a24
x3
a33
a34
x3
a32
a33
x4
a42
a44
x4
a43
a44
then we find that where DI
Y3 = D1Y1 -Y2,
=EMa Y=
1
Do.
(8.1.1 2)
(8.1.13)
aaii
In other words Mii is the (i, i) minor of Do, (8.1.11), obtained by striking out the ith row and ith column of Iaz5I. The expression (8.1.12) for y3 is not as elegant as the result in (8.1.13), where it is noted that Mii = aDo/aaii, but it is clear that (8.1.12) can likewise be expressed in this manner, for a Y3
aa44
+
a
aa33
+
a
(8.1.14)
aa22/ y2.
Note that (8.1.13) cannot be obtained from (8.1.9) simply by applying the differential operator in (8.1.14) to (8.1.9), because this operator does not generally commute with d/dt. The form (8.1.14), however, does suggest that the next useful function might be
-Y2= a2
Y4 =
i=2,3 aaiiaajj j>i
xl
a12
x1
a13
x2
a22
x3
a33
+
xl
a14
x4
a44
(8.1.15)
236
'Moderate-order' systems
Indeed we find that
where D2 = Y,
Y4 = Y1 D2 - Y3,
i = 2.3
j>i
az
Do.
(8.1.16)
aaiiaajj
That leaves only the equation for y, (= x,) to be determined. From the form of (8.1.11), (8.1.13) and (8.1.16), it is natural to expect that a3
where D3 =
.Y1 =D3Y1 -Y4,
Do.
(8.1.17)
i> j>kaa,,aajjaakk
Exercise 8.2 Confirm (8.1.16) and (8.1.17) and note that D3 is simply the trace of the matrix (D0). Collecting these results, we first set z4 = Y11 zk = Yk+ 1 (k = 1, 2, 3). Then we find that
the general fourth-order linear system (8.1.9), can be put in the form _10..,4
Dk-tz4-zk-1
dt
(8.1.18)
J
CZ 0
where 4
Dk =
ak
Do = det a; j .
D0;
E
i> j>...
(8.1.19)
aaiiaajj...
For an nth order system we only need to replace 4 by N in (8.1.18) and (8.1.19). Thus
there are N independent control parameters, (D0,... ,DN_1), and the general characteristic equation, for the characteristic exponents A, is N-1
(8.1.20)
(DN-1 = 1)
1)kDkAk = 0 k=0
The fixed point is stable if all roots of (8.1.18) have negative real parts, and unstable
if any root has a positive real part. In particular, a theorem, due to Hurwitz (e.g., Lancaster (1985) or Gantmacher (1959)), states that the necessary and sufficient condition for all roots, A,, of ai, 2" + a 1 A" -1 + + a" = 0 to have negative real parts is that all Ok > 0 (k = 0, ... ,), where ao = 1 and A2=
a1
O
a3
A3=
a2
a1
a3
a5
ao
a2
a4
0
a1
a3
(ak = 0, k > n). ai
Linear systems
237
Case B Just as in the case (A), one of the important properties of a solution y°(t), is its stability; that is the behavior of solutions of (8.1.1) which have initial conditions in
some neighborhood of y°(t). Fig. 8.1 shows a schematic illustration of an initial neighborhood of y°(t), which propagates to another location, shrinking in one direction, expanding in another, and remaining unchanged in yet another. We saw in the last chapter how these changes in three dimensions can be characterized in terms of Lyapunov characteristic exponents, )'1,)2,A3. The same description can clearly be used in any dimension, but the practical problem of computing the exponents becomes more difficult.
=o
It should be emphasized again that the Lyapunov exponents only measure the asymptotic exponential divergences of initially nearby solutions. The asymptotic statement means in the limit t - + oo, and nearby means in the limit e =I x(0) 1-+ 0 (in (8.1.2)). These two limits are not interchangeable; the E --> 0 limit must be taken first.
That is the reason why the variational equations (8.1.6) should be used, as was emphasized in the last chapter. Moreover, since a Lyapunov exponent does not give any indication of a separation going as t", we have seen from the power spectra of various Rossler attractors and the 'sticky-Island' effect (Karney, 1983), important
long-time (if not strictly asymptotic) properties are not detected by Lyapunov exponents. Indeed, such fundamental properties as irreversibility may be more directly related to `local exponents' in phase space (Mistriotis, 1985), rather than the
asymptotic Lyapunov exponents. Nonetheless they are of major importance in describing the behavior of a system.
Case C Floquet theory, which is discussed in many books (e.g., Yakubovich and Starzhinskii, 1975), concerns the case when the coefficients of the variational equations (8.1.6) are periodic functions of time x = a(t)x;
A(t + T) = A(t),
where A is an (n x n) matrix. A very important example of this in R2 is the Mathieu
equation (MacLachlan, 1964), which also arises in the coupling of modes in
'Moderate-order' systems
238
R 2" (Sait6, Hirotomi and Ichimura, 1975). We will return to this in the next chapter.
We will only note here that this case can be transformed, according to Lyapunov's reducibility theorem, into the case of constant coefficients (Case A) by introducing `normal coordinates', rather than the coordinates x(t) in (8.1.2). The idea is to first make a transformation to new variables (0, z1, Z2.... , z" _ 1) such that y(t) = y°(t) + B(O)z.
Here B(O) is an (n - 1) x n matrix satisfying B(0 + 2n) = B(0). This can be done in such a way the equations (8.1.1) become
0=w+g(0,z) 1= h(0, z)
where w = 2n/T and g(0, 0) = h(0, 0) = 0. These conditions reflect the fact that y = y°(t)
(i.e., z = 0) is a solution of (8.1.1) with period T. The assertion is that B(O) can be chosen such that the (n - 1) x (n - 1) matrix r = (ah(0, z)/az)Z=o
is independent of 0. This means that the variational equations associated with these normal coordinates now have constant coefficients 9 = w;
d(8z)/dt = F8z
and the eigenvalues F are known as the Floquet exponents. It is, of course, one thing to know that B(0) exists, and another thing to find it! Nonetheless the formulation of the dynamics in the (0, z) variables greatly simplified perturbative analyses (Moser, 1967).
Case D The case when the coefficients of the variational equations are almost periodic functions of the time is of great importance in perturbation theory. An almost periodic function f (t) can be expressed in terms of a Fourier series
f(t)=Eakexp(ik-cot)
k=(k1,...,ks)
(8.1.21)
k
such that the series Lk I ak 12 is convergent. Here we use the notation klwl + k2w2 +
+ ksws,
(8.1.22)
where s is some finite number, the {kj} are integers (positive or negative) and the {w1} are rationally independent frequencies (i.e., k-w = 0 only if k = 0). Such functions
Linear systems
239
of t arise naturally when a system of coupled oscillators (e.g., planets ) are analyzed by perturbation theory. Unfortunately very little is known about the convergence of such formal perturbation series, and it has been found useful, in order to obtain some
convergent results, to restrict the class of functions to so-called quasi-periodic functions (Siegel and Moser, 1971, Section 36). Quasi-periodic functions are a special subclass of almost periodic functions, whose Fourier series converges very rapidly. Specifically, if we measure the magnitude of k by IkI=Ikl+...+lksh
(8.1.23)
then f (t) is a quasi-periodic function of time provided that
Iakl < Mexp(- Iklp)
(8.1.24)
in (8.1.21), for some positive constants (M, p). Series of the form (8.1.21) are very common in perturbation analysis simply because,
when some approximate solution exp (ik (ot) is substituted in a nonlinear equation to obtain the next approximation, we obtain terms like (exp(ik'-(ot))m(exp(ik"-wt))" = exp [i(mk' + nk")-wt].
Since (mk' + nk") has only integer components, this again generates a function in the same class (8.1.22). However this term acts as an inhomogeneous (`forcing') term in obtaining the next approximation x(t), which then is given by
z = f (t).
(8.1.25)
An important question is, if f (t) is quasi-periodic, is the solution of (8.1.25) also quasi-periodic? In other words, will a perturbation series only generate functions which are quasi-periodic? The answer is generally no, but it is yes if the frequencies (01'..., ws satisfy a sufficient (but not necessary) condition, namely that for any nonzero (integer) vector k
CIkI',Ik-wI-1
(8.1.26)
for some positive constants (C, µ). This condition insures that we do not encounter the classic problem of small divisors in the perturbation analysis. More details on these subjects can be found in Appendix L concerning the KAM theorem. To see that x(t) is quasi-periodic if (8.1.24) and (8.1.26) are satisfied, note that the solution of (8.1.25), where f (t) is given by (8.1.21), is the same as the solution of Ei w;
where
X(wt)-x(t),
ae = F(9),
(8.1.27)
i
F(6)=1: akexp(ik-9),
0 = (w,t,...,cost)
'Moderate-order' systems
240
(and ao = 0). The solution of (8.1.27) is X(O)
Y
ak
k*Oi(k-w)
We need next the following fact (see Appendix L); for any k > 0, p > 0, 6 > 0 the following inequality holds: I k Iµ < (p/e)" exp (I k 16)6 - µ. Now, because of (8.1.24) and (8.1.26), we see that the coefficients bk satisfy IbkI < CIk1"'Mexp(- I k I p) < M'exp(- I k I p').
In other words the divisors do not become too small for any integer component vector k. Hence the solution x(t) = X(wt) is indeed quasi-periodic (and real, if f (t) is real).
The condition (8.1.26) not only requires that the set of frequencies (w, .... 1 (0.1) are rationally independent, but moreover that they belong to an irrational class which cannot be `very easily' approximated by rationale (i.e., using k with a small magnitude). For a brief discussion of this see Berry's review of chaotic dynamics (1978), or the
continued fraction study of Olds (1963). A classic example of approximating an irrational number is the theorem due to Hurwitz (1891; e.g., see Old, 1963), that states that any irrational number /3 has an infinite number of rational approximations, p/q, satisfying 1
< 51/2g2
(q
q
and that this is false if 51/2 is replaced by any larger number. While the condition (8.1.26) is not satisfied for all frequencies {w;}, and hence it is not generally possible to insure that the solution of (8.1.25) is quasi-periodic, one can draw upon Diophantine approximations to show that CIkIs+'
(8.1.28)
for `nearly all' w,, ... , w, - that is, the measure of the set {w; } not satisfying (8.1.28) goes to zero as C - 0 (see Arnold, 1963, p. 98, and Appendix L). The above concepts
will be encountered again when we consider the Kolmogorov-Arnold-Moser theorem.
8.2 Turing's linear chemical morphogenesis system In 1952 Turing published a paper which has greatly influenced research in a variety of fields, such as biology, ecology, and chemistry. The purpose of Turing's model was to explore how forms, structures, and patterns might grow from a rather homogeneous (nearly spatially uniform) situation - as, for example, in the growth of an embryo.
Turing's linear chemical morphogenesis system
241
To this end, he proposed a simplified dynamical model to investigate the process of morphogenesis (the evolution of morphological characteristics - the formation of physical structures). The first half of his study involved a system of linear equations, which we will consider here. Some nonlinear aspects of his theory (in which he made one of the early uses of a computer) will be discussed in a later section, which details Smale's very different analysis of such systems. Fig. 8.2
Turing considered a one-dimensional chain (Fig. 8.2) of identical cells (k = 1, ... , N),
each of which contained various chemicals (or enzymes), which he referred to as `morphogens'; that is, form producers. Generally each cell contains m such morphogens, where m may be a large number. The dynamics within each identical cell is described by a system of m coupled linear equations, whereas the coupling between the cells is modeled by a simple diffusion process through the cellular membrane.
As an example, assume that each cell has only two morphogens (m = 2), with concentrations (Xk, Yk). Consider small deviations from the equilibrium state, so that Xk = X ° + Xk, Yk = Y° + Yk (X ° >> xk, Y° >> yk). Then Turing's linear model is described by the (variational) equations )4 = axk + byk + p{(Xk+ l -xk) -(xk - xk- 1)]
Yk=Cxk+dYk+VC(Yk+1-Yk)-(Yk-Yk-1)I (8.2.1) where k = 1, ... , N, and boundary conditions will be specified later. Here p > 0 and v > 0 are `diffusion' coefficients for the morphogens x and y respectively. The equations
have been left in a form which makes it clear that this `diffusive coupling' between cell k and k + 1 is, for example, p(xk+ 1- xk). In other words, the coupling is nearest-neighbor, vectorial (.zk is only coupled to xk+1, not Yk±1), and' linear.. The coefficients (a, b, c, d) are constants, which we will assume yield stable dynamics for the isolated cells (p = v = 0). This means (Chapter 5) that the following conditions are satisfied
p=a+d<, 0;
q=ad-bc>0.
(8.2.2)
If the inequality holds, then the uncoupled dynamics in the morphogen phase spaces (Xk, yk) is either a stable node or focus. If p = 0 it is a center (see Chapter 5). An example is illustrated in a `mixed' representation involving both physical space and
the cells' morphogen phase spaces. The illustrated dynamics (Fig. 8.3) is for the uncoupled cells (p = v = 0), satisfying 4q > p2.
'Moderate-order' systems
242
Fig. 8.3
Morphogen phase spaces Y3
Y2
y1
oe.--r X
y4
00000'
Physical one-dimensional space
x2
3
4
Now everyone `knows' that diffusion 'smooths out' spatial inhomogeneities in a system - because it tends to `equalize the concentration' in adjacent regions. The very interesting and surprising result which Turing obtained - a result which probably contradicts most people's intuition and `obvious expectations' - is that this simple diffusive coupling can lead to a growth in an infinitesimal spatial inhomogeneity. In other words, instead of producing a spatial `smoothing' of the concentrations in the cells, this coupling can produce an increase in a spatial structure with time (a form of morphogenesis).
The proof of this is not difficult, and will be outlined below, but it is perhaps of greater importance to recognize why our intuition may have failed us. One reason is that most `diffusive' processes which we are familiar with (e.g., involving the density, or `heat content' of particles) concerns the transport of a scalar quantity. The dynamics
described by (8.2.1), however, involves a `diffusive coupling' of a vector quantity (x, y) - and we don't have much intuition (experience) along those lines. We could, alternatively, attempt to understand (8.2.1) by drawing upon our experience with interacting particles (in a plane). However, (8.2.1) does not fall into that category, because particles are not normally coupled in that vector `diffusive' fashion. Therefore we should simply recognize that at least some of us have little (no?) valid intuition about such systems, and proceed to develop some understanding. Following Turing, we will use periodic boundary conditions, so that Xk(t) = xk+N(t),
Yk(t) = Yk+N(t).
(8.2.3)
This can also be viewed as a single ring of N cells (connected ends), which, however,
has no common physical application. Other boundary conditions could easily be used (e.g., a finite system xo(t) __ x1(t), xN+1(t)
xN(t) in (8.2.1)), which would change
traveling waves into standing waves, but does not significantly influence stability properties. The standard method of analyzing such systems is to introduce `normal modes',
Turing's linear chemical morphogenesis system
with amplitudes
243
qs(t)), N-1
z
(xk, Yk) _ Y, (Ss, qs) exp (2niks/N)
(8.2.4)
S=O
which satisfy the boundary conditions and possess an orthogonality relation N
E exp (27ik(s - s')/N) = k=1
N (ifs - s' = 0, N), 0
(otherwise).
Using these in (8.2.1) we readily obtain the dynamic equations for the normal modes
_ (a - 4µ sin2
bq5
rls = cps + (d - 4vsin2 (ns/N))qs.
(8.2.5)
In contrast to (8.2.1), the equations (8.2.4) are now decoupled (for different s values), which are much easier to analyze. The value of s is related to the wavelength of the inhomogeneity in the concentrations, as seen from (8.2.4). We introduce the notation
a=sin 2(ns/N) as = a - 4µQ,
and look for the characteristic exponents characteristic equation
(l>a>, 0) da = d - 4va
(8.2.6)
q - exp (At)), using (03.2.5). This yields the
A = -(a,, + dQ) ± 2[(a, + da)2 + 4(bc - a,,d,,)] 112.
(8.2.7)
The behavior of the normal modes can therefore be analyzed in the same fashion as done in Chapter 5. Now, however, the control parameters are pa = (ao + da),
q,, = (aad, - bc),
and according to the conditions (8.2.2) the uncoupled system lies in the lower right (stable) quadrant of the control space. Examples of the (x, y) flows associated with the various regions of the control parameter space are illustrated in Fig. 8.4. As (µ, v, a)
are increased above zero, the parameter p, decreases, whereas qQ may change in a variety of ways, depending on the variation of a,d,,. Because of the behavior of p, the condition for the instability of the normal mode a is 0 > q, or (ad - bc) - 4a(dµ + av) + 16µva2 < 0.
(8.2.8)
We will refer to this change in the sign of qQ as the Turing (linear) bifurcation. Note that the fixed point becomes a saddle point rather than a spiral, as in the case of the Hopf bifurcation. The Hopf-like character of Turing's bifurcation can apparently be recovered in nonlinear systems (see Smale's analysis, in Section 8.12). Assume for the present that µv > 0 (neither diffusion is zero). Since q = (ad - bc) > 0,
244
'Moderate-order' systems
Fig. 8.4
PQ
by (8.2.2), (8.2.8) is satisfied only if the left side vanishes for some real value of a af
= d±+ av
± 8-v [(dµ + av)2 - 4,uv(ad - bc)] 1/2.
The unstable modes lie in the range a + > a > a-, and exist only if the lower root is real and lies between zero and one. It is useful to introduce
B= b/µ,
A= a/µ,
P=A+D,
C= c/v,
D= d/v
Q=AD-BC=q/µv>0.
Note that this does not simply amount to a renormalization of the constants in (8.2.1). The condition for instability, 1 > o-_, can then be written 8
P - (P2 - 4Q)'/2 > 0.
(8.2.9)
Reality requires that P2 > 4Q. The above conditions require P > 0 and Q > 0, whereas
the inequality (8.2.9) is satisfied for all Q if 8 >, P. If P > 8, then it requires that 4P > Q + 16. The resulting unstable region in the (P, Q) control space is illustrated in Fig. 8.5. The instability region is `nearly' the same as the unstable node region of the system x = (a/µ)x + (b/µ)y;
y = (c/v)x + (d/v)y.
(8.2.10)
Put another way, the ring of cells satisfying (8.2.2) is unstable only if (8.2.10) has an unstable node at the origin. Perhaps some `intuition' could be developed along these lines(?).
In the unstable region, the range of unstable modes is min (a+, 1) > a - sine (ns/N) > a _
which indicates that the intermediate wavelength formations are unstable. Note that, if 16 + Q > 4P then 1 > a+, and modes near s -- N/2 are stable (see Fig. 8.6).
Turing's linear chemical morphogenesis system
245
Fig. 8.5
Fig. 8.6
If v = 0 (for example), the system is unstable only if d>0
and
1 > (ad - bc)/4dµ.
(8.2.11)
This condition cannot, of course, be represented in the (P, Q) control space. Exercise 8.3 Ifav > 0, determine under what conditions there is at most one unstable mode with = 0m > 0. Obtain its o-. and Amax
Exercise 8.4 It is interesting to determine whether simpler cellular dynamics can exhibit Turing's `diffusive instability'. Assume that the cellular dynamics is simply X = - x (one-dimensional). Determine whether the following coupled cells can have an unstable origin:
(A) z= -x+D(y-x); y= -y+D(x-y) (B) x= -x+D(y-x)+D(z-x), y= -y+D(x-y)+D(z-y), i= -z+D(x-z)+D(y-z),
'Moderate-order' systems
246
where D > 0. For more complex cellular dynamics, see Smale's analysis in Section 8.12.
Having established that (8.2.1) can be unstable, there are a number of interesting questions which can be explored. One of the most important is to determine what `structure' is formed by the nonlinear system. This will be discussed in Section 8.11. Since Turing's original study there have been a number of generalizations of his linear analysis, such as Prigogine and coworkers [J. Chem. Phys. 46, 3542 (1967); ibid., 48, 1695 (1968); in International Conference on Theoretical Physics and Biology
ed. M. Marois, p. 23-52, (1969)], and the very general formulation of Othmer and Scriven (1971).
8.3 'Integrable' Hamiltonian systems Before discussing some specific models, we will consider a few general dynamical features of Hamiltonian systems Pk = - aH/aqk;
qk = aH/oPk
(k = I__ , n).
(8.3.1)
Generally, of course, it is not possible to integrate these equations explicitly, to obtain solutions of the form Pk = fk(t; p°, q°);
qk = 9k(t; p°, q°)
(8.3.2)
where the (p°ER", q°ER") are the initial conditions. There are, on the other hand, exceptions to this general rule - among them the so-called integrable Hamiltonian systems. These exceptions are, in fact, the examples which are invariably used to illustrate Hamiltonian dynamics. However these integrable Hamiltonian systems
are not common ('generic') even within Hamiltonian systems, nor are they structurally stable. This illustrates Thom's idea (1975, p. 29) that computability and structural stability may be incompatible requirements. A simple example of an integrable system is when the Hamiltonian is only a function of the pk, H(p, q) = K(P)
(PER").
(8.3.3)
Since
Pk = - aHlagk = - aK(P)lagk = 0;
qk = aH/apk = aKlaPk
all pk are constants, and hence we readily obtain solutions of the form (8.3.2) Pk(t) = pk;
qk(t) = (aKlaPk)Pot + qk,
(8.3.4)
so that we essentially have a generalized `free particle' situation. The Hamiltonian (8.3.3) is sometimes said to be in `normal form'.
'Integrable' Hamiltonian systems
247
Another standard example of an integrable system is a collection of n harmonic oscillators (with unit masses) n
H(P, q) =
1
2
2
k
(8.3.5)
+ wk2 q 2k ).
k=1
Then Pk = - wk 9k;
9k = Pk,
with the solutions (8.3.2) now of the form Pk = Ak COS (0Jkt + 4k);
qk = (Ak/wk) sin (wkt + 0k)
where 0)2
gklpk = tan Ok;
1)2)1/2.
Ak = ((P )2 + (q )2)1/2.
(8.3.6)
The `normal form' Hamiltonian (8.3.3) can be recovered in this case by using the canonical transformation to the 'action-angle' variables (4, Ok) qk = (21k/wk)112 sin O.
Pk = (2wklk)112 COS Ok;
(8.3.7)
The Hamiltonian (8.3.5) then becomes simply H(I, 0) _
(8.3.8)
(0kIk k=1
which is now in the normal form, (8.3.3), with the solutions
Ik(t)=lk;
Bk(t)=(Wkt+Bk
(k = 1,...,n).
(8.3.9)
Note that the variables ek are defined only modulo 2n by the definitions (8.3.7). That is to say that 9k = ek + n2n (n = ± 1, ± 2,...) are all the same points in phase space. The Bk are therefore referred to as cyclic variables. If n = 1, a trajectory in a suitable phase space (see Fig. 8.7) is simply a circle with radius (2w1°)1/2. This is an example of a one-dimensional torus, denoted as T1. If n = 2, the motion (8.3.9) is in four-dimensional phase space, but it is restricted
Fig. 8.7
P
'Moderate-order' systems
248
to a surface which satisfies the two conditions
(pt)+(wkgk)=2WkIk
(k = 1, 2).
Now any set of points xeR4 which satisfy the conditions
xi+x2=A,
x3+x4=B
(A, B: constants)
is called a two-dimensional torus, and denoted by T2. Fortunately this surface can be represented (embedded) in R3, because T2 is simply the surface of a doughnut (see Fig. 8.8.),
x1 =(r1 +r2sin02)cos01 x2 = (r1 + r2 sin 02) sin 01 X3 = r2 COS 02.
Fig. 8.8
Here the radii, rk, can be taken to be 2w111 and 2w212, with the largest energy being associated with r1 in Fig. 8.8 (e.g., r1 = 2(o111 > r2 = 2(0212). The dynamics can also be represented in the (01, 02) plane, using the fact that these are cyclic variables. Thus
the points at Ok = 2n are identified with points 0k = 0, keeping the same value of 0;(j :A k). A trajectory in this representation looks something like shown Fig. 8.9. There are two distinct possible types of trajectories which can occur on T2, as given by (8.3.9). The frequencies w1 and w2 are called rationally independent if Fig. 8.9
02
01
249
'Integrable' Hamiltonian systems
m1w1 + mZw2 = 0 (mk: integers) has no solution except m, = m2 = 0. Simple examples are, w, = 2112, w2 = 4, or w, = (12)1/2, W2 = 5. In this case the trajectory is everywhere dense on the surface of T2. That is, it can be found in any small region of the surface, and the motion is said to be ergodic on T2. Also, since the motion is simply the sum
of periodic parts, it is called quasi-periodic motion. On the other hand, if w, and w2 are rationally dependent, so that m,w, + mZw2 = 0 has a solution for nonzero integers (m1, m2), then the motion is periodic. An example of this is when w, = Fig. 8.10
2112, w2 =
(18)112.
02
01 02
01
These two cases are illustrated in Fig. 8.10. In either case the limit lim 01(t)/02(t) = w,/w2 = p.
(8.3.10)
exists and is called the rotation (winding) number of the trajectory on T2, as discussed in Chapter 6. One can obviously generalize this to the case of the motion in the 2n-dimensional phase space (8.3.7), which takes place on n-torus, T", that is, the set of points
T"= {peR",qER":pk +wkgk =2wklk, k
1,...,n}.
(8.3.11)
Unfortunately, if n > 2, it is not possible to represent T" in our R3 space. Although it cannot be done, it is fun to try to represent T 3 in R 3. Just as T 2 = T 1 x T 1 (the
direct product of two one-dimensional tori), we have T 3 = T 2 x V. If we do something like this in R 3 (see Fig. 8.11), we see that points in R 3 correspond to two points on T3. (o") are rationally independent, so that there are no solutions of If the "
Y_ mkwk = 0
(8.3.12)
k=1
= M. = 0, then we again have quasi-periodic motion which ergodically covers P. If the frequencies are rationally dependent, then the motion except m1 = m2 =
is periodic, and a trajectory will not visit some regions of T".
'Moderate-order' systems
250 Fig. 8.11
A less obvious integrable system is the 'Kepler problem' in R", defined by the Hamiltonian n
n
H = 12 Y_ pk -
(E q2)
k=1
-1/2
(8.3.13)
k=1
This system is obviously invariant under many rotations, and correspondingly the system has many constants of the motion, such as
Ki;=piqj -gipj
(I<_ i<j
(8.3.14)
Moreover these constants are clearly independent of one another (since they contain different independent variables). However these functions do not enjoy the property
of being `in involution' with each other. Two functions (F, G) are said to be in involution if their Poisson bracket vanishes (identically)
(aFaG aF aG _
{F,g}
aqi api
(8.3.15)
0.
api aqi
It will be recognized that, corresponding to two such classical functions there are two quantum observables which can be observed simultaneously (because the commutator of their operators vanishes). The concepts of independence and involution are unrelated. Thus (piq; - p;qi) and (piqj - pjgi)2 are two constants of the motion, which are in involution, but are clearly not independent. Moreover, the converse situation has just been noted. In the case of the system (8.3.13) a set of n independent constants of the motion, which are in involution, is (Moser, 1981) K 1 = H;
K. =
Y
(pigj - pjgi)2
(m = 2, 3,.
.., n).
1,i<,j5m
If n independent, smooth, global and single-valued constants of the motion can be
found, all of which are in involution, then it is possible to determine the other n
'Integrable' Hamiltonian systems
251
constants of the motion (e.g., Whittaker, 1944, p. 322 ff.). Such Hamiltonian systems
are now called, by definition, `integrable' systems (e.g., J. Moser, 1973, p.41; G. Birkhoff, 1927, p. 255 ff.). As Birkhoff noted, however, it is well to keep in mind Poincare's dictum that a system of differential equations is only more or less integrable. More specifically, it is rarely the case, even for integrable systems, that one actually proceeds to obtain the explicit solutions (8.3.2). However, for an integrable system, the n constants of the motion, K;(p, q) (i = 1, ... , n), define an n-dimensional surface in the phase space which is an n-dimensional torus, T" (e.g., Arnold, 1978). In other words the surface defined by K.(p, q) = ci (constants), is topologically equivalent to
the set (8.3.11). It is the information that the trajectories of an integrable system lie on such a smooth n-torus, T", which is of primary interest in many cases, rather than the explicit expressions (8.3.2).
In case you are left with the impression that all integrable systems are simple, consider the following example (due to Moser). Exercise 8.5 Consider the functions Gk(p, q) = apk + (b + c)Pkgk + dqk - (ad - bc) Y,
(Pigk
*k
ak
Pkgi)
2
ai
where k = 1, ... , N, (a, b, c, d) are constants, and {a;} are distinct constants. Determine for what (a, b, c, d) these functions are independent (i.e., when the differential d Gk are
linearly independent). Show that all the Gk are in involution. Finally, consider the Hamiltonian N
H(p,q)= Y- PkGk(p,q), k=1
Show that, for any choice of the constants, Pk = - aHlagk;
I'k,
the equations
qk = aHlaPk
are integrable.
Other physical examples of integrable systems, notably the Toda lattice, will be considered in following sections. It has been conjectured for some time that if all adjacent initial states only separate essentially linearly in time (i.e., averaging out the oscillatory motion), then the system is probably integrable. This has not yet been proved, but the converse is easy to show (Casati, Chirikov, and Ford, 1980). In other words we want to show that the rate of separation of adjacent states is, on the average, linear in time for bounded integrable Hamiltonian systems.
'Moderate-order' systems
252
Assume that the motion Pk=
0H
OH
qk=-
-apk
(8.3.16)
is bounded and integrable, so that (by definition) there exists an analytic canonical transformation (global) q, = Qk(1, 0);
Pk = Pk(1, 0)
(8.3.17)
such that
_OK(I)0;
1,=
=aK(I)=w , (I)
(8.3.18)
0, = w,(i°)t + 00.
(8.3.19)
O,
a0,
Oil
where K(I) = H(Pk, Qk). Then
I, = I°
and
Let (plc (t), q,°c (t)) be the solution of (8.3.16) corresponding to the initial conditions (I0,00 ) in (8.3.19), so P1c(t) = Pk(1°,w(1°)t + 00) (8.3.20)
qk (t) = Qk(10, (O(I°)t + 00).
To study the behavior of a nearby solution qk = qk + Sqk,
Pk = Pk + bPk;
(8.3.21)
where ON, bqk) are `small', we can use Poincare's variational equations, which were discussed in Section 8.1. The variational equations associated with (8.3.16) are z
pk
C(
H
)o(a2H
a9q)obql
agk PI 2
dtbgk = ' rLCaPk
2
H
H
P,/obP +
(8.3.22)
)Obql]
CaPkq,
where the coefficients (e.g., (a2H/agkap,)o) are evaluated at (p° (t), q°(t)). The solution of (8.3.22) can be expressed in terms of the derivatives of the functions (Pk, Qk), in (8.3.20), with respect to the initial conditions (1°, 0°). This was also shown in Section 8.1, (8.1.8). That is, the general solution of (8.3.22) is bPk =
y
010 0
bqk= [(LI +
()(tAwl + ee°)] (8.3.23)
() 0
Kolmogorov-Arnold-Moser theorem
253
where Awl - Y_k(0(0,(1°)/alk )Alk, and (AI°, A0°) are arbitrary constants. The pretty feature of (8.3.23) is that, for a bounded integrable system the functions (pk (t), qk (t)) are at worst almost periodic, so that the same is true of the differential coefficients in (8.3.23). Therefore, provided that some Aw, # 0, (8.3.23) shows that
superimposed on this almost periodic motion is a linear growth in the separation (Pk - Pk = bpk, 9k - qk = bqk) Note that this growth does not occur for adjacent orbits in harmonic systems, where all Aw, = 0. This shows that bqk `typically' looks something like Fig. 8.12. As we will see in our study of the Toda lattice (Casati, Chirikov, and Ford, 1980). Fig. 8.12
sqk
t
8.4 The Kolmogorov-Arnold-Moser theorem consider some of the ideas which enter into the Kolmogorov-Arnold-Moser (KAM) theorem. More details can be found in In this section we will
Appendix L.
We consider a near-integrable system, with a Hamiltonian (1, 0ER")
H = Ho(1) + EH1(1, 0)
(8.4.1)
and Ik
= - aH/a0k;
ek = aH/alk
(k = 1,..., N).
`Near-integrable' means that E is sufficiently small for the following results to be useful (e.g., for series to converge). Here H 1 is periodic in 01,. .. , 0" and 2
fH,(I,
0)d01...d0"=0,
J0 so
H1(1,0)=
(8.4.2) k40
Now, in the spirit of the Hamilton-Jacobi theory, we look for a generating function, S(1', 0), of a canonical transformation (8.4.3)
254
'Moderate-order' systems
so
0'=0+eaS,/al'+
(8.4.4)
which transforms the Hamiltonian to only a function of the new actions I'. We obtain
Ho(I) = Ho(I') +
4_
c1. ,
a0
+
al' a0
+ O(c3)
(8 . 4 . 5)
and E2
EH , (I , B) = EH , (1' , 0) +
aH, aSt + O( e 3) . al ae
(8 . 4 . 6)
Adding these, the term proportional to c can be made to vanish provided that H,(I', 0).
(0(1') as_-a0 0)
(8.4.7)
.
If Ht is analytic in a strip about the real 0 axes and periodic in 0, then we know (see Appendix L) that (8.4.2) is a quasi-periodic function. Moreover, we know from the discussion in Section 8.1 that the solution of (8.4.7) also gives a quasi-periodic function for St = E Ak(1')exp('k'B)
(8.4.8)
k
provided that, for all integer component vectors k lk.wi>C-IIkl-µ
(8.4.9)
In this case Ak(I') =
ihk(I') -
Ck'w(I')j
where
w(I) = aHo(I)/al.
(8.4.10)
The fact that S, is quasi-periodic insures that the series in (8.4.8) in fact converges, rather than being just a formal solution. Moreover, the subsequent iterations for S2, S3, ... will similarly all give quasi-periodic functions. It is important to note these frequencies can be made to satisfy (8.4.9) provided that they are independent functions
(nonconstant) of I. Namely that the following Hessian does not vanish* 02Ho(I )
al;alk #
0.
(8.4.11)
*From the implicit function theorem, the equations w(I) = co', where w° satisfy (8.4.9), can be solved for I = I(w°) if I aw;/alk10 0.
Kolmogorov-Arnold-Moser theorem
255
If (8.4.11) holds then most of the tori will satisfy (8.4.9) (with µ = N + 1), with the measure going to one as a - 0. However, densely scattered in the phase space are the resonant tori, for which (8.4.9) is not satisfied, and for which the series (8.4.3) diverges. This is connected with PoincarFs theorem which will be discussed in the next section.
What Moser (1962) and Arnold (1963) proved was a theorem suggested by Kolmogorov (1954). The principal problem is to show that for most of the tori which satisfy (8.4.9), the infinite series of quasi-periodic generating functions (8.4.3) actually converges for small (but finite) e. This result yields the famous theorem (Siegel and
Moser, 1971; Arnold, 1978), one version of which is (for another variant and for further details, see Appendix L):
Kolmogorov-Arnold-Moser (KAM) theorem Let Q be an open set of R" and let H(1, 0, e) be a real analytic function of (1, 0, e) for all IEQ, 0 ,<0 ,<2n, and e near c=0Assume also that H(1, 0, e) has period 2n in each 01, 02, ... ION, and that Ho(I) __ H(I, 0, 0)
(8.4.12)
is independent of 0. It is further assu med that for I E Q a2Ho(I )
(8 . 4 . 13)
a1kal, and that the corresponding frequencies w(1) __ aHo/al satisfy
Ik.wI >, CIkr-"
(8.4.14)
for all integer vectors
IkI + I k" I >1 1. Then, for sufficiently small a there k11+ exist solutions of (8.4.1) on `invariant tori', defined by
0'=0+F(0,s) I'=I+G(0,e)
(8.4.15)
where F and G are real analytic functions of e, and 0, having period 27r in each 01, ... , 6N and which vanish for e = 0. Moreover on these tori the flow satisfies
0k = wk =aHo/aIk
(k = 1,..., N).
.(8.4.16)
Finally, the measure of the states on the energy surface which lie on such invariant tori approaches one as a -> 0. In other words, for `sufficiently small' S. the measure of the states on invariant tori is `large' (the quoted expressions, however, are not quantitative).
As important as this theorem is, it does not give a means of actually determining
the relationship between the measure of invariant tori and the parameter B. In particular no quantitative values can be related to the above `large' and `sufficiently small' descriptions. Thus many practical questions remain unanswered, but in some
256
'Moderate-order' systems
particular systems (e.g., see the Henon-Heiles analysis) quantitative measures of the limits of this invariance have been attempted. We will now compare this result with a very influential theorem due to Poincare, and Fermi's 'theorem'- which he later helped to essentially show is incorrect! (See the Fermi-Pasta-Ulam phenomena.)
8.5 Poincare's and Fermi's theorems; Arnold diffusion In 1890, Poincare published a theorem which has had a great influence on the thinking of many scientists concerned with ergodic properties of systems, among them E. Fermi (1923). The general tendency among scientists was to draw conclusions from Poincare's theorem which are unjustified (Jackson, 1963), which is particularly
evident when the theorem is considered in the light of the KAM theorem, and to some extent even in an earlier result due to Whittaker (see below). Poincare's theorem (see 1957), Vol. 1, p. 233, or Whittaker, 1944, p. 385 states essentially that, if we have a Hamiltonian of the form assumed in the KAM theorem, H = Ho(I) + EH1(I, 0)
(8.5.1)
which is periodic in all the 0; variables, and whose Hessian of Ho does not identically vanish (see (8.4.13)) a2Ho(I) 00,
alkal,
(8.5.2)
then there exists no analytic, single-valued constant of the motion (aside from H) of the form (D_ Y E"4"(1, 0)
(8.5.3)
"=1
where the functions 4"(I, 0) are also periodic in the 0;, and differentiable in the I.. It should be emphasized that the family of constants of the motion which Poincare's theorem includes, (8.5.3), is very limited in its form - in particular, they are smooth functions of their variables.
Fermi (1923) built upon Poincare's theorem, but he went a step further in his assumption of smoothness. He was interested in establishing that most perturbed systems (E 0 0) are ergodic. A system is ergodic if all invariant sets of states on the energy manifold have only measure zero or one. Put another way, a system is not ergodic if there exist two invariant sets on the energy manifold, both with nonzero
measure. Fermi showed that the boundary between two such invariant sets is determined by a constant of the motion, ch(I, 0, E). He defined a `kanonische Normalsysteme' to be one which satisfies Poincare's conditions (8.5.1) and (8.5.2). He
Poincare's and Fermi's theorems
257
then made the assumption that the boundary between two invariant sets on the energy surface must be a member of the family of smooth constants of the motion, (8.5.3). The immediate conclusion is that a kanonische Normalsysteme is ergodic, since no such boundary can exist, because of Poincare's theorem. This conclusion,
however, is based upon Fermi's unjustified assumption that the boundary of an invariant set of positive measure must belong to the family (8.5.3) (Jackson, 1963; Moser, 1978, 1981; Petrosky and Prigogine, 1982). The possible importance of such a `nonsmooth' character of integrals of the motion was suggested long before (1916) by Whittaker's discovery of the `adelphic integrals'
(see Whittaker, 1944). He obtained constants of the motion in the form of formal power series in E, for a system of coupled harmonic oscillators, Ho = co111 + (0212, which does not satisfy Poincare's condition (8.5.2). For this system he obtained a convergent series, if a)1/(J2 is irrational, and two different series if w1/c02=m/n is rational, depending on whether H1(11,12,01,02) does, or does not contain a term cos(nO1 -m62). While he could not prove the convergence of these latter series in general, for all particular Hamiltonians he considered these series likewise converged.
Thus Whittaker proved that for at least these Hamiltonians there exist constants of the motion which have different analytic forms in (11, 12, 9,, 92) on two dense sets in the parameter space a)1/(o2, and a third form on a select set depending on the form of H1(I, 0). While this result is not directly applicable to Poincare's Hamiltonians, satisfying (8.5.2), it is very suggestive that when this condition is satisfied, so that the frequencies cok(I) = 3H0/3Ik vary independently, it will no longer be possible to obtain integrals which are analytic in the I;. Indeed the preserved tori of the KAM theorem
are only obtained by first fixing the frequencies, just as in Whittaker's case. These comparisons are, of course, only suggestive, but they perhaps add another insight into the KAM theorem. The KAM theorem shows that the system (8.5.1) and (8.5.2), for small E, has smooth
manifolds of dimension (2N - 2), namely the sufficiently nonresonant tori (8.4.14). These sets, however, are not smooth over the energy surface, but have a Cantor set
structure transverse to the tori. Stated differently, the orbits which comprise the (2N - 2)-dimensional boundaries of the invariant sets on the energy surface are dense
on this manifold. This is, rather foolishly and highly schematically, illustrated in Fig. 8.13 (it is difficult to draw sets of dense orbits!). The illustration, however, does partially convey the correct impression that the energy surface is `full of holes'. We might loosely describe it as a `Cantor sponge', that is a dense tangle of limited invariant regions. The `water' which leaks through the Cantor sponge consists of some of the `resonant' orbits which do not satisfy (8.4.14). Arnold (1964) in fact established that some of the unstable orbits can slowly escape to regions which are far from the invariant torus near which they start, and this effect has become known as 'Arnold diffusion'. It has been extensively studied by numerical
'Moderate-order' systems
258
H(p,q)=E
H(p,q)=E
Fig. 8.13
If invariant set then boundary is member of smooth family (8.5.3)
Fermi's
A Cantor sponge:
boundaries of invariant sets not members of smooth family
assumption
methods by Chirikov (1979) and others (Chirikov, Ford, and Vivaldi 1980). For small values of a the terminology `diffusion' may be misleading, however, since the Cantor
sponge is very dense in that limit, and the rate of increase of 11(t) - 1(0) 1 for the unstable orbits is very slow. For example Nekhoroshev (1971, 1977) has shown that, under certain conditions, all solutions of Ik
= - 8H/80k;
Ok
= 8H184
(8.5.4)
may remain near their initial location for `exponentially long' periods of time. More specifically, let IER" and p be a manifold in this space (R"), with a dimension > 0. Consider the gradient of Ho(I), restricted to p, grad Ho 1µ. Roughly, if this does not vanish for any p, then Ho(I) is called `steep'. For such systems, if M = max [cH1(1, 0)], then there are positive constants (a, b) which depend only on Ho, such that I 1(t) -1(0)1 < Mb
for all t < M-' exp(M-°).
(8.5.5)
In other words, for small M, 1(t) remains near 1(0) for very long times (- exp (1/M°). However, Chirikov, Ford, and Vivaldi (1980) found in a numerical study that they obtained, due to `technical difficulties', an Arnold diffusion which is much more rapid than Nekhoroshev's theory seemed to indicate. The need for the steepness requirement is illustrated by Nekhoroshev's example of channels of superconductivity, in the case
H-Ho+sH1=2(I;-I')+esin(0,-02).
(8.5.6)
Note that Ho satisfies the KAM condition, (8.4.13), 0,
so there exists a measurable set of invariant tori, when a is small. However, for (8.5.6), the equations (8.5.4) have the solution
I1= -to-et= -12
and
01 =02=0o-lot-Zet2.
Fermi-Pasta-Ulam phenomenon and equipartitioning
259
These solutions escape more rapidly than (8.5.5) along the 'superconducting channel' 1, + 12 = 0, along which Ho is constant (zero), and hence not `steep'. Note, however, the Ho in (8.5.6), is not a typical physical system. Other bounds have been obtained (e.g., Benettin, Galgani, and Giorgilli, 1985), and the interesting relationship to the holonomic constraints of classical mechanics has been studied (Benettin, Galgani, and Giorgilli, 1987); but we now turn to one of the earliest indications of such bounds, the Fermi-Pasta-Ulam phenomenon.
8.6 The Fermi-Pasta-Ulam (FPU) phenomenon and equipartitioning In 1954 Kolmogorov addressed the International Congress of Mathematicians (see an Appendix of Abraham and Marsden, 1978, for a translation of this address), in which he outlined the proof of the stability of most near-integrable solutions, and which
later became the famous KAM theorem. Halfway around the world, Fermi, Pasta and Ulam were concurrently investigating the 'energy-sharing' properties of the linear normal mode energies in certain nonlinear lattice models, with the help of Mary Tsingou and the MANIAC I computer at Los Alamos. These two investigations, from very different perspectives, were in fact closely related to each other. The FPU investigation was directly related to Fermi's longstanding interest in statistical mechanics, as illustrated by his early theorem (last section) together with many subsequent contributions. He was particularly interested in the process of relaxation to thermal equilibrium. The study of nonlinear lattices, in which the linear modes are coupled by nonlinearities, seemed to be an ideal system for a manageable study of the relaxation of some nonequilibrium distribution of these mode energies back to the presumed equipartitioning of their energies. This equipartitioning of the mode energies was expected, of course, because of the presumed ergodic nature of a nonlinear lattice (this general association of equipartitioning with ergodic properties is now appreciated to be incorrect - see Section 8.9). It will also be recalled that Fermi thought that he had proved the ergodic character of such lattices, so the question which FPU sought to investigate was not whether the system relaxes to equilibrium, but simply how long it takes for this relaxation to take place. To investigate this phenomenon, they studied the dynamics of several one-dimensional
nonlinear lattices. The equations of motion involved nearest neighbor interactions d2xk/dt2 = F(xk + 1 - xk) - F(xk - xk - 1)
(k = 1, ... , N)
(8.6.1)
where xk is the displacement of the kth particle of unit mass. They investigated several forces between the particles, F2(z) = z + Az2 F3(z) = z + Bz3
(8.6.2)
'Moderate-order' systems
260 Fig. 8.14
z 11
or a `broken linear' force, with the same S-shape as the cubic force, F3. These are illustrated (Fig. 8.14) for A > 0 and B < 0 ('soft' nonlinearity), and a similar broken linear force.
Introducing the linear modes x = (2/N + 1)'12Eak sin (kn)
k=lir/(N+1)
(1=0,...,N)
(8.6.3)
(8.6.4)
the frequencies of the linear equations are found to be wk = 2 sin (k/2)
(8.6.5)
and the energy of the linear modes are Ek - 2(ak +
(02
2
ak) = wklk.
(8.6.6)
FPU studied the dynamics given by (8.6.1) and (8.6.2), for the cases A = a and A = 1, and for several values of N (in particular, N = 32). They initially put all of the energy into the lowest mode, k = 1, and wanted to determine the time required for this energy
to become equally distributed (on the average) over the other linear modes. Thus what was expected is something like shown in Fig. 8.15. What was actually found for the quartic force is shown in Figs. 8.16(a) and (b), one for A = a and the other for the stronger nonlinearity A = 1. These results are a striking indication that the energy does not become rapidly distributed over the available 32 modes of this system, but instead is shared by only a few modes, and in the case A = 4, nearly all returns to the origin mode. If we represent the longitudinal displacement, xk(t), of the particles vs the particles' equilibrium position (for easier visualization), then we get the FPU figure (8.17). The numbers on the curves are w,t/2n, which is the number of periods of the harmonic lattice in the excited mode. We see that although the initially simple sine-displacement
becomes badly distorted, it all `recovers' in about 156 periods, in accordance with Fig. 8.16.
This calculation was repeated much later by Tuck and Menzel (1972), who showed that this near-recurrence of the energy of the first mode became poorer in the next recurrence. They found, however, that after seven progressively worsening recurrences,
Fermi-Pasta- Ulam phenomenon and equipartitioning
261
Fig. 8.15
t
the process reversed, and after fourteen recurrences the first mode's energy was actually nearer the original value than after the first recurrence. In other words, they observed a form of 'super-recurrence'.
The study of the dynamics with A = 1, or other initial conditions and/or other interparticle forces, did not all show this simple form of recurrence. However, even the extremely nonlinear case of a broken linear force between particles failed to indicate that there was any tendency to produce equipartitioning between all of the linear modes. Fermi felt that these results'... constituted a little discovery...', as Ulam later recounted (1965, 1983), but unfortunately Fermi died (November, 1954) prior to the writing of their report (May, 1955). He had also intended to talk about this topic at the Gibbs Lecture, which he had been invited to give, but he became ill prior to this meeting of the American Mathematical Society. It would have been most interesting to have had Fermi's more considered judgement of the physical importance of this `little discovery'. As it was, this report was never published, except years later in Fermi's collected works (1965). How times have changed! However, Ulam (1982) did recall another reaction of Fermi which is quite interesting: 'Fermi expressed the conjecture that in problems like the one we studied (the
'Moderate-order' systems
262 Fig. 8.16
40
160
120
80
wl t/2Tr (a)
8
20
40
60
wl t/2a (b)
80
Fermi-Pasta- Ulam phenomenon and equipartitioning
263
Fig. 8.17
Position of the mass point
vibrating nonlinear string cited earlier), the phase space may consist of a number of separate large regions connected by narrow bridges, so that the rate of approach to situations envisaged in statistical mechanics might be extremely slow'. Remember, this is all long before the KAM theorem was known, much less such concepts as Arnold diffusion, or the results of Nekhoroshev (end of the last section). The long relaxation times of Fermi and Nekhoroshev are very important in nonequilibrium process, as will be illustrated shortly (lattice heat conduction). Much later, Northcote and Potts (1964) discovered a very nonlinear lattice which exhibited equipartitioning of the energy among the modes, provided that the energy is sufficiently large. The lattice model they considered consisted of a 'harmonic-plushard-core' force, namely
F(z) = - oo
(z = - d):
F(z) = µz
(z > -d).
(8.6.7)
Among other cases, they investigated the same lowest mode excitation which was used by FPU, taking the initial energy to be E = NE. Here s = is the potential energy per particle required for a hard-core collision. In other words, there is sufficient energy in the system for all particles to simultaneously make a hard-core collision. They found that, when 1,500 hard-core collisions had taken place in a lattice with N = 15, the percentage of the time-averaged energy of the 15 modes is distributed in the way which is shown in Fig. 8.18. The figure on the right shows the distribution under the same circumstance except that the initial condition involved putting all of the energy in the highest mode (k = 15). It can be seen that there is essentially no 2µd2
'Moderate-order' systems
264
Fig. 8.18 100(Ek)/E
100(Ek)/E 10
10
5
5
1
5
10
15 k
1
5
10
15 k
difference in the two cases, indicating again the general nature of this equipartitioning process.
The Northcote-Potts model clearly exhibits another feature, which remains an active area of interest. If the energy in this system is less than E = (2µ)d2, no hard-core
collisions can take place, so there can be no equipartitioning of this energy among the modes. Clearly some type of transition towards equipartitioning must occur as E is increased (Saito, Ooyama, and Aizawa, 1970). A fundamental question is whether there is, for `generic' (usual) lattices, a critical energy, E, such that most initial states
lead to equipartitioning when E > E, and how Ec(N) depends on N in the `thermo-dynamic limit', N - oo. Unfortunately this is the type of question upon which
the KAM theorem sheds no light, but research continues on these basic questions (e.g., Budinsky and Bountis, 1983; Livi, Pettini, Ruffo, Sparpaglione, and Vulpiani, 1985).
It is important to emphasize that, even though a lattice exhibits equipartitioning of energy among its linear modes, it does not imply that this system is ergodic, much less that it is mixing. A simple example of this fact is the case of two modes, whose time-varying energies are El = 1 +a cost (t);
E2 = 1 + a sine (t)
which obviously do not go to many parts of the energy surface, even though the energy is equipartitioned. This example is quite trivial, but a nontrivial example (Ford, Stoddard, and Turner, 1973) will be discussed in Section 8.9.
It is worth remarking that the study by Fermi, Pasta and Ulam shared in the age of computer studies of nonlinear dynamics. As we have already seen many times in these studies, this interplay between analysis and computers, to further our understanding of nature, has now become general practice. The importance of this interplay, which Ulam (1960) emphasized, and referred to as `synergetics', has been adopted by and adapted to several research philosophies (Zabusky, 1981; Haken, 1983). However,
Fermi-Pasta- Ulam phenomenon and equipartitioning
265
the importance of the computer, and the lessons we are learning from it, may even exceed the importance anticipated by `synergeticists'. There is good reason to believe that the field of computer science, which has historically been viewed as a `computing
service', may now have reached a level of maturity where it can contribute its perspective to the description of nature, and thereby shine a new light on to the fundamental formulation of physical theories. Some of these points will be discussed in Chapter 10. Before considering such aspects of computers, we will see how they were used to
investigate the significance of the FPU phenomena on the historic problem of `irreversibility'. As mentioned above, Fermi's interest in these studies was to see an explicit example of the relaxation to equilibrium of a simple conservative, deterministic system; in short, to see an `irreversible process' in a `dynamically reversible' system. The fact that this was not observed revived the old, and very basic problem of irreversibility in nonequilibrium statistical mechanics, and elsewhere. In addition to the relaxation of a lattice to a state of equipartitioned energy between
its modes, there are other irreversible lattice processes which can be investigated. One such classic process is the heat conduction of a lattice. In the real world, a three-dimensional solid is placed between two plates, which are maintained (e.g., by electrical heating, etc.) at different temperatures, T1 : T2 (Fig. 8.19). The objective of the study is to see how much energy per unit time ('heat Fig. 8.19
J k= 1
2
T1
3...
.-0 N
x=0
x=L
flux') is exchanged between the plates, as a function of T1, T2 and the type of solid. Only the steady state (time-independent) situation will be considered here. If J is the heat flux, the standard equation which relates these variables is known as Fick's law of heat conduction,
J = - K dT(x)/dx,
(8.6.8)
where K is the coefficient of heat conductivity (dependent on the type of solid). This is a well-accepted, noncontroversial `law' of solid state physics. Since J is constant everywhere in the solid (conservation of energy), this predicts that
T(x) = T1 - (J/K)x
(T(0) = T1),
(8.6.9)
and if T(L) = T2, the heat flux must be J = (K/L)(T1 - T2). Thus J > 0 if T1 > T2 (see Fig. 8.19).
266
'Moderate-order' systems
The FPU system we will consider, drastically simplifies this real system, and replaces
it with a one-dimensional lattice. This can be, and has been generalized to two-dimensional lattice computations (see lattice dynamics references for more general
discussions). We will only illustrate here the effect of the FPU phenomena on the heat conduction of one-dimensional lattices. As we will see, the FPU phenomena is as dramatic in this situation as it was in their original study. As can be seen from (8.6.9) the spatial variation of the temperature (time-averaged kinetic energy of the particles) is expected to be a linear function of x. When a harmonic lattice of 100 particles is placed between the reservoirs, and the equations of motion are integrated for a long time, the computed `temperature' is found to be as illustrated in Fig. 8.20 (Jackson, Pasta and Waters, 1968). The `applied gradient' Fig. 8.20 1.5
1.4 1.3
1.2 1.1
1.0
0.9 0.8 0.7
0.6 0.5
0
10
20
30
40
50
60
70
80
90
100
Particle number
refers to the solution (8.6.9), and clearly the observed temperature shows no such gradient. Indeed the observed symmetry is remarkable (these are time-averaged values).
The space-time behavior of this system is shown in Fig. 8.21 where a vertical wiggly line is the trajectory of a particle. The transmission of pulses (hence energy packets) across the lattice can be clearly seen. These pulses slow down when they cross each other, but there is no obvious `irreversible' (scattering) effects on this spatial scale. The cause of this delay in a harmonic lattice (where solutions superimpose hence do not interact) is discussed in Exercise 9.1. The above result for a harmonic lattice has not yet been predicted by any analysis, which illustrates the important of computer solutions. However, the fact that Fick's law is not satisfied by a harmonic lattice is not surprising. What was unexpected is
Fermi-Pasta- Ulam phenomenon and equipartitioning
267
Fig. 8.21
Particle number
Fig. 8.22 1.5
1.4 1.3 1.2 1.1
1.0
0.9 0.8
0.7 i 0.6
0
10
20
30
40
50
60
Particle number
70
80
90
100
268
'Moderate-order' systems
how hard it is to obtain `normal' heat conduction in a one-dimensional nonlinear lattice. Figure 8.22 illustrates a small gradient of the temperature when there is a large
nonlinear force (F = Z + 10x2 + (200/3)x3), but there remains a large temperature jump at the boundaries (small effects of this type in fact do occur). Moreover the appearance of a temperature gradient does not imply that the system has a normal heat conduction (see Section 8.9). The dynamics of the pulses now `straighten out', as illustrated in Fig. 8.23. We will see that these are associated with the phenomena Fig. 8.23 350
0
Particle number
of solitons. They again do not represent an irreversible effect. How all of these phenomena are interrelated is presently unknown. Thus the FPU phenomenon is exhibited in several ways, and continues to present a challenge to our understanding of irreversible phenomena in such systems.
8.7 Molecular models As a precursor to the study of some lattice systems in more detail, we consider some
very simple, but instructive, (one-dimensional) molecular models illustrated in Fig. 8.24. The analysis in this section is largely based on the observations of Lunsford and Ford (1972) and Ford, Stoddard, and Turner (1973).
Molecular models Fig. 8.24
269
v3
X1
V1
Y2
V2
Three unit masses are connected by springs, whose equilibrium lengths (11, 12, 13) satisfy 11 + 12 = 13
(8.7.1)
and which have potential energies V1(x2 - x1 -11), V2(x3 - x2 -12), and V3(x3 x1 -13). We eliminate the Ik (k = 1, 2,3) by simply setting x1 = Q1, x2 =11 + Q2, x3 =13 + Q3, using (8.7.1) in V2. Since this is a simple point transformation, the new momenta are Pk = Qk, so the Hamiltonian is
H=2k=1
Pk + V1(Q2 - Q1) + V2(Q3 - Q2) + V3(Q3 - Q1)
(8.7.2)
Notice that the variables Qk do not indicate the relative ordering of the particles in space, as do the variables Xk.
Exercise 8.6 Consider an infinite, one-dimensional `lattice', with a common interaction potential, 1(Qk+ 1 - Qk), between the particles. If the initial conditions of this system satisfy the `periodic boundary conditions' Q4 = Q1, P4 = P1, Qo = Q3, PO = P3, determine the relationship which must exist between 1(r) and the above potentials Vk(r) (k = 1, 2, 3), in order for three lattice particles to have the same equations of motion as the above molecular particles, (8.7.2). The above selection of variables is not the simplest selection, because it is obvious that this system has two trivial dynamic variables, the location of the center of mass and the total momentum. That is, since d
3
(klPk)=o
(8.7.3)
there is a canonical transformation such that the `center of mass' coordinate b l = a(Q1 + Q2 + Q3)
(a = some constant)
(8.7.4)
is an ignorable coordinate. We can accomplish this transformation with a simple point (contact)
'Moderate-order' systems
270
transformation, using the generating function F(Q,'i) = E aijQigj, i.j
so that OF
Pi =OQi - = E jaijnj and
j=
OF aa
Vnj
= Y_ aijQ,.
(8.7.5)
i
Comparing the last expression with (8.7.4), we want all = a21 = a31 =a.
(8.7.6)
We would also like the new Hamiltonian to be quadratic in the new momenta 2Pk = k
k
i,j
akiakjfi1j =
,
so the matrix (ai j) will be taken to be orthogonal I akiakj = ai j.
(8.7.7)
k
A matrix which satisfies (8.7.6) and (8.7.7) is _2-1/2
_6-1/2 + (2)+ 1/2
A= +2-1/2
(8.7.8)
_6-1/2
The inverse matrix, A -1 = (ai j), which satisfies Y_ aikakj = Sij k
is, according to (8.7.7), simply the transpose of A, (i.e., ask = aki). We therefore have from (8.7.5)
bj = E ajiQi so
> akj 1J = Y
(akJa;i)Qi = Qk J
Molecular models
271
Specifically, we are interested in the combinations K
Q2 - Q1 = (a22 - a12)b2 + (a23 -
Q3 - Q2 = (a32 -
/
KK
2- 1/252 + \2)1/2S3
(a33 - a23)SZz3 = 2- 1/2xx2 - (3)1/253
Q3 - Q1 = (a32 - a12)b2 + (a33 - a13)S3 = 2+ 1/2b2
Therefore the new Hamiltonian is H = 1 Y, nk + V1(2- 1/252 +(3)112 253) 2k=1
+
V2(2- 1/22
- (2)1/23) + V3(2+ 112b2)(8.7.9)
We see that
does not appear in (8.7.9) (it is an ignorable coordinate) and therefore
which is a restatement of (8.7.3). We will therefore not explicitly retain q , in the following (we are in the center of mass coordinate system). We next consider some possible potentials Vk(r). An important class of potentials is the polynomial potentials V(r) =' K2r2 +
3K3r3
+
4K4r4
(8.7.10)
which, of course, are simply the first few terms of a Taylor expansion of a general function V(r) about its equilibrium position, (aV/ar) = 0. Even the restricted class (8.7.10) has not been thoroughly studied, and we will at first restrict our considerations to the case V(r) = !r2 + 3r3
(8.7.11)
If a is negative, the spring is easier to stretch ('soft' nonlinearity) and more difficult to compress than the harmonic spring (see Fig. 8.25), and vice versa if a > 0 ('hard' nonlinearity).
Fig. 8.25
V(r)
a=0
1 1
'Moderate-order' systems
272
To make a connection with the Henon-Heiles Hamiltonian, discussed in Chapter 6, we consider the rather unusual case Vl (r) = V2(r) = 2 r2 + r3;
V3 (r) = 2 r2
3 r3
(8.7.12)
3
In other words, if a is positive, the two inner springs are `hard' to expansion, whereas the outer spring is `soft' to expansion (note the relationship with an infinite lattice, Exercise 8.6). The system as a whole is dominated by the outer spring, and hence is
unstable to large enough expansion. Using (8.7.12) in (8.7.9), we obtain the Hamiltonian H = 1 Y, nk + 2k=2
2+ 2(
3)
+ (3a/2'I2)(S2S3 - 3S2)
and introducing scaled time and space variables i = 31j2t,
q2 = (a/2'/2K2,
q1 = (a/2112)S3
one obtains the (scaled) Henon-Heiles Hamiltonian H=
z(p2 + p2 + q2 + q2) + g2g2 - 3q3
(8.7.13)
which will be discussed further in Section 8.9. Note that positive values of q2 correspond to expanding the springs (if a > 0). The potential energy has a saddle point at (q 1 = 0, q2 = 1), so if H > i - 3 = 6 the solutions are unbounded.
Exercise 8.7 Obtain the Hamiltonian for the case where V3(r) is the same as V1(r) and V2(r) in (8.7.12). Above what value of H is this system unbounded, and what are the asymptotic values of (Q1, Q2) when there is an instability? Is this model of a molecule unrealistic for unstable solutions? Exercise 8.8 Show that if we `only' change the sign of one term in the Henon-Heiles Hamiltonian, so that now
H=2 1+p2+q2+q2)+giq2+3821
(8.7.14)
then this system can easily be integrated. If it is not clear that this Hamiltonian can be put in the form H1(p1, q1) + H1(p2, q2) - in other words, it separates - return to the equations of motion, and look for Qk, that are linear combinations of q1 and q2, and are separable. If V3(r) = Zr2 + 3/3r3, what is the relationship between /3 and a of V1(r) = V2(r) which yields this integrable case?
Other less obvious integrable cases, when
H=2(p2+p2+Aq2+Bq2)+q q2+3q3
(8.7.15)
Molecular models
273
are C = 6 (any A, B) and B = 16A, C = 16. By a suitable selection of V3(r) we could
obtain either of these Hamiltonians (see the section on the Painleve property and integrability conjecture). Another important example is the so-called Toda potential V(r) _ (alb) a - n. + ar Fig. 8.26
(ab > 0),
(8.7.16)
V(r)
r
where a and b may either be both positive or negative (see Fig. 8.26). In the case of
the three-particle molecule we take V, = V2 with (a = b = + 1), and V3 with
a = b = - 1. Again introducing the variables T = 3112 t, q2 = 2- 1/22, q, = 2- 1/223, we b
obtain the Hamiltonian
H = 2(p + P2) + i
3
[exp (q2 + 3112q 1) + exp(q2
- 3112q 1) + exp(- 2q2)] - I(8.7.17)
In contrast with the previous `molecule', (8.7.13), this molecule always has bounded motion. As we will see in the next several sections, the Toda potential, (8.7.16), occupies a very
special position in dynamical systems, because of the existence of global analytic constants of the motion, and the `regularity' in the dynamics of many systems with this form of interaction. It should be noted that potentials of the form (8.7.10) and (8.7.12) are not realistic for `large' values of r, since all particles have infinite repulsive cores at some finite distance, and neither of these model potentials exhibit this feature. One consequence of this feature of (8.7.10) and (8.7.12) is that the relative spatial ordering of the particles
(in one dimension) is not always properly taken into consideration. For example, a large enough negative value of S2 = 2- 112(x3 - x1 -13) corresponds to x3 < x1, in other words an inverting of the spatial order. This change in spatial ordering is both possible in energetic dynamics described by (8.7.13), (8.7.15), and yet not properly accounted for in the forms of the potentials (8.7.10), (8.7.14). Indeed, once an implicit
274
'Moderate-order' systems
ordering has been selected, the subsequent ordering is a mute point, since length scales (such as the Ik) no longer appear in the formalism. A much more realistic potential is the `Lennard-Jones' type
V(r)=K[(ro/r)m-(ro/r)"]
(r=lxi-x;I)
(8.7.18)
where m > n are integers, such as m = 12, n = 6. Other values of m and n can be reasonably used for lower-dimensional motion. Notice that now a scale length is retained (ro ), because the particles' positions, Xk, are used in V(I x, - x; 1), rather than
the variable Qk. The equilibrium separation is governed by ro, and the `binding energy' by the magnitude of K. Since V(r) -+ oo as r -- 0, the potential retains the ordering of particles in one dimension, and accounts properly for particle orientations Fig. 8.27
-(3k
in any dimensions. The potential is illustrated schematically in Fig. 8.27, where a = (m/n)"m-"). Dynamic studies involving this potential have been rather limited.
8.8 Toda's solitary waves in a lattice One of the consequences of the Fermi-Pasta-Ulam investigation was to stimulate attempts to obtain analytic solutions for nonlinear lattice systems (see Chapter 9 for a brief history of these studies). One of the spectacular successes of such searches was Toda's discovery (1967) that the rather unlikely looking potential, (8.7.16), V(r) = (alb) exp (- br) + ar,
(8.8.1)
yields a system of equations which can be solved analytically. The clever reasoning which led Toda to this discovery is explained by him in Chapter 2 of his book (1981). If we introduce the relative displacements rk = xk+ 1 - xk, and subtract between
equations of motion of the lattice (8.6.1), we readily obtain (for an infinite lattice) rk = F(rk + 1) + F(rk -1) - 2F(rk ).
(8.8.2)
Toda's solitary waves in a lattice
275
Inserting the Toda force F(r)=a(1 - e - br), this becomes Yk =
2ae-hrk
-
ae-
nrk+ 1
-
ae-hrk- 1.
By scaling the time and distance, br, r, (ab)l12t -+t, this simplifies to Toda's equation Yk = 2e-rk - e-rk' 1 - e-rk- 1.
(8.8.3)
Toda has lucidly discussed the many solutions of this equation in great detail in his book (1981), so our treatment will be relatively brief. Following Toda, we introduce the variable nk = e - rk - 1
(8.8.4)
into (8.8.3)
d
dt2ln(1 +rik)=nk-1 +nk+I -2i1k
and integrate, to obtain
dtln(I +N)=sk-I +sk+l -2sk,
(8.8.5)
where Sk =
J?lk(t)dt.
(8.8.6)
Since sk = rlk, (8.8.5) leads to Toda's final form §k AI +Sk)=Sk+I +Sk -
I - 2Sk.
(8.8.7)
It is not obvious that (8.8.7) has any manageable solutions, but Toda obtained exact solutions in terms of elliptic functions. We will consider only a few, more specialized, solutions to illustrate his results. We note that the right side of (8.8.7) is simply the sum of a few functions, whereas the left side is the ratio of some polynomial of functions. Thus, if we are to find a solution, the sum of the functions must equal a rational expression in these functions. One possibility is to note that tanh(u ± v) = [tanh(u) ± tanh (v)]/[ 1 ± tanh (u) tanh (v)],
which is a rational expression. Thinking of the particular form of the right side of (8.8.7), we might next consider (if we are as clever as Toda) fl[tanh (u + v) + tanh (u - v) - 2 tanh (u)].
(8.8.8)
'Moderate-order' systems
276
Using the previous identity, and sech2 (u) = 1 - tanh2 (u), a little algebra shows that (8.8.8) equals
- 2$ sinh2 (v) sech2 (u) tanh (u)/[ 1 + sinh2 (v) sech2 (u)],
(8.8.9)
which indeed looks like it could equal the left side of (8.8.7). Clearly we are anticipating that is essentially tanh (something). If we consider E(u) - /3 tanh (u), then
sech2 (u),
E'(u)
and 2$ sech2 (u) tanh (u).
e"
Hence E"
2$ sech2 (u) tank (u)
1+E'
1+/3sech2(u)
The right side is of the form (8.8.9), except we are missing some factors of /3 ' sinh (v), for it to be a solution of (8.8.7). However, if we set Sk(t) = E(/3t + ak),
with /3 also in the argument, we find that sk/(1 + sk) = /32E"/(1 + /3E') _ /3[tanh (u + v) + tank (u - v) - 2 tank (u)]
=Sk+1+Sk-1-2ikf provided that /32 = sinh2 (v)
and
v = a.
Hence one solution of (8.8.7) is sk(t) = /3 tank (fit + ak)
(/3 = ± sinh (a)).
(8.8.10)
Returning to variables rk = Xk + 1 - Xk, and using (8.8.4), and (8.8.6), this solution can be written
e-'k- 1 =sk=$2sech2(/3t+ak). exp -(xk+1 - xk) - 1
Fig. 8.28
--13<0
13>0
000 -Qt/a
k
(8.8.11)
Toda's solitary waves in a lattice
277
This result shows that rk < 0, so that the lattice is locally compressed. The nonlinear compression, (8.8.11), is schematically illustrated in Fig. 8.28. The dimensionless velocity of the shock wave is vo = - (3/a (or a dimensional velocity (a/b)1J2vo) which, according to (8.8.10), may be either positive or negative. Exercise 8.9 Show that an infinite succession of pulses,
e - 'k - 1 = E /32 sech2 [ft - a(k - Al)], is also a solution. Determine A. Such nonlinear periodic waves are called cnoidal waves (Fig. 8.29). This terminology was introduced by Korteweg and deVries (see Chapter 9). Fig. 8.29
A cnoidal wave
Exercise 8.10 The total compression of the lattice is C = limk- (x _k - x+k). Verify that Xk = In [(1 + exp (2a(k - 1) + 2#t))/1 + exp (2ak + 2ft)] + constant satisfies (8.8.11). What relationship between a and /3 is needed for this result? Obtain the relationship between the total compression, C, and the speed, vo, of the solitary pulse. This solution, for an infinite Toda lattice, is interesting, not only because it is exact,
but because the traveling disturbance does not change shape as it moves. This is not the usual behavior for disturbances in solids, liquids or gases; usually they spread
out (disperse) as they travel. This will be discussed further in Chapter 9 (see Exercise 9.1), where we consider other solitary solutions (solitons and nonsolitons). Historically Toda's solutions followed the discovery of solitons in continuous systems, but his results gave an important connection between this physical phenomena and the dynamics of discrete particles. A second, and much more interesting solution unfortunately requires a considerable amount of algebra to prove (to say nothing about cleverness, to find). A brief outline (see Toda, 1981) will be given here to illustrate the considerable subtlety of even the simpler solutions. q Toda first introduced the function that (8.8.5) yields (8.8.12)
'Moderate-order' systems
278
Because of Toda, we look for a solution of the form =1n [cosh (Kn - (3t) + B cosh (µn - yt + S)]
(8.8.13)
where (K, /3, u, y, 6, B > 0) are unknown constants. We can take K > 0, µ > 0 without loss in generality. Substituting (8.8.13) into (8.8.12) and suffering through some algebra, one finds that the arguments of both logarithmic functions on the two sides of (8.8.12), can be put in the form A, cosh2 (u) + A2 cosh2 (v) + A3, where
u - Kn - fit
and
v-µn-yt+6.
(8.8.14)
Equating these arguments, and treating u and v as independent variables, yields three conditions which the constants must satisfy, /32 - sinh2 (K) + y2B2 - B2 sinh2 (µ) = 0
2+ /32+y2-2 cosh (K) cosh (p)=0 fly = sinh (µ) sinh (K).
Expressing (/3, y, B) in terms of (K, p), we find that there are two possibilities; (a) /3 = 2 sinh (K/2) cosh (µ/2);
y = 2 cosh (K/2) sinh (p/2)
B = sinh (K/2)/sinh (p/2), (b) /3 = 2 cosh (K/2) sinh (p/2);
y = 2 sinh (K/2) cosh (p/2)
(8.8.15)
B = cosh (K/2)/cosh (p/2).
Actually (/3, y) can be replaced by (- /3, - y) in these expressions, but this just reverses all of the velocities.
Once again, returning to the displacements, r, and using (8.8.13), (8.8.14), some algebra produces
exp (-
1 = {sinh2 ()c) + B2 sinh2 (p) + 2B cosh (u + v) sinh2 ((K - µ)/2)
+ 2B cosh (u - v) sinh2 [(K + p)/2] }/[cosh (u) + B cosh (v)] 2.
(8.8.16)
Note that all the time dependence is contained in (u, v), (8.8.14). Also, again, all
<0. As t-- oo, if we consider values of n(t) such that JuJ/JvJ goes to either 0 or oo, then the right side of (8.8.16) vanishes, so rn(t) - 0. In other words, the disturbance has gone away. In order for rn(t) not to vanish, we must change n (i.e., move along the lattice) in such a way that, when Itl -> oo, either
u+v=c
or
u-v=c,
where c is a constant. That is, there are two possible ways to move along the lattice and continue to observe a disturbance; so these are now two isolated disturbances, for large t.
Toda's solitary waves in a lattice
279
For example, let n vary such that u = v + c. Then, as t - ± oo (I v i -> oo), (8.8.16) reduces to
exp (-
1-> 2B sinh2 ((ic - p)/2) cosh (2v + c)/[cosh (v + c) + B cosh (v)]2. (8.8.17)
The limiting behavior of the right side depends on whether v > 0 or v < 0 which, according to (8.8.14), is related to the limits t - + oo or t - - oo. To determine the sign of v as t - ± oo, consider first the case u - v = c, so that
(K-µ)n-(/3-y)t= c', or n=
I'-y K-µ It+c".
(8.8.18)
Therefore
L K-µ
(8.8.19)
K-µ
J
In case (a), (8.8.15), µ/3 - KY = 2 [µ sinh (K/2) cosh (u/2) - K cosh (K/2) sinh (µ/2)]
so we conclude that sign
- I < 0; K-µ 111
(Yfl
KY
hence sign (v)
sign (t).
The same conclusion is obtained for case (b). On the other hand, if u + v = c, we obtain from (8.8.16)
v=1 i
\ K+ µy)t+const.,
K+
(8.8.20)
from which it can be shown, in a similar manner, that sign (v) = sign (µ - K) sign (t). Therefore (8.8.16) yields two spatially separated compressional pulses (u ± v = c) as ItI ->oo,
lim [exp (- r) - 1 ] _ t
±00
2 ) sech2 [(u - v) + sign (t) In B]; sinh2 (K_u J µ )sech2(4(u+ v) + Z sign (t) sign (p - K) In B) sinh2 CK
2 (8.8.21)
which are valid for both case (a) and (b), (8.8.15). The structure of each separated disturbance is similar to the single disturbance (8.8.11), which is obviously reasonable.
What is not `obviously reasonable' is that two nonlinear disturbances can interact, but retain their same shapes in the limit t - + oo which they had as t - - oo; as is illustrated by the example (8.8.21).
'Moderate-order' systems
280
We note that the solution (8.8.21) is invariant under the interchange of u and K (note B in (8.8.15)), so we can set K > y without loss of generality. Thus sign (K - µ) > 0,
and In B > 0, in (8.8.21). Hence we can write these two asymptotic pulses more concisely as
lim (e-'^ - 1) = sinh2 (K
'u
) sech2 [2(u ± v)
sign (t) In B]
(8.8.22)
2 This finishes the algebra. It remains to physically interpret such solutions. Since
u±v=(K±µ)n-(/l±y)t+6
(8.8.23)
the two disturbances (8.8.22) have dimensionless velocities /3 ± y = 2 sinh ((K + p)/2)
(K ± p)
K ± It
x
+ 1 case (a) ± 1 case (b)
(8824)
In case (a) we have two disturbances, both with positive velocities, one faster, (/3 + y/K +,U), than the other, (j3 - y/K - p). In case (b) the fast disturbance has positive velocity, while the slow has a negative velocity. [Recall the remark following (8.8.15).] Note that, from (8.8.22), the faster pulse has the larger amplitude, so these two cases look schematically as in Fig. 8.30. Fig. 8.30 Case (a)
I
Case (b)
t<< 0 n
,P t
>0
I
n
As t goes from - oo to + oo, the arguments of sech in (8.8.22) shift by + In (B). Recalling (8.8.23), we conclude that the centers of these localized compressions are shifted by An = ± 2 In B/(K ± y). Hence the fast pulse has a smaller positive shift than
the negative shift of the slow pulse. This is illustrated in the `n-time' figure (8.31), where the lines (8.8.25) (u + v) + sign (t) In B = 0 z are plotted. According to (8.8.22), these are lines with maximum I r 1. Of course, these
lines are meaningful only for large It 1. If suitable masses are associated with the fast and slow pulses, the `center of mass' experiences no shift (see Exercise 8.11). S simply locates the `origin' of the interaction of these two pulses. It should be noted that this
Dynamics of various Toda lattices t
Fig. 8.31
-In B+6 K-p
281
`Center of mass'
/
/-In B-S K + t1
/
//
/ j
K-11
figure is not exactly a space-time diagram. An example of the latter can be seen in Section 8.6.
Exercise 8.11 Obtain the `n-time' diagram for the case (b), (8.8.15), noting the change in (8.8.24). Determine whether the phase shift is larger in case (a) or (b); Ion(K,µ)Icase(a)'< Ion(K,µ)Icase(b)? What must the mass of the fast and slow pulse be proportional to, in order for their `center of mass', mfnf + msn*, not to experience a phase shift? Here n* is where I rn I is maximum.
For many other interesting results, see Toda (1981). We now turn to a second spectacular discovery about Toda lattices (finite as well as infinite).
8.9 The dynamics of various Toda lattices A very important aspect of Henon and Heiles' results, discussed in Section 6.11, was
the observation that the area of the islands of coherent motion rapidly decreases, once the energy is above a critical value E, = 0.11 (Fig. 8.32). The largely regular behavior of the dynamics for E < Ec is a dramatic illustration of the KAM phenomena, namely the preservation of most of the invariant tori. Indeed the Henon-Heiles result
was the first quantitative indication of how much anharmonic energy might be required to break up the invariant tori. A second very important application of both Poincare's surface of section and Lyapunov exponents was made by Ford and coworkers, who investigated various Toda lattices of equal and unequal masses. We discuss first the results of Ford, Stoddard and Turner (1973) for the three-particle, equal-mass Toda lattice. They found that the Poincare map on the surface of section always yielded smooth curves,
'Moderate-order' systems
282
Fig. 8.32
1.0
0.9 0.8 0.7
0.6
0.5
0.4
0.3
0.2
0.1
01 0
0 02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Energy
Fig. 8.33
P2
9
q2
7DY E= 1
E=256
even for very high energies. The variation in these curves is illustrated in Fig. 8.33 for two energies, E = 1 and E = 256. They found that these curves remain smooth
for energies up to 56,000, which was the maximum for which they could retain computer accuracy. These results are obviously dramatically different from those found by Henon and Heiles.
Dynamics of various Toda lattices
283
A second important check which they made on the dynamics involved the separation of initially adjacent solutions. They established, for both three- and six-particle Toda lattices, that the separations 1/2 (I)
-P
(21 2
)
]
t/2.
(1)
-qt
(21)2
]
(8.9.1)
for two different solutions (q(l ), p(1)) and (q(2), p(2)) grow linearly in time (on the average), as illustrated in Fig. 8.34. This linear growth in time occurs for all energies Fig. 8.34 50
L#T .I
Ii
I
20
to
10
20
30
40
50
Time (s) 50
40
I0
I
10
I
20
I
I
30 40 Time (s)
I
50
which they tested (e.g., 1024 and 132 for three and six particles respectively). As was discussed in Section 8.3, this linear separation of adjacent states is characteristic of integrable Hamiltonian systems. Recall from Section 8.8 that this three-particle Toda lattice can be reduced to two degrees of freedom. They found that this system, with the Hamiltonian
H = (pi + pi) + (6) [exp (q2 + 31/2g1) + exp(q2 - 3112g1) + exp (- 2g2)7 - 2
i
284
'Moderate-order' systems
has a second independent constant of the motion, C=2(Pi -3Pi)P1 +{(P1 +3112P2)eXp[(g2-311281)
+(p1 -3112P2)exp[(g2+311291)]-2p1eXp(-282)}. Moreover C is in involution with H, because in general dC dt
aaC
at
+ {C, H}
and since 8C/at = 0 and dC/dt = 0, then {C, H} = 0. Thus this system is integrable (Whittaker, 1944).
Exercise 8.12 Show that dC/dt - 0. Think about how this constant differs (in character) from II, and how you might use it to better understand the dynamics of this system.
Based on these results and similar results for a six-particle Toda lattice, Ford, Stoddard and Turner suggested that the Toda lattice might be (generally) integrable. Shortly after this prediction Henon (1974), Manakov (1974) and Flaschka (1974) independently proved that the Toda lattice is integrable. Recalling that the Toda equations of motion are Qn = Pn (8
Pn = C[exp (Qn
.
9 . 2)
-1- Qn) - exp (Qn - Qn+ 1)],
or in terms of the variables
XnCexp(Qn-Qn+1);
-Xn;
(8.9.3)
Xn=(Pn-Pn+1)Xn.
(8.9.4)
The periodic boundary condition are Pn+N = Pn'
Xn+N = Xn.
(8.9.5)
In terms of these variables, Henon gave the following expressions for the n independent
constants of the motion which are in involution In=EPi,...Pjk(-Xii)...(-X;,)
(k+21=n)
(8.9.6)
where all indices i 1 , i2, ... , ik, 11 , j 1 + 1,... , j,, j, + 1 are distinct, and permutations are not counted in different terms. To illustrate for the lattice of three particles:
I1=P1+P2+P3 12=P1P2+P2P3+P3P1-X1-X2-X3 13 = P1 P2 P3-P1X2-P2X3-P3X1.
(8.9.7)
Dynamics of various Toda lattices
285
I1 and I2 are simply related to the classic constants of the motion, total momentum and energy (21; - 12), whereas 13 is a nonclassic constant of the motion. As important as this result is mathematically, it is worth repeating Birkhoff's emphasis of Poincare's dictum - to the effect that equations are only more or less integrable. The point here is that these beautiful expressions for constants of the motion have yet to shed any further light on the physical behavior of the Toda lattice (i.e., in addition to its nonchaotic behavior). That is rather disappointing, and perhaps in the future we will learn how to use such constants of the motion. The nonclassic constants share an important distinction from the usual ten classic constants of the motion (energy, linear momentum, angular momentum and the center of mass dynamics), namely they are not additive constants of the motion. If we double the size of a Toda lattice, the Henon constants do not simply acquire a new additive part (plus a negligible interaction term), but suffer `massive' revisions. The degree to which these constants are not additive (thereby indicating how `massively' they must be changed when systems are joined together) is indicated by Toda's form of the constants of this system. For example, his first three constants of the motion are N
T1 = E Pk k=1
N
T2 = Y (ZPk + Xk)
(8.9.8)
k=1
T3
N
k1 C3Pk+Pk(Xk+Xk+1)1 The first two classic constants are, of course, strictly additive in the variables (Pk, Xk) and nearly additive (except for one boundary term) in the variables (Pk, Qk). That is, from (8.9.3), the variable X involves Q + . Therefore, if two systems, A and B are combined then T2(A + B)
T2(A) + T2(B).
However the equality nearly holds. T3 shows a minor nonadditive feature in the (Pk, Xk) variables due to one `boundary' term, (i.e., PN, XN+ 1). The higher order Toda constants, T,,, which have leading terms Y_Pk, are successively more complicated and less additive in structure.
An important lesson concerning ergodicity can be learned from this Toda lattice,
as was emphasized by Ford, Stoddard, and Turner (1973). They considered the six-particle Toda lattice, and plotted the harmonic normal mode energies Ek(t)=2(pk +wkgk),
in the same way that Fermi, Pasta and Ulam had done. Figs. 8.35 and 8.36 illustrate
'Moderate-order' systems
286
Fig. 8.35 1.2
1.0
0.8
cy 0.6 0.4
0.2
0
5
10
15
20
25
30
35
40
Time (s)
0
5
10
15
20 25 Time (s)
30
35
40
their results for two different energies, E = 1.32 and E = 132. Whereas, in the case E = 1.32 (Fig. 8.35) we see the typical FPU recurrence phenomena, the higher energy system in Fig. 8.36 appears to share the energy freely among all the degrees of freedom. In
other words, the almost periodic behavior has disappeared at higher energies and it appears that the system tends to a time-average equipartitioning of the energy. An equipartitioning of the normal mode energies had also been noted for the very different hard-core-plus-harmonic (HCH) lattice of Northcott and Potts (Section 8.6). The difference here is that, since the Toda lattice is integrable, the motion can not be ergodic, so that the observed energy sharing in the Toda lattice cannot be associated
with ergodicity. Schematically we can only make the relationship illustrated in Fig. 8.37. Just as in the case of the hard-core-and-harmonic (HCH) lattice, this system
Dynamics of various Toda lattices
287
Fig. 8.37
Toda monatomic lattice Low energy lattices
shows a transition from nonenergy sharing to energy sharing as the total energy is increased (also see Saito, Ooyama and Aizawa, 1970). Whether the HCH lattice ever becomes ergodic (at high energies) is unknown, but it certainly does not occur for the monatomic Toda lattice. Historically the concept of ergodicity has played a central role in the development of statistical mechanics. This was due to the desire to equate the physically observed time averages with the theoretically predictable phase averages (based on ensembles
of systems). Not only are the physical time averages quite finite in duration (particularly for nonstationary processes), which distinguishes them from the asymptotic time averages of ergodic theory, but also the phase function may bear little direct relationship with the macroscopic observables (a relatively minor, but instructive, example of this are the normal phase functions, discussed by Khinchin, 1949). The present results for a Toda lattice also suggests that ergodicity may not be an important concept in discriminating between very different macroscopic behaviors (e.g., the nonsharing of normal mode energies vs their quasi-equipartitioning). Similarly, it may be of limited importance (or irrelevant) in understanding whether certain systems will exhibit `standard' forms of irreversible behavior (e.g., see `lattice thermal conductivity' in the index).
In contrast to the integrable monatomic Toda lattice, the diatomic Toda lattice
(unequal masses) affords the opportunity to study the KAM transition from near-integrability (low energy lattice, E < Ec,) to large scale stochastic behavior in
phase space, when E>Ec. Here E.(mt/m2) is a critical value of the total lattice energy, which roughly separates these two types of dynamical behavior. To study this transition, Casati and Ford (1975) considered the behavior of the two-atom Toda `lattice', with the Hamiltonian
H=iCpi/mt+pz/m2]+exp(-qt)+exp[-(q2-qt)]+exp(g2)-3.
(8.9.9)
They first investigated the way in which two solutions, initially close together,
288
'Moderate-order' systems
separated as a function of time. To do this they computed the distance D(t) = [(Pi - Pi )2 + (P2 - P2)2 + (q1
- qt )2 + (q2 - q2)2] 1/2
(8.9.10)
between two solutions (p, q) and (p', q'). They showed that it is possible to clearly distinguish between situations where D(t) increases (on the average) linearly in time, similar to integrable behavior (Section 8.3), and situations where D(t) increases exponentially in time. The latter corresponds to dynamics with a positive Lyapunov exponent, and therefore characterizes chaotic dynamics. Two examples, which they found for E = 7 and m2/m1 = 0.54 (but initiating from different parts of the energy surface - see below), are shown in Figs. 8.38 and 8.39. Fig. 8.38
10 X 10-6+
50
100
200
150
250
Time (computer units)
Fig. 8.39
50
100
150
Time (computer units)
200
250
Dynamics of various Toda lattices
289
The two solutions in each of these examples initially differed by only 10-6 in the values of the single variables q,. The exponential separation is, of course, in sharp contrast to what Ford, Stoddard and Turner had found for the monatomic Toda lattice (m2/m1 = 1), and shows that the lattice is nonintegrable if m2/m1 : 1, or m2/ml :0 (taking m1 >1 m2).
The `degree of nonintegrability' of the Toda lattice can be roughly characterized by the fraction of the area of the surface of section in which stochastic motion is observed. `Observed' is an important qualification, since computer solutions can only sample a limited number of solutions to a limited degree of accuracy. On the other
hand no apology needs to be made for this qualification, since it is a fundamental characteristic of all empirical sciences. Using such computer studies, Casati and Ford explored the degree of integrability
a two-particle Toda molecule both as a function of the energy and the mass ratio, m2/ml (taking m1 = 1). Figs. 8.40 to 8.42 show the surface of section (q2 = 0, P2 I> 0) for m2/ml = 0.54 and the three energies E = 2,7, 30. Fig. 8.40
Pi
q1
E=2
q1
E=7
The smooth curves are freehand curves connecting points produced by the intersection
of simple trajectory. At the lowest energy, no chaotic motion was observed, but presumably the system is nonetheless (in a mathematical sense) nonintegrable. At E = 7, a large portion of the surface of section is shown to be covered by the stochastic motion of a `single' trajectory (a single numerically computed trajectory,
'Moderate-order' systems
290 Fig. 8.42
P1
q
E=30
which is characteristic of some exact erratic trajectory). The linearly separating solutions and exponentially separating solutions in Fig. 8.40 came from the two points q1 = 1.74, pl = 1.5 (q2 = 0, P2 = 2.72) and q1 = - 0.66, pl = - 0.037 (q2 = 0, P2 = 4.92) on this surface of section. The former is in the right `island', the latter in the `heart' of the chaotic region. Finally, at E = 30, most of the surface of section appears to be associated with stochastic motion. If the mass ratio is changed to m2/m1 = 0.33 the corresponding three surfaces of section are illustrated (Fig. 8.43). It can be seen that the `degree of integrability' of this system is greater than when m2 /m 1 = 0.54, at least in the cases E = 7 and E = 30.
In order to study the dependence of this `degree of integrability' on both E and m2/ml, Casati and Ford considered the initial states (p1,g1,p2,g2), given by 1
P1 = 2E,
ql=E,
=a
P2
7g,
z
q2=5E
(8.9.11)
They fixed the value of m2/ml, and changed the value of s, thereby changing the energy. Considering a nearby initial state p'= pi, q'2 = q2, q 1 = q, + 10', they then determined whether the distance D(t), (8.9.10), grew linearly or exponentially with time. Having determined the approximate `critical energy', E, separating these types of D(t)-dynamics, they repeated the process for other values of m2/ml. The results of
Fig. 8.43
P1 P1
9
41
(a)
E=2
(b)
E=7
E= 30
'Moderate-order' systems
292
Fig. 8.44 100+ rr
L
50+
01 0
0.5
1.0
m2/m1
these calculations are shown in Fig. 8.44. The full dots represent linear separation, the
asterisks represent exponential separation. The dividing curve is only meant to be suggestive, but it clearly illustrates the trend toward integrability as m2/m, tends to
0 or 1. It should also be made clear that this curve is related to these different behaviors only for the initial conditions (8.9.11). It is clear from the previous figures that the value of E, depends on where the trajectory crosses the surface of section, but its qualitative dependence on m2/m, is presumably the same almost everywhere.
These results of Casati and Ford suggest that irreversible effects might also be found in diatomic Toda lattices; specifically, it might be possible to observe normal thermal conductivity in such systems (see Section 8.6). The answer to this question is computationally quite difficult to obtain, because it involves many interactions with (statistical) thermal reservoirs, and the transmission of that energy through a lattice of 'many' particles. The first indication that the diatomic Toda lattice has a normal heat conductivity was found in the numerical study of Mokross and Buttner (1983). For lattices of 17 and 33 particles, they found reasonably linear temperature gradients away from the thermal reservoirs (Fig. 8.45), from which they could estimate the coefficient of heat conductivity (see Section 8.6). Subsequent calculations (Jackson and Mistriotis, 1989) with much larger lattices, and for different mass ratios, r = m2/m,, have strongly indicated that this coefficient of heat conductivity, K, is an intensive property of the system (independent of N, for large N), provided that r = 0.5. If we define the particles' 'temperature' in the lattice to be twice their average kinetic energy, Ti = m,, then Fig. 8.45 illustrates a typical temperature gradient, when r = 0.5 and N = 300. For larger values of r, this intensive property no longer
appears to hold, as is illustrated in Fig. 8.46. However, at present there is no precisely known r-demarcation between an intensive and nonintensive K (if it exists).
Dynamics of various Toda lattices 100r
Fig. 8.45
i Fig. 8.46
150
L4
293
'Moderate-order' systems
294
These results for the diatomic Toda lattice raise interesting and basic questions concerning this irreversible process. The classic theory of lattice heat conductivity is based on Peierl's famous `umklapp' processes between lattice normal modes (phonons).
It can be easily shown that the three-phonon umklapp process, which is the basis of Peierl's theory, cannot occur in diatomic one-dimensional lattices (e.g., J. M. Ziman
Electrons and Phonons, Oxford University Press). Thus the present irreversibility cannot be explained on the basis of this standard understanding of normal heat conduction (Jackson, 1978). More sophisticated forms of irreversibility need to be understood, even in this classic situation.
(1962),
8.10 The Painleve property and integrability conjecture In Chapter 2 we saw that nonlinear ordinary differential equations (ODE) can have singular solutions. A simple example is dx/dt = x2;
x = xo/(1 - xot),
where x0 = x(0). The singularity in this case is a simple pole at t = 1/xo. Since the location of the singularity depends on the initial condition, it is called a movable singularity. This type of singularity can be contrasted with the case dx/dt = x/(1 - t);
x = xo/(1 - t)
which has a singularity at a fixed location, t = 1, regardless of the initial condition, x0. This is called a fixed singular point, and it can be determined a priori by simply looking at the differential equation. This is not the case with movable singularities. The singularity of a solution may, of course, be a higher order pole, as in the case
dx/dt = x("+ a/";
1/N
x=
xo
N
1 - (xal"t/N)
It may also involve other types of singularities. If we generalize our considerations to a complex independent variable, zeC (complex plane), then the solution of dx
dz
= x3;
x = xo/(1 - 2xoz)1/2
has a branch point singularity in the complex z plane, whereas
dx/dz=ex;
x=1n(µ-z)-',
9=e-x0
has a logarithmic singularity. This generalization to a complex independent variable
certainly does not appear natural from the point of view of physical dynamical problems, where time is real. However, let us follow this `mathematical abstraction'
Painleve property and integrability conjecture
295
a little further. In any event, any singularity of the solution of an ODE which is not a pole (of any order) is called a critical singular point. One of the ways that ODE were characterized in the last century was based on whether or not their general solutions have movable critical points (see Ince, 1956, for a history of these investigations). Fuchs showed that, in the class of first-order ODE,
dx/dz = F(x, z)
(Z C- C)
where F(x, z) is analytic in z and a rational function of x, the only equations which
do not have solutions with movable critical points are the generalized Riccati equations dx/dz = ao(z) + al(z)x + a2(z)x2.
The generalization of this characterization of ODEs to second-order equations was carried out by Painleve and his collaborators (see Ince, 1956). Second-order ODEs can have solutions with movable essential singularities, instead of movable poles or critical points. Thus d2x/dz2 = (dx/dz)2(2x
- 1)/(x2 + 1)
(8.10.1)
has the general solution x = tan [In (az + b)],
where a and b are arbitrary constants. In the complex z plane, as z approaches (- b/a) along an arbitrary line, x(z) approaches no limit. Restricting the class of second-order ODE to the form d2x/dz2 = F(dx/dz, x, z)
(zE C)
where F is analytic in z, and a rational function of dx/dz and x, it was shown that there are `only' 50 canonical equations whose general solutions do not have movable critical points. Only six of these equations had not been previously solved, and the solutions of these six equations therefore define new functions, which are now called the Painleve transcendents. All six equations are nonautonomous. The three simplest, which Painleve actually discovered, are (P1) d2x/dz2 = 6x2 + z
(P2) d2x/dz2 = 2x3 + zx + a
(8.10.2)
(P3) d2x/dz2 = (dx/dz)2x-' - (dx/dz)z-'(ax2 +b)z-1 + cx3 + dx-1.
Unfortunately, however, there does not presently exist any systematic way to determine if a particular second-order ODE can be transformed into one of these 50 canonical forms. Moreover it should be noted that six of these equations required
296
'Moderate-order' si,stems
the introduction of new trancendental functions in order to be `integrable'. The significance of `integrability' in this case lies in the regularity of these functions over the entire complex plane. Why this regularity should be of particular importance to the physical observations on the real time axis is not clear.
Returning now to physical dynamical systems, there is the general question of whether a given system of equations is `integrable'. By the term `integrable' we will mean that the nth order system possesses n global, single-valued analytic integrals of the motion. This is a generalization of the concept of `integrable Hamiltonian' system, where we require n/2 global, single-valued, analytic, and time-independent integrals, which are independent and in involution. In either case there is presently no known general method which permits us to decide whether a system of equations
is `integrable'. It should be repeated that it is not clear how all of these various mathematical conditions defining `integrability' are relevant to properties of physical
interest - the integrability conditions are, after all, much more restrictive than the conditions for the existence of unique solutions. An important result in this direction was due to the discovery of S.V. Kovalevskaya,
involving the solution of Euler's equations for the motion of a rigid body about a fixed point I. d1,/dt + (/k - Ij)S2kfl j = Mg(x jµk - xkµj) dµ;/dt = f2kµ j - S2jµk,
(8.10.3)
where (i, j, k) = (1, 2, 3), (3, 1, 2), (2, 3, 1). Here the /k are the principal moments of inertia
at the fixed point, M is the mass of the body, g the gravitational constant, (x 1, x2, x3) the location of the center of gravity relative to the fixed point, and µk the direction cosines of the principal axes with the gravitational force. These satisfy the geometrical relation µi + µ2 +,U2 = 1. In addition, we have the energy integral 3
Ikf2k - 2Mgxkpk = E k=1
and the angular momentum along the direction of the gravitational force 3
I 'kµkS2k = L. k=1
Because the above six equations are autonomous, and df2;/dt does not involve f2;, it can be shown that it is only necessary to obtain four integrals of the motion in order to complete the integration of these equations (e.g., V.V. Golubev, 1953). Since three integrals are generally known (above), the complete integration is insured if one can determine one more integral. Prior to the study of Kovalevskaya, the only systems for which the general solutions had been obtained were the cases x1 = x2 = x3 = 0
Painleve property and integrability conjecture
297
(Euler-Pointsot) and I1 = I2, x, = x2 = 0 (Lagrange-Poisson), in which cases the fourth integrals are polynomials similar to the general three integrals above. Jacobi later showed that the general integrals in these cases are expressed in terms of elliptic functions of the time. Kovalevskaya, for the first time, introduced the concept of a complex time variable into the study of dynamics, and she noted that all of the integrals of known integrable systems are meromorphic functions in the entire complex t plane (that is, the integrals are single-valued analytic functions of t except for poles, which are movable). The mathematical generalization of Jacobi's elliptic functions to complex variables had been studied previously by Liouville, but the idea of using a complex time variable
to classify dynamical systems was the invention of Kovalevskaya. She chose to determine all the cases for which the integrals of the above six equations are single-valued, meromorphic functions of (complex) t. In so doing, she discovered in 1888 the integrable case 1, =12 = 213, x3 = 0, and was awarded the Borden Prize by the Paris Academy of Sciences. This case involved the first application to dynamics of hyperelliptic integrals, defined by
fX(aksk)_I/2ds= t, where the polynomial has six real roots (Golubev, 1953). More recently, it has been suggested (Ablowitz, Ramani, and Segur, 1978) that the integrability of partial differential equations (PDE) may be related to this classification of Painleve and Kovalevskaya. To be specific, we first introduce the definition: If the only singularities of the general solution of the ODE dx/dz = F(x, z)
(xeR", zeC)
in the complex z plane are movable poles (of any finite order), then we say that the ODE has the Painleve property, is a P-type ODE.
Put another way, the general solution of a P-type ODE has no movable critical points. Curiously, a P-type ODE can have a particular family of solutions with a movable critical point. Thus d2x/dz2 = - x3 dx/dz + x(dx/dz)(4 dx/dz + x4)1/2
has the general solution
x=Atan(A3z+B) which is free of movable critical points, yet it has the one-parameter family of solutions x = (4/3(z - C))113 with a movable critical point.
298
'Moderate-order' systems
The original conjecture of Ablowitz, Ramani and Segur (ARS) was essentially that a nonlinear PDE is `integrable' (solved directly by IST; see Chapter 9) only if every nonlinear ODE which is obtained from that PDE by some reduction process has the Painleve property. In still more recent works (e.g., Bountis, Segur, and Vivaldi 1982; Tabor and Weiss 1981), this idea has been applied directly to any system of ODEs, without any considerations of PDEs. In this form, the ARS conjecture is a direct extension (to arbitrary ODEs) of Kovalevskaya's classification, and draws upon her use of a complex time variable. Moreover it is strengthened by her demonstrated
usefulness of this scheme of classification of ODE systems. Perhaps the most conservative form of this conjecture, which we will refer to as the integrability conjecture, is Integrability conjecture Any system of ODE
dx/dt = F(x, t)
(xeR teC),
which has the Painleve property, is integrable. The converse of the integrability conjecture does not hold. That is, there are integrable systems of ODE which are not of the P-type (e.g., Dorizzi, Grammaticos, and Ramani, 1984; Ramani, Dorizzi, Grammaticos and Bountis, 1984). We will, however, only be
concerned here with examples which illustrate reasons for believing that the integrability conjecture is correct. We will now discuss an algorithm (Ablowitz, Ramani and Segur, 1980) which gives necessary (but not sufficient) conditions for an ODE to be of the P-type. For a nice review, references, and extensions of this method, see Bountis (1984). The algorithm does not detect the presence of essential singularities, as in the above example, but it does give conditions under which the equations may have only movable poles. The algorithm consists of three steps: (1) Find the dominant singular behavior. (2) Find the `resonances' - the powers (degrees) associated with arbitrary coefficients. (3) Determine all of the constants of integration. We now discuss these steps briefly.
(1) The dominant singular behavior We assume that the singularity occurs at some to, and set i = t - to. We look for a solution in the neighborhood of to which goes as x - Ax° as T -> 0, where a < 0. As t 0, two or more terms in the equations may balance, depending on the values of A and a. Those terms which must balance to insure the lack of movable singularities are called the leading terms of the equations. There may be several possible values
Painleve property and integrability conjecture
299
of a which balance these terms. If any of these values of a are not integers, and if x - Ara is actually asymptotic (see below), then this term represents the dominant behavior near a movable algebraic branch point at T = 0 of order a. In this case the equation is not P-type. To prove that x - AT° is asymptotic, consider a new function u(T) = x'la, and obtain its differential equation. It has the property that u(0) = 0 and du/dt is finite at T = 0. If the solution of the ODE for u is an analytic function of r, then the original ODE in x has a branch point at T = 0 if a is not an integer (hence it is not P-type).
Example 1 As a simple example, consider the Riccati equation
dx/dt = x2 + bx + c. Using the above substitution AaT°- 1 - A2T2a + bATa + c,
and since a < 0 the leading terms are AaTa-1 -A2T2a
which requires that a - 1 = 2a or a = - 1, and A = a = - 1. This suggests that x - - T-' is the asymptotic behavior of x(T) near T = 0. To check this, set x Then
u-'.
du/dT = 1 - u + cue
and it is clear that this has an analytic solution satisfying u(0) = 0 and (du/dt)o = 1. Since this is a first order ODE, there is only one constant of integration, namely to, so we do not need to proceed to steps (2) and (3). Hence this ODE is P-type.
Example 2 For the ODE d2x/dT2 = x4
we obtain, using x - ATa,
Aa(a -
1)Ta-2
A4T4a
so a-2=4a, or a= - 3 which suggests that it is not P-type. To check this, we set X=U - 213, and obtain the ODE for u - 3u(d2u/dT2) +
q(du/dT)2 = 1
which has an analytic solution in T satisfying u(0) = 0, u'(0) = (-L )112, and finite u"(0).
Hence x - AT - 213 is indeed the asymptotic behavior as r -- 0, and this ODE is not P-type.
'Moderate-order' systems
300
(2)-(3) Resonances and constants of integration We now look for the next higher order term in the possible Laurent series solution X(T) = T° E 00 AkTk
(Ao = A)
k=0
(8.10.4)
in the neighborhood 0 < IT I < R. The values of k which have arbitrary coefficients, Ak (constants of integration), are called resonances. To obtain the general solution in the above form, there must be (N - 1) such resonances, where N is the order of the ODE. to is, of course, the Nth constant of integration. To proceed, we look for the first resonance by setting
x-AT a+BT°+' where r > 0 (because a was assumed to be the leading term). If any root for r is not a real integer (and Re (r) > 0), then the ODE is not P-type, and the process can be terminated (once again applying the above test for the asymptotic behavior). If, for every (A, a) of part (1), there are fewer than (N - 1) real positive integer roots, r, then these solutions are not general, which suggests that the above Laurent series misses part of the general solution. Example 3 Consider d2x/dt2 = 6x2
which is easily integrated, but we will use it for practice. Beginning with x - ATa we obtain the leading terms a(a - 1)ATa-2 , 6AZT2',
so a= -2, A=l. Next we set x - T -2 + BT' -2 and obtain (r - 2)(r -
3)BT'-4
(r+
1)(r-6)BTr_4=0.
= 12BTi-4
or
This shows that the possible resonant terms (for which B is arbitrary) are r = - 1, and r = 6. The root r = - 1 is, however, not allowed, since T-2 is the leading term (i.e., only the roots r > 0 are considered), so the only possible resonance is r = 6, and B is the arbitrary second constant of integration. We need to prove that the resonance condition is satisfied, and proceeding in the series (8.10.4), x - T -2 + BT4 + CTS + , we find that both s and C are determined, and the remaining series is consistent.
Example 4 We consider here Painleve's first transcendent (8.10.2), given by the solution of d2x/dT2 = 6x2 + t = 6x2 + T + to.
(P1)
Painleve property and integrability conjecture
301
Beginning as in Example 3, we again obtain the leading term x _ T - 2, so we next set x - T-2 + BT -2 '. Then the leading terms are (r-2)(r-3)BT'-4- 12Brr-4+to
so r = 4 and B = - to/l0. In the next order x - T -2
- (to/10)T2 + CTS and
s(s- 1)Ts-2 ti 12CTs-2 +T
so s = 3, C = - 6. We have not yet obtained the resonant term (with arbitrary coefficient), so we proceed further in the series (8.10.4) (q > 3).
x - T-2 - (t0/10)T2 - (6)T3 + DT9,
We now find that all terms are linear in D, which characterizes the resonant term
q(q -
1)DTq-2
12DTq-2
Since q > 3, the resonant term is q = 4, and D is arbitrary. The subsequent terms in
the Laurent series are determined in terms of (to, D), yielding Painleve's first transcendent. (For practice, the next term is (to/300)T6).
It should be noted that, in the last example, we did more work than is required to determine the existence of resonant terms, since we also determined (r, B) and (s, C), which do not influence the resonant term. In the present method of testing for a P-type ODE, it is first useful to check for the resonances to see if there are a sufficient number of arbitrary coefficients, and after that check to see that the remaining coefficients of the Laurent series (8.10.4) can be determined consistently.
Example 5 We illustrate the determination of the resonance, without determining the intermediate coefficients. Consider Painleve's second transcendent (8.10.2) d2x/dt2 = 2x3 + xt + a = 2x3 + XT + tox + a.
The leading term is x -AT a, yielding a(a - 1)ATa-2 - 2A3T3a. Hence a = - 1 and A = 1. Next we set
x-T-1+BTb
(b i0)
and look for the resonant B. The linear equation in B is b(b - 1)BTb- 2 '
2(3BTb- 2),
and, since b > 0, the resonant term is b = 3 (i.e., B is then arbitrary). Since there are two arbitrary constants, (B, to), this indicates that the general solution can be obtained
in the form (8.10.4), and hence is P-type. It remains to check that the remaining coefficients in this series can in fact be consistently obtained. This is usually easily established by obtaining a recurrence relation.
'Moderate-order' systems
302
Exercise 8.13 Determine whether the following ODEs pass the above necessary test for being P-type: (A) d2x/dt2 = 6x2 +X (B) d2x/dt2 = 2x(dx/dt)2/(x2 - 1) (C) d2x/dt2 = (dx/dt)2[(2x -1)/(x2 + 1)].
We now turn to a more physically interesting, and a priori unknown case of the generalized Henon-Heiles Hamiltonian system. Many other cases are discussed in detail in the literature (for details and reviews see, e.g., Bountis, Segur, and Vivaldi 1982; Bountis and Segur 1982; Dorizzi, Grammaticos, and Ramani 1984; Ramani, Dorizzi, Grammaticos, and Bountis 1984). The generalized Henon-Heiles Hamiltonian is
with the equations of motion
4i=Pi,
P2f22q2-q1+eq2*
42=P2,
Here 52,, S22 ands are arbitrary constants. The original Henon-Heiles Hamiltonian
had S2;=S2z=E=1. We write the above equations as a pair of second order equations deg1/die = - iq, - 2glg2 d2g2/dT2 = - n2 q2 - q1 + eq2 I n l owest
(8.10.5)
ord er we set
q, -
q2 = Bib
AT',
(8.10.6)
yi e ldi ng
a(a - 1)ATa-2 _ - 2ABTa+b and
b(b -
1)BTb-2
- A2z2a + EB2T2b.
The first equation requires b = - 2, and the last equation then allows two possibilities:
Case (I): a=b= -2,B= -3,A 2 = 9(2 + r)
(8.10.7)
Case (II): a > b = - 2, B = 6/E, a(a - 1)c = - 12, A: arbitrary. In order for the system to pass the present necessary test for being P-type, these two cases must (jointly) yield no movable critical points, and have four arbitrary constants. We expect, of course, that this will only be the case for particular values of e, fl, , and f12
Consider first case (I). We look for resonant terms by setting
q,-AT-2+CTi-2,
q2_-3T-2+DTi-2
(r>0)
(8.10.8)
Painleve property and integrability conjecture
303
which yields the linear terms C(r-2)(r-3)Tr_4- -2[-3C+AD]Ti-4
D(r - 2)(r - 3)Tr-4 - - 2A CT,-4 - 6DETr-4 This yields the resonant condition
(r - 2)(r - 3) - 6
2A
2A
(r - 2)(r - 3) + 6E
=0.
Let R - (r - 2)(r - 3), then (using A2 = 9E + 18)
(R-6)(R+6E)-72-36E= (R - 12)(r + 6 + 6E) = 0. One root is R = 12, with the resonance (r > 0)
r=6 representing a second arbitrary constant (in addition to to). The other root, R = - 6(1 + E), yields the possible resonances r=i±i[-23-24E]1/2
In order that this be an integer root, E can only have the values
g = - [(2n - s)2 + 23]/24
(n = 1, 2, ... ,).
Some of these Es yield two resonances and some only one. To see which are possible,
we next turn to case (II). In this case Ea(a - 1) = 12 and a = m > - 2 (an integer). Combining this with the last condition yields e = - 12/m(m - 1) = - [(2n - s)2 + 23]/24 which gives the following possibilities:
E_ - 1; a=m=4, n=2 or 3 E_ -6; m=2, n = 8. If E
1, there are the resonances n = 2, 3, and r = 6 yielding three arbitrary
coefficients, plus the arbitrary to. Note that a = 4 in (8.10.6) is the same as r = 6 in (8.10.8), and hence does not yield a new arbitrary constant. Hence these are the four arbitrary constants needed for a general solutions. A further detailed calculation, to insure that there are no contradictions in determining the remaining Laurent coefficients, shows that the square frequencies must be equal, 522 = S22 (see Exercise 8.8). If z
6
,
the resonances m = 2, n = 8, and r = 6 now all yield independent
arbitrary coefficients, so that (with to) there are the required four arbitrary constants. In this case it turns out that S21 and 522 can be arbitrary, and another integral is 2 (8.10.9) G=q1 +4gig2+4p1(P1g2-p2g1)-4521gi2 g2+(4521 -(22)(Pi+521gi)
304
'Moderate-order' systems
obtained by J. Greene. Since G is quadratic in the momenta, it can also be obtained by a method discussed by Whittaker (1944, Section 152). Finally, other integrable cases of (8.10.5) are discussed in the literature involving special 521, 02 (e.g. S2, = 1,
522=4; ore=-16,522=16521). A number of other physically interesting systems have been shown to be integrable using the Painleve test. We mention here only a few examples (see the above references for more examples and other references). Two coupled quartic oscillators
H =2(pi +p2 +Q2gi +Q2 q2)+qi/4+ oqi/4+pg2g2/2 when a = p = 1, or a= 1, p=3. Fixed-end, two-mass Toda lattice 2
H
P1
2
+ p2 +exp(-bq1)+exp[e(g1-q2)]+exp(g2) 2m2
2m1
when MI/M2= 1, 6 = e = 1, or M,/M2=11 6= 1, B= 2, or MI/M2= 3, S = 1, E=12' Free-end, three-mass Toda lattice (center-of-mass frame) 3
H=
2
kYl Zmk
+ exp [6(q 1 - q2)] + exp(g2 - q3)
(m3 = 1)
when m1 = e(2e - 1)/(2 - e) and m2 = (2e - 1), 2 < e < 2. Generalizations to n degrees
of freedom have also been obtained by Moser (1975) and Bogoyavlenski (1976). Moreover unbounded integrable cases are known (Bountis Segur and Vivaldi, 1982).
8.11 Chemical oscillations and dissipative open-system structures We have seen in Section 8.2 that diffusive coupling between cells, which each have dynamics in a two-dimensional phase space, can produce an instability in a spatially homogeneous state. This instability initiates the growth of some inhomogeneous spatial structure (the Turing instability). As Turing pointed out, nonlinear effects are
necessary to control this linear instability, which then produce finite amplitude inhomogeneous states (either time dependent, or stationary). He in fact developed a nonlinear example involving two morphogens per cell, with concentrations (X, Y), which have the equations of motion
dXk/dt=F(Xk, Yk)+/i(Xk+l -2Xk+Xk-1)
(8.11.1)
dYk/dt = G(Xk, Yk) + v(Yk+ 1 - 2Yk + Yk- 1),
(8.11.2)
Chemical oscillations and dissipative structures
305
where k = 0,1, ... , N + 1, and F(X, Y) = 32 (- 7X 2 - 50X Y + 57)
(8.11.3)
(Y> 0; otherwise G - 0)
G(X, Y) = 32(7X2 + 5OXY- 2Y- 55)
and p = Z, v ='-n. Using the University of Manchester computer, on a system of twenty cells (N = 20) in a ring, XO(t)=XN(t),
XN+1(t)=X1(t),
YO(t)=YN(t),
YN+1(t)=Y1(t) (8.11.4)
He found that various stationary configurations could develop, but only after the concentration of Y became zero in a number of cells (up to that point his nonlinearity led to `catastrophic' instabilities). The patterns obtained by Turing were stationary. He concluded that three or more morphogens are required in the cells, in order for traveling wave patterns to occur.
Another example of coupled cellular dynamics was proposed by Prigogine and Lefever (1968). The cellular dynamics involve a model chemical oscillation, which has subsequently become known as the `Brusselator', from the city of its birth. It is based on the hypothetical set of chemical reactions
A-.X;
2X+Y--*3X;
X-E
involving the net reaction
These initial and final products are assumed to be homogeneously distributed, whereas
the reactants X and Y are spatially controlled by Turing's diffusion mechanism, (8.11.1) and (8.11.2). Thus Turing's `morphogens' are here identified with the chemicals
(X, Y). In addition, the parameters (A, B) explicitly account for the coupling of the system with the outside world. That is, the system is an open system exchanging matter and energy with some infinite `reservoir', or supply. The dynamics in an uncoupled cell (p = 0, v = 0) is governed by the equations for the concentrations (X > 0, Y > 0) in this Brusselator model by dX/dt = F(X, Y);
dY/dt = G(X, Y),
(8.11.5)
where
F(X, Y) = A - (1 + B)X + X 2 Y
(8.11.6)
G(X,Y) = BX-X2Y. This has cubic nonlinearities, in contrast to (8.11.3) and other more realistic models of chemical oscillations (e.g., see the `Oregonator'). The final products, D and E, of course do not appear in the dynamic equations.
'Moderate-order' systems
306
Fig. 8.47
Y
51
X 5
The fixed point (X = A > 0, Y = B/A > 0) of (8.11.6) is unstable (if B > A2 + 1), and it is enclosed by a limit cycle. This is illustrated in Fig. 8.47 when A = 2, and B = 6, for the two initial states (X, Y) _ (0, 0) and (2, 3.1). As one example of coupled Brusselator cells, Prigogine and coworkers investigated the case of two coupled cells (a 'reaction-diffusion' system), dXkldt = F(Xk, Yk) + DX(X; - Xk) dYk/dt = G(Xk, Yk) + Dy(Y; - Yk)
(j # k = 1, 2).
(8.11.7)
This system has a static homogeneous solution
Xk=A,
Yk=B/A.
(8.11.8)
which is, of course, independent of (D.,, Dr). On the other hand, the stability of the homogeneous solution does depend on (D.,, Dy), as in Turing's case (Section 8.2). The value of two of the parameters in equations (8.11.6) and (8.11.7) were fixed,
D,,=1,
A=2,
and they studied the nature of the dynamics as a function of the other two control parameters, (B, Dr). The result of this investigation (Lefever 1968, Glansdroff and Prigogine 1971) is illustrated in Fig. 8.48. Region I The static homogeneous state is stable if 5 > B, and 3(2+Dy)/Dy > B. Region II The static homogeneous state is unstable to homogeneous pertubations (i.e., the uncoupled cells have an unstable fixed point).
Chemical oscillations and dissipative structures Fig. 8.48
CQ
4
9
6
3
Dy (arbitrary units)
2.5
1.5
20
10
30
Time (arbitrary units)
3
1
3
1
0
0.5
i/N
1
307
'Moderate-order' systems
308
Region II, III, V No time independent stable state exists.
Region IV Perturbations of homogeneous systems develop into stable inhomogeneous steady states. Note that in the region 5 > B (the shaded region), the uncoupled cells are stable. The middle figure shows the approach to the inhomogeneous steady state in region IV, starting from the near-homogeneous equilibrium state X 1 = X2 = 2, Y1 = 2.6, Y2 = 2.64.
The system (8.11.1), (8.11.2) was also investigated for larger cellular rings (periodic
boundary conditions, (8.11.4)). The lower figures illustrate the inhomogeneous stationary state which was obtained for N = 50. Moreover they noted that there exists a multiplicity of these inhomogeneous solutions, depending on the initial conditions (see Nicolis and Prigogine, 1977). Such spatial organization, which arises from the exchange of energy and matter with the environment of a system, Pregogine called a dissipative structure (also see Prigogine and Stengers, 1984). The description `dissipative structure' may be confusing at first, for several reasons. First of all `dissipative' is an adjective indicating an activity, whereas a (stationary) `structure' is a fixed, nonactive noun. Of course, what is `dissipative' is the dynamics of the cellular system, so `dissipative structure' - `dissipative'-system's structure.
The second possible point of confusion is the adjective, `dissipative', which means to be wasteful of something (e.g., energy or matter in dynamic systems). On the contrary,
autocatalytic systems make efficient use of outside energy/matter. Thus, in the Brusselator model (8.11.5) and (8.11.6), the cell's dynamics is autocatalytic if B> A2 + 1, whereas it is dissipative (in the usual sense) if B < A2 + 1. As noted above, what `dissipative' refers to is the exchange of energy/matter with the environment of the system. Such systems may also be referred to as open systems, without any reference
to the character of their dynamics. Therefore, in greater detail, `dissipative structure' = a spatial structure produced by an open system open-system structure
In addition to these structures, there is also the interesting area of the dynamic behaviors which occur in regions such as II, III, and V above. The reader may justifiably wonder what is the difference in the character of the dynamics in these regions, and wish to explore this matter. We will return to such dynamic aspects of coupled cells in the following sections, and in Chapter 10. In his 1952 paper, Turing made the observation that
Most of an organism most of the time, is developing from one pattern into another, rather than from homogeneity into a pattern. One would like to be able to follow this more general process mathematically also. The difficulties
Chemical oscillations and dissipative structures
309
are, however, such that one cannot hope to have any very embracing theory of such processes, beyond the statement of the equations. It might be possible, however, to treat a few particular cases in detail with the aid of a digital computer.
These remarks stimulated Martinez (1972) to investigate the character of the
,open-system' structures produced by coupled Brusselators, for different numbers of cells. He considered the system (8.11.1), (8.11.2), using either the periodic boundary conditions, (8.11.4), or fixed ends
X0=O,
Yo=O,
XN+, =0,
YN+1 =0.
(8.11.9)
He chose A = 3 and B = 9.75, which produces stable dynamics for a single cell, and p = 0 and v = 1, which yields an unstable homogeneous steady state for a ring of N cells. It should be noted, however, that these values of y and v are physically extreme values.
Exercise 8.14 Determine the conditions for the stability of (8.11.1), and (8.11.2), and (8.11.4) with periodic boundary conditions, for the homogeneous state (8.11.8), of the Brusselator model, (8.11.6).
In the case of cellular rings, if N is not a prime number, there are a number of stable inhomogeneous configurations, composed of suitable multiples of the divisor configurations (e.g., if N = 6, there are configurations consisting of two N = 3 configurations, or three N = 2 configurations). In addition it is sometimes found that there are other stable inhomogeneous configurations which are not composed in this
fashion (e.g., also for prime N rings). At present there does not appear to be a systematic study of all such possible inhomogeneous configurations (Nicolis and Prigogine 1977). Several examples, many found by Martinez, are illustrated in Fig. 8.49 for N = 1, ... , 6, and the prime case, N = 11. Not all of these configurations appear to arise from small perturbations of the homogeneous state (8.11.8) (Martinez, 1972),
but they are the attractors of other initial states. The manner in which these configurations vary or cease to exist, as p and v in (8.11.1) and (8.11.2) are varied, remains to be determined. The importance of determining the dynamic interrelations between such patterns becomes clear once we consider the possibility of `growth' in the number of dynamic cells, N. A simple model of such a `developmental' process was discussed briefly by Martinez. The basic idea is to introduce a third morphogen in each cell, Z;, whose concentration decreases unless (X; - Y;) > 0, in which case it increases. Martinez's equation for Z is dZ dt
(
a(X - Y) - bZ
1-bZ
ifX > Y ifX 1< Y
(8.11.10)
'Moderate-order' systems
310 Fig. 8.49
N=4
i
N=5
N=1 N=2. N=3
N=6
®X ®Y If the concentration of Z exceeds some critical value, Z, that cell divides, retaining
the same concentration of (X, Y) in the daughter cells. The future dynamics is significantly dependent on the concentration of Z in the daughter cells, and a systematic study of this effect has not yet been made (Boon and Noullez 1986). As
pointed out by Turing, such models do not constitute theories of actual cellular growth, until such morphogens can be related to real biological components. The development of patterns in two dimensions have been made using diffusively coupled Brusselators (Martinez, 1972), and other dynamics which apparently is more relevant to patterns in biological contexts. Many of these latter studies are based on the introduction of localized `activators' a, and longer range `inhibitors', h, governed by nonlinear diffusive partial differential equations (Gierer and Meinhardt 1972, Meinhardt 1982)
as at
cat h
82a
ua + Da ax2
ah=ca2-vh+Dha2 h at
(8.11.11)
aX2
The longer range influence of the inhibitor is obtained by taking D,, > Da, but with v > It. These studies employ diffusively coupled cellular dynamics whose morphogens are activators and inhibitors (Swindale 1980, Boon and Noullez 1986). An example of the development of a pattern (t = 320) from noise (t = 0) is illustrated in Fig. 8.50
Smale's analysis of Turing's morphogenic system
311
Fig. 8.50
't F r
-Uh
jr dfefr
111
Jul
uh11" 9
JAB%
P
.
i
1P
t = 320
IP
r
rJL "fir fsy.
$i '4w 7
f1P .
L9u.
'Qai
U.
'Ib rIL f°4f" nlfi 1Ie..h
'!.i.r
it
d'91F"' -Ji
i5'
'
I
u2111I,
g4pow
5fit. Tii
i !Y
ii
gIIMEiiib .a
gJe'r-
r
M,
i
,
J
-
dim.,0 :'jIb== W (b)
(a)
(Swindale 1980). One of the apparent differences between the dynamics described by the partial differential equations and those of coupled cells is the freedom which the latter systems have to possess different time scales between the spatially local (e.g.,
a cell) dynamics, and the diffusive interactions. Much remains to be understood about such processes.
8.12 Smale's analysis of Turing's morphogenic equations In this section we will examine another perspective of nonlinear analysis - one which does not involve any deep analytic, numerical, graphical or mapping considerations, and yet establishes a structurally stable, global, interesting and surprising dynamic result. This is quite a different type of analysis from those that we have considered
up to now. It is not an uncommon line of reasoning for mathematicians, but this application is particularly nice because of its possible usefulness and its unexpected physical effect.
The analysis by Smale (1974) is based on Turing's idea that morphogenic processes
may be understood by describing cells in terms of their chemical (enzyme, or morphogen') concentrations z', z2, ... , z', where zk , 0, which are coupled to each other by a diffusive process through the cellular membranes. Thus the kth cell is treated as a mathematical point (no spatial structure is considered), with a phase space P={ZkER',Zk=(Zk,...,Zk),Zk,0,1=1,...,m}
governed by dynamic equations of the form dzk dt
= Rk(Zk) + µ(Zk+ 1 - 2Zk + Zk - 1 ),
(8.12.1)
'Moderate-order' systems
312
where y is an m x m diagonal matrix which represents the diffusive coupling between adjacent cells (written here for one-dimensional system). Most of Turing's analysis was limited to the case where the functions Rk(zk) are linear functions of their argument. In contrast, Smale was interested in obtaining dynamical features which are global and nontrivial in character, requiring decidedly nonlinear functions Rk(zk). The functions Rk(zk) are assumed to have a unique singular point zk > 0, so that Rk(zk) = 0. Such a fixed point can then be viewed as a `dead' state, since there is no
time-dependent dynamics. One of Turing's chief concerns was to show that the diffusive coupling could produce an instability in a uniform collection of cells in such
stationary states, and thereby produce (for example) stationary wave patterns in cellular `rings' (periodic boundary conditions). Smale's interest was to extend this to a dynamic situation, where the stable fixed points of two uncoupled cells are changed by diffusive coupling in such a way as to produce a stable periodic limit cycle (dynamic `life') in the coupled dynamics. This means that the system can have a (supercritical) Hopf bifurcation as the diffusion coefficient is increased. The surprising ('paradoxical') feature, of course, is that in this system the diffusive effect, which tends to equalize the concentrations between cells, makes such a uniform state unstable and produces
stable oscillations. To over-dramatize it a bit (Fig. 8.51), `life' is produced by a 'dissipative' process from a nearly `dead' state (any perturbation of zk).
Fig. 8.51
Diffusive coupling
To establish that this behavior is possible, Smale considered only two identical cells (R1 = R2 = R), each with four morphogens (m = 4), zke R° (k=1,2). These numbers were selected for convenience. However the results hold for larger systems, but possibly not for fewer morphogens (this is not clear). He specifically wished to show that there exist systems, R, with the following properties: (R 1) If dz/dt = R(z), zeP, then R(z) is required to be such that all z(t) - z, where R(z) = 0, as t --+ oo. In other words, the dynamics of an isolated cell is globally asymptotically stable to the (unique) stationary state zeP (note again that zEP, (8.12.1), implies that all z` 3 0). Moreover, R(z) is required to be structurally stable. This means that the system dz/dt = Ro(z) must have equivalent dynamics if Ro(z) is a C1 perturbation of R(z).
Smale's analysis of Turing's morphogenic system
313
(R2) For the coupled system
dz1_ dt dZ2
R(zl)+µ(z2-Zl); R(z2) + µ(zl - z2);
dt all z(t) = z1(t), z2(t)EP
.0
(0
) ;,U > 0
P1 X P2, except a set F of measure zero, tends to a nontrivial periodic solution y(t), as t -+ oo (so y(t + T) = y(t), for all t). Smale calls such a system a `global oscillator' in P. (R3) Boundary conditions: If zo = 0 for some component (k) of a z0EP, then the kth component, Rk(zo), of R(zo) is positive. This insures that, if the initial conditions
satisfy z' > 0, these conditions will be satisfied for all time (note that µ is a positive diagonal matrix in the above equations). To show that such a system R can exist, Smale first considered another system (Q) with the properties: (Q1) dz/dt = Q(z), with zER4. The origin is the unique fixed point, Q(0) = 0, and all solutions tend to it, z(t) - 0 as t --> oo. Note that the phase space is no longer restricted to positive zk. (Q2) The system d
d
dtl = Q(zl) + µ(z2 - z1):
is
a
global
oscillator,
so
dt = Q(z2) + µ(zl - z2)
that z -+ F(t) (zER8) where K* > II z(t) II
(Q3) There is a K > K* > 0 such that, if zER4 and II z II >, K, then Q(z) = - z. Smale next noted that, if the system Q can be found, then the system R can be constructed. To see this, pick some z such that ?> K, so that zEP if II z - i II < K. Set R(z) = Q(z - z), then R(z) satisfies (R1) and (R2) - because the global oscillator F(t) satisfies K > II z(t) II, so y(t) is in P x P. Finally, if zo = 0, then zo is on the boundary of P, so II zo - z II > K and therefore Q(zo - z) (zo - z). Therefore, if zo = 0, then Qk(zo - z) = z* > 0, so that (R3) is satisfied. Hence from Q(z), zER4, we can construct R(z), zeP.
In order to construct the Q-system, Smale considered yet another system, S. The system S satisfies:
(Si): same as (Q1);
(S2): same as (Q2)
but (Q3) is temporarily ignored. First we note that the set of points
A= {(zl,z2)ER4 x R4Izl =z2} is invariant under the flow dtl = S(zl) + µ(z2 - zl ):
Z2
dt
= S(z2) + µ(z1 - z2).
(8.12.2)
'Moderate-order' systems
314
That is, the condition z, = z2 is preserved for this set. Moreover the set A tends to the origin, because of (Si). Now suppose that we select S(z) to be antisymmetric, S(z). Then the set S(- z)
A' = {(zl,z2)ER4 x R41z, _ -z2} is also invariant under the flow (8.12.2), which now becomes
dz_ dt
S(Zk) - 2y zk
(zkeA').
(8.12.3)
The idea now is to find a function S(z) = - S(-z) such that the set Al is an attractor for (8.12.2) in R4 x R4, and on the other hand the dynamics (8.12.3) is a global oscillator in R4 (the set A'). Thus most of the solutions in this set tend to a stable limit cycle, F(t). Of course there may be a set of measure zero in R4 x R4 which does not tend to this limit cycle (such as those in A). This is illustrated rather poorly (because of the limited dimensions!) in Figure 8.52. To obtain such an S(z), we begin by considering S(y) (yeR4), where y and z are related by some linear change of coordinates, y = Az, which will be determined shortly (Fig. 8.52). Let S(Y) = SiY + Sa(y)
with
1 +a Si _
1
ya
0
-1
a
0
ya
-ya
0
2a 0
0
Fig. 8.52
Z2 (R4 )
-ya 0
2a
Smale's analysis of Turing's morphogenic system
315
and
(8.12.4)
S3=(-(y1)3,0,0, 0),
where (a, y) are unspecified constants for the present. The choice (8.12.4) is hardly obvious, but it will become clear soon. First of all 9(-y) = - S(y), as desired. Next we want to show that the origin is a global attractor of dy/dt = S(y). This is done by showing that y2 decreases with time if a < -1 (in Si). This follows from considering the dot product (Y, S(Y)) = (1 + a)(Y1)2 + a(y2)2 + 2a(y3)2 + 2a(y4)2 - (yl)4,
so dy2/dt < 0 if a < - 1.
Next, we would like a matrix µ such that
dy/dt = S(y) - 2µy
(8.12.5)
is a global oscillator in R4. We note that, if
2µ =
a
0
ya
0
0
a
0
ya
- ya
0
-2a
0
0
-ya
0
-2a
then
S(Y) - 2µy =
1
1
0
0
-1
0
0
0
0
0
4a
0
0
0
0
4a)
1 Y + S 3(Y)
Because y3 - 0 and y4 - 0 as t -> oo (a is negative), the subspace (y', y') is an attractor in R4. Moreover, in this subspace
dal=Y1+Y2-(Y1)3; dal=-Y1
or
Y'
dt
d(1-3(y1)2)-yl. d
This is the well-known van der Pol equation (which explains the selection of S3(y)), which is a structurally stable, global oscillator in R2. It follows that (8.12.5) is indeed a global oscillator in R4, which moreover is structurally stable. This last analysis was relatively easy because we used the coordinates y, where we could select the matrix P. The concentration variables z must now be related to the
y variables by a linear transformation, y = Az, such that the matrix µ will be
'Moderate-order' systems
316
transformed into the desired matrix y. Multiplying (8.12.5) on the left with the matrix
A', we obtain dz
dt
= A-' S(AZ) - 2A -' µAz
so we will identify
S(z) = A -'S(AZ)
(8.12.6)
µ = A -' µA.
(8.12.7)
and require that Since the elements of µ are required to be positive, the eigenvalues of the matrix µ must also be real and positive. The characteristic equation associated with µ is readily found to be [(a - A)(2a + A.) - (ya)2]2 = 0
so that the eigenvalues are real and positive provided that 2" < y < 3/2. This establishes the possible values of y in (8.12.4).
Note that none of the properties of the flow (8.12.5) are changed by this linear transformation, and we do not need to explicitly obtain the matrix A, nor the function
S(z). We only need to establish that it has the desired properties. So far we have established that it satisfies (Si) and that (8.12.5) is a global oscillator in R. We next turn to the property (S2), and note that if z, = - z2, then y, = - Y2. Therefore the set t is invariant under the transformation y = Ay. We want to show that A' is a global attractor in R4 x R4. We have dal=S(Y1)+f2(YZ-Yi);
dd2S(Y2)+µ(Yi-Y2)
so
d
-(YI + Y2) = S(Y1) + S(Y2) = Sl(Yl + Y2) + S3(Yi) + dt
Then 2dt(Y'
+Y2)2 = [(Yi +Y2),Si(Y1 +Y2)] -(Yl +Y2)[(Yi)3+(y )2]
We have shown that (z, 91 (z)) < 0, and since (a + b)(a3 + b3) , 0, it follows that d(y, + y2)2/dt 1< 0, so that y, - -Y2 (i.e., all solutions tend to the set A'). Therefore Al is a global attractor in R4 x R4. Finally we note that the set of points I which do not tend to the limit cycle is six-dimensional, and hence has zero measure in R4
4x R.
To obtain Q(z) we need to alter S(z) so as to satisfy (Q3). This is done by introducing
a function such that 4(r) = 0, if r < ro, and 4(r) = 1 if r > r, > ro. These two values
High-order dissipative systems in lower dimensional R"
317
can be connected by a Cx function if we take (for example)
0(r)=exp[-1/(r-ro)]eXp[1/(r1-ro)](1 -exp(-1/(r1 -r)))
(r1
r%ro)
Now set
Q(z)=(1-/(IIzII))S(z)-0iizii)z. If ro is large enough for II z II < ro to contain the van der Pol limit cycle, then (Q1) and (Q2) are still satisfied. Moreover, if K > r,, then (Q3) is satisfied. Therefore it has been shown that the Q system, and hence the R system, does exist. Smale's very pretty analysis is not intended to be a realistic example of a physical system (e.g., a `smoothing function' such as 0(r) has no physical basis). The purpose of this analysis was to produce an example of a system with the interesting properties (R). Its generalization to study the topological contributions of a many-cell system, or the influence of different numbers of morphogens, to say nothing about more realistic cellular dynamics (R(z)), offers many challenges for the future.
8.13 Embedding the dynamics of high-order dissipative systems in lower dimensional R" In general the study of the dynamics of high-order systems is largely impossible in any detail. However, if the high order system is dissipative so that its autonomous dynamics leads to a contraction of volume elements in phase space, the nonautonomous (driven) dynamics may tend asymptotically to occupy a rather limited
number of dimensions of the phase space. It clearly would be interesting to experimentally determine whether this in fact happens in specific situations, such as `chaotic' fluid flows, chemical or plasma dynamics - all of which have dissipative features. It is, of course, not a priori clear what ratio of dissipation to applied force is necessary to `reduce the dynamic dimension' to a level which makes the resulting dynamics more or less tractable. Thus it is essential to have an empirical method to explore this possibility, which has fortunately been established by several people (Packard, Crutchfield, Farmer, and Shaw (1980), and Takens (1981)). The first problem is to make clearer what is meant to `reduce the dimension' of the description of the dynamics. What we want is a `faithful' (one-to-one, continuous) representation of the dynamics which is occurring in the phase space (R": large n), in another Euclidean space, R' (m << n). This map, which is a homeomorphism between two sets guarantees that the two dynamics are topologically equivalent (by definition).
If we require that this map be differentiable (a diffeomorphism), then this map is referred to as an `embedding' of the dynamics into the space R. This is discussed in slightly more detail in Appendix B dealing with topological concepts. We limit the discussion here to some simple examples of embeddings.
'Moderate-order' systems
318
Suppose that the phase space is three-dimensional (e.g., as in the case of the Lorenz system), and assume that the trajectory of some equations satisfies x = Y2;
Z
= Y2 + Y
Fig. 8.53
as illustrated in
Fig. 8.53. Can this dynamics be `faithfully' described in a
lower-dimensional space by a diffeomorphism
(seR')? The answer of course is `yes', simply by setting
x=s2,
y=s,
and
z=s2+s.
This is not the only possible map, but that is not the point. The point is that all of the points of the trajectory in the original phase space are associated uniquely (and
differentiably) with seR'. We note that the set of dynamic points in R3 is one-dimensional, as is the embedding space, seR'. As a second example, assume that the dynamics yields the closed floppy circle
x2+y2=1;
z=3-y2
as illustrated in Fig. 8.54. This set of points is again one-dimensional. Can we embed
this set in a one-dimensional Euclidean space? We can represent this set with the
Fig. 8.54
ai,_ONDI
High-order dissipative systems in lower dimensional R"
319
help of a single angle variable, 0, simply by setting
x=sin0;
y=cos0;
z=3-cos20.
However 0 is not in R', because 0 and 0 + 2n must be the same point of the set, which is not the case if 0 is in R'. Thus, for this one-dimensional set, we need a two-dimensional Euclidean space (say, x, y) for the diffeomorphic association between
the two sets, both in Euclidean spaces. Note, for example, that specifying y alone does not uniquely determine the original point. These simple examples illustrate the fact that a smooth curve (a one-dimensional set) sometimes can be embedded in R', but can always be embedded in R2. So, even though the curve is in a three-dimensional space, it is not necessary to use R3 to give the topological information about the set (in the new space one loses, of course, information about the original distances, etc., until the inverse mapping is performed).
There is a famous embedding theorem (see Appendix B) which ensures that something like this is true in general, namely: a smooth manifold (set of points) of dimension n in a phase space can be mapped by a diffeomorphism into R', provided that m 3 2n. To make it clearer that 2n dimensions are generally necessary (as we saw above, in the case of the closed curve) consider a Klein bottle. It is a two-dimensional manifold, which we can attempt to represent in R3, as illustrated in Fig. 8.55. This is, however, not a faithful representation, because there is a two-to-one association Fig. 8.55
of points along the closed curve. Thus this two-dimensional set can only be embedded in R4. On the other hand, a spherical surface (two-dimensional) can be embedded in R3, and a disk (two-dimensional) can be embedded in R2, so the full 2n dimensions are not generally required for embedding. While many of us might appreciate the fact that the dynamics of dissipative systems
tends toward regions of lower dimension than that of the original phase space, not many of us would have any idea of how we might take advantage of this, and reduce the description of these systems. Clearly we cannot obtain a one-to-one association
'Moderate-order' systems
320
by simply projecting the motion onto some arbitrary subspace of the phase space. Even if there existed some such select subspace, we would not know how to find it without solving the dynamic problem - which is impossible, of course. Thus a clever idea is needed, which was supplied by Packard, Crutchfield, Farmer, and Shaw (1980) and Takens (1981). The idea is to construct the new embedding phase space by using only one variable of the original system (say the density, or some velocity component,
etc.) but measured at different times. Thus we may take as the coordinates of the new space
(x(t), x(t + T), x(t + 2T)..... x(t + mT)),
(8.13.1)
where x(t) is the selected variable, and T is essentially an arbitrary fixed time. This Euclidean space has dimension (m + 1). If the original dynamics is on a smooth manifold, the dynamics described by (8.13.1) will be a `faithful representation' of the dynamics in the original phase space, provided that m is sufficiently large. More specifically, the above representation of the dynamics is an embedding provided that m in (8.13.1) satisfies m > 2n, where n is the dimension of the smooth manifold on which the dynamics takes place in the original phase space (Takens, 1981). This process will be useful, of course, only if the value of m which is needed turns out to be fairly small (say, 6 or less). Part of the problem in obtaining this reduction is to determine the value of m which will in fact make the new representation an embedding.
An empirical approach to this problem will be discussed shortly. Before considering some experimental applications of this idea, let us construct the dynamics (8.13.1) from some simple known trajectories, (x(t),y(t),z(t)) in R3. For example, if we take a circular orbit x(t) = cos (7rt );
y(t) = sin (art),
z(t) = 0
(8.13.2)
and construct the trajectories in RZ (cos (7rt), cos (lr(t + T)))
(8.13.3)
for several values of T, we obtain Fig. 8.56. We see that, except for the singular case
Fig.8.56
T=3
T=3
T=1
x(t + T)
x(t + T)
x(t + T)
X(t)
High-order dissipative systems in lower dimensional R"
321
T = 1, all of these curves are embeddings and hence are topological circles. Moreover we would obtain similar figures using (y(t), y(t + T)). This is not the case if we used
(z(t), z(t + T)). However that again is a singular case, caused by the fact that the original circle is both planar, and lying in a plane which is orthogonal to a measured variable (z(t)). Thus, aside from these very singular ('nongeneric') cases, the curves (8.13.1) are a faithful representation of the original curve. Moreover, even though the circle has dimension n = 1, we did not need the full (2n + 1) = 3 dimensions to obtain an embedding. Next we will see that (8.13.1) does not generally give an embedding unless the full
(2n + 1) dimensions are used. To show this, we again consider a one-dimensional manifold which is a torus knot, Tp q, in R3, x(t) = cos 8.,(t)(1 + rcos Oq(t)) y(t) = sin 9p(t)(1 + r cos 9q(t))
z(t) = r sin
(8.13.4)
Oq(t),
where 1 > I r 1, OP = 27rpt, and Bq = 2ngt. Such knotted cycles occur in the Lorenz system,
as discussed in Chapter 7, so their consideration is not simply academic. We will consider the following `representations' of (8.13.4) (x(t), x(t + T));
(y(t), y(t + T));
(z(t), z(t + T)).
(8.13.5)
These differentiable maps of the set (8.13.4) into R2 are usually examples of so-called immersions. Immersions are maps which are everywhere diffeomorphisms in local
regions, but globally may be many-to-one maps. If an immersion, f, is globally one-to-one (i.e., f -' is everywhere unique), then it is an embedding. Thus the case T = 1 in (8.13.3) is `not quite' an immersion, because it is not one-to-one in the neighborhood of the points t = 0, 1. Except for these points it is an immersion, even thought it is usually a two-to-one map. Next we consider the trefoil knot, T2,3, with r = 0.8 in (8.13.4). If we take T = i in (8.13.5), we obtain the three very different looking curves in Fig. 8.57. Fig. 8.57
x map
y map
z map
'Moderate-order' systems
322
Exercise 8.15 What is the many-to-one character of each of the maps (8.13.5), when T = 2? That is, how many points of (8.13.5) have multiple pre-image points on the torus knot, and how many pre-image points are there? Which map is an immersion, and which are not immersions? In the neighborhood of what values of t do these maps fail to be diffeomorphisms?
If we change to T = 1.1 in (8.13.5), we obtain the three immersions in Fig. 8.58. While it is clear that the x and y maps have isolated many-to-one points, the z map may appear to be one-to-one. However, it is actually everywhere three-to-one, and Fig. 8.58
x immersion
y immersion
z immersion
it is the `generic' z immersion obtained from (8.13.5). This degeneracy is not the fault of (8.13.5), but rather the very special (nongeneric) character of the torus knot (8.13.4);
just as (8.13.2) is not a typical circle in R3. If we make the torus `fatter' in some regions, by setting r = ro + r, cos 6P(t) Fig. 8.59
(ro > rl ),
z immersion
then the z immersion (ro = 0.6, r, = 0.3, T = 1.1), is as illustrated in Fig. 8.59, whereas
the x and y immersions are not substantially changed. These examples hopefully make it clear that (8.13.1) is an embedding of (8.13.4) (dimension n = 1) only if we
High-order dissipative systems in lower dimensional R"
323
use the full 2n + 1 = 3 dimensions. Hence, in general, we cannot expect to do any better when using (8.13.1). As the first empirical use of the embedding (8.13.1), we consider the dynamics
(B(t), B(t + T), B(t + 2T))
(8.13.6)
in the case of the Belousov-Zhabotinskii chemical oscillations and chaos (Roux, Simoyi, and Swinney 1983). Here B(t) is the concentration of bromide ions (the bromide-ion potential) and T was selected to be T = 8.8 s. The reaction takes place in a stirring container, into which the reactants flow at a determined rate (Fig. 8.60). Fig. 8.60
C 7) X1(t) X2 (t)
X,\, (t )
to computer
The bromide-ion potential is monitored by a probe, and stored in a computer for later analysis. The temporal behavior of the potential depends on the flow rates through the container, as illustrated in Fig. 8.62. On the left there is regular periodic oscillations, with a set of sharp frequencies in the power spectrum (lower figure). This
system can have a complex bifurcation sequence between periodic states (not period-doubling) as the flow rate is increased. The observed `winding number' (- number of peaks/total number of peaks) vs the flow rate can behave similar to a devil's staircase (Maselko and Swinney, 1985), as illustrated in Fig. 8.61. Another example of such a staircase was also discovered by Harmon (1961) in an analog simulation of neurons. But we digress! Here we will focus on the chaotic oscillations, illustrated on the right in Fig. 8.62, with their corresponding broad power spectrum. Before the use of the embedding technique, this was typically information one could obtain about such behavior. Questions were raised as to whether the observed chaotic dynamics was due to external influences (an amplification of some form of noise), rather than due to any intrinsically generated chaotic motion. This important fundamental question can be answered with the help of the embedding representation, as we will soon see.
'Moderate-order' systems
324
Fig. 8.61
0.13
0.12
0.11
0.10
0.09
Residence time (hr)
We consider first the influence of selecting different values of m and Tin the space
(B(t),B(t+ T),...,B(t+mT)).
(8.13.7)
In the case of periodic oscillations, with measurements taken every 0.88 s, the two-dimensional portraits in Fig. 8.63 were obtained using m = 1. The different portraits were obtained by using different values of T = 0.88 K (seconds), where the value of K appears in the figures. If the resulting figures were embeddings of the
dynamical set, then they would all have to be topologically equivalent, which is clearly not the case. Moreover, of course, no intersections can occur in an embedding,
since it cannot occur in the original dynamical set. On the other hand, it appears very likely that (8.13.7) yields an embedding of this dynamics in three dimensions (m = 2), since the above figures look like a simple projection of a closed curve in R3 onto a plane in that space. Indeed the above immersions are considerably simpler than the previous x or y immersions of the torus knot, from which it is not obvious that the set (8.13.1) yields an embedding in R3. Note that the question is not whether
a set can be embedded in R3, since that is true of any periodic orbit, but instead whether the particular map (8.13.7) yields an embedding when m = 2. In the present case that is clearly true.
High-order dissipative systems in lower dimensional R"
325
Fig. 8.62 (a)
185
185
0
0. 170
E
12
18
24
30
0
6
12
18
24
30
Time (min)
Time (min)
1
-3 I
-5
5
0.00
0.01
Fig. 8.63
0.03 0.02 Frequency (Hz)
0.04
0.05
20
,.._
0.00
0.01
-__r__- -_-T----1 0.02 0.03 Frequency (Hz)
0.04
0.05
326
'Moderate-order' systems
In the case of chaotic dynamics, the resulting set is not a smooth manifold, but nonetheless this embedding method is very useful. The set obtained by Roux, Simoyi, and Swinney, when they used m = 2, is illustrated in Fig. 8.64. Assuming that this representation is an `embedding', as it essentially appears to be, one can investigate
the dynamics by constructing suitable Poincare surfaces of section, as illustrated in the left hand figure. The dynamics is illustrated in the adjoining figure, showing the flow towards the `surface' in one direction, and the divergence of nearby trajectories in another direction (positive Lyapunov exponent). They obtained the intersection of this `surface' with a series of Poincare surfaces which each make an orthogonal intersection with the `surface' at various locations, as illustrated by the numbers in Fig. 8.65. The resulting sets on the Poincare surfaces (Fig. 8.66) clearly indicate the stretching action (1 to 2) and folding action (3 through 8), followed by compression (8 through 1), which is characteristic of strange attractors. The similarity between these results and such simple models as those due to Rossler (see Chapter 7) illustrates clearly the generality of these basic dynamical properties. Moreover these results clearly indicate that the chaotic motion is an intrinsic (deterministic) feature of these systems, and not governed to any extent by external noise effects. Another illustration of the use of this technique is to determine the dimension of a turbulent state in a hydrodynamic system related to Taylor vortices (Brandstater et al., 1983). The experimental arrangement involves two concentric vertical cylinders, with water contained in the gap between the inner and outer cylinder (radius, r; and ro). When the inner cylinder is rotated with an angular velocity f2, the resulting fluid
flow depends on a variety of factors, among which is the Reynolds number,
High-order dissipative systems in lower dimensional R"
327
Fig. 8.65
B(ti)
Fig. 8.66
2
5
7
- Fold
R = r;S2(ro - r,)/ v, where v is the water's kinematic viscosity. For R < R, (a critical value), the fluid flow has the same rotational axis as the cylinder. As R is increased, the flow goes through a series of bifurcations, the first yielding a stack of regular Taylor vortices (rings). These are illustrated in the experimental Fig. 8.67, and also in Fig. 8.68(a) photograph (Fenstermacher, Swinney and Gollub 1979; also see Swinney
'Moderate-order' systems
328
Taylor vortices (rings)
Fig. 8.67
Laser beam
Photo-tube i(t)
1-4 - r; I
Laser beam ro
Fig. 8.68
(a)
(b)
(c)
(d)
High-order dissipative systems in lower dimensional R"
329
and Gollub). These photographs are obtained by suspending small flat flakes (e.g., fish scales or aluminum filings) in the fluid; the flakes align with the flow, so that bright regions correspond to vertical flows, and dark regions are radial flows. As R is increased further the flow bifurcates to a traveling-wave vortex structure
(R/R, = 6 in (b)), with 4 wave lengths around the cylinder. A further increase (R/R, = 16, R/R,: = 23.5) yielded the turbulent flow incorporated in the vortex structure shown in (c) and (d). As Coles (1965) discovered many years ago, these vortex states are not uniquely determined by R/RC. That is, if we express the radial fluid velocity in the form vr(r, t) _
Am p(t, r) cos (pk I z) cos (mO)
he found, for a given R, 26 distinct flow states with 18 5 p < 32 and 3 < m < 7. Thus this system has multiple basins of attraction, and where it ends up depends on the
history of its preparation. These fascinating features are discussed in the above references.
We return to the problem of determining the dimensions of one of these `turbulent' traveling-wave vortex states, using the embedding technique (8.13.1). As a dynamic variable, Brandstater et al., used the radial fluid velocity at a fixed point in space, as determined by the Doppler shift of scattered laser light from small suspended particles (see schematic Fig. 8.67). They recorded Vr(tk), for tk = kAt, k = 1, ... , 32768 and At = 6 ms. They then constructed the sets of points in spaces of different dimensions, m, (vr(x, t), Vr(X, t + T), . . . , ur(x, t + mT)).
(8.13.8)
where T > 20 At. They then considered cells of size e in these spaces, and determined the number of data points which occurred in the cell, N(E). By decreasing the size of the cells, s, they obtained an estimate of the dimension of the set in that space, using log N(E)
D =1im E'o IlogEI
(8.13.9)
The estimate of this limit can be obtained by plotting log N(E) vs log e, and determining the limiting slope. They did this for a number of dimensions m = 1, 2, ... , 8, with the results shown in Fig. 8.69.
An indication of the dimension of the turbulent dynamics in the original phase space can be obtained by increasing m until the dimension of the set in the `m space' no longer changes (at which point, the m space is an embedding space). As Fig. 8.69 shows (for R/RC = 13.7), the dimension of the turbulent dynamics appears to be slightly greater than 3 (see insert), and the Euclidean space (8.13.8) needed to represent (embed) this dynamics has to have at least seven dimensions (m = 6).
'Moderate-order' systems
330
...
Fig. 8.69 3
'
3
2
l
Z 2
MI.
0
2
4
6
8-
to
0
1
M
=1
m2
.0
..
.
0
0
1
log e
8.14 Some dynamics of living systems Certainly one of the most challenging forms of dynamics involves the organization, evolution, and interaction of living systems - whatever `living systems' means! When we look out the window, we all will generally agree on which objects are living, and which are inanimate. But when we attempt to define the term `living system' in some dynamic detail, we find that it is very difficult. Of course the variety of living systems which we see may cloud the issue at first. After all, nature has been evolving various forms of living systems for roughly 3.5 billion years, so there now is a vast range of complexity in such systems. This complexity is obviously very important, but it is not the basic issue in understanding the dynamics which constitutes a `living system'. We cannot presently hope to understand the detailed dynamics of complex systems, but instead we can only hope to model some aspects of selected dynamic units, such as reproducing molecular chains, metabolizing cells, collections of insects or animals, etc.
One of the basic features of living systems, which differs from many (but not all!) inanimate dynamic systems, is that they are open systems. Living systems exchange energy and matter with their environment, in order to sustain themselves (metabolism) and/or `organize' matter from their environment, in order to communicate genetic
information (reproduction) and interact in various ways with the environment, including other living systems, causing `survival of the fittest'. When living systems are modeled, some organized unit must be selected as the elementary unit of the dynamic system, without further consideration about how its subunits generate its dynamic properties - e.g., metabolic process, mobility, reproduction, etc. Thus, much of the `internal' complexity of `living unit' is frequently suppressed and we are left
Some dynamics of living systems
331
Table 8.1 Environment
Living systems
Primordial soup:
Organized structures: (M) Metabolic processes
Prerequisite molecules; temperatures, radiation; random/chaotic structure
M>
* (R) Reproduction-replication
4 Cells-protection of replicators-
Coherent features; daylight, temperature variations, food supplies,
genes.
4
seasons
`Survival machines' (Dawkins) organisms; mobility; local, diurnal, and seasonal reactions
Immune systems
Instincts Coherent elements: objects, obstacles, `dangers' Living systemssmall groups -'neighbors'
Memory, adaptive dynamics, learning
predator-prey
Emotions Homo sapiens
Social hierarchies
Fire, tools, cooking, clothes, weapons. Larger social units, societies', nations, religions, astronomy,
literature, philosophy, art, music
4
4 Reasoning, inventiveness, creative thoughts
Spirituality Introspection, esthetics
Science -experimentation, empiricism; machines, psychology, sociology, economics
`Extrasensory' perception
Computer interaction Interplanetary environments
Telepathy
b
332
'Moderate-order' systems
with the question of whether we can understand this unit's dynamics in terms of its interactions with other units at the same level of complexity. In order to obtain some appreciation of where the dynamics of this section falls within this broad range of complex systems, consider Table 8.1. It outlines, in a rather crude manner, some of the relationships between the environment and the evolution of complex behaviors in living systems. It should be viewed as simply a starting point for a controversial discussion! Once living systems become more complex, they of course become an active part of the environment of other systems, and some form of sociological evolution develops (Laszlo, 1987). None of the present models address this level of complexity. The modeling of living systems we will consider in this section relates to two types of dynamic units. One involves replicating molecular chains, while the second concerns units which have some group interactions ('predator- prey'). These two very different
units are indicated by * in the above table. All other complexities concerning these units (i.e., how they do what they do) are ignored. This selection is quite arbitrary, since considerable progress has also been made in such areas as memory, adaptive
dynamics, and learning. On the other hand, the most basic dynamic stages of metabolism and replication are presently controversial, at best (see references). In any case, these interesting topics are outside the scope of this study. To illustrate a model of reproductive dynamics, which includes effects due to mutations, an outline will be given of Eigen and Schuster's hypercycle model (1979). The system consists of N types of self-reproducing `polynucleotides'. Let xk(t) >1 0 (k = 1, ... , N) be the concentration of these units (Fig. 8.70), which contain v(k) digits Fig. 8.70
of information (molecular symbols). Thus xeR' _- {xeR" I xi >, 0 (i = 1, ..., N)}. The probability of an accurate reproduction of any digit is represented by 1 > q > 0. The probability for reproducing the unit is Qk = q"(k) (assuming independence), which is called the quality factor. The reproduction rate is denoted by Ak, which depends on the concentration of 'energy-rich' building materials. The decomposition rate is Dk < 1 ('death' rate). Finally, the population of species k can change due to mutation errors in the 1 population which is a `near relative'. Let Wk, be the rate at which an I unit contributes to a k unit. Of course I can only contribute to some k, provided it makes an error in its reproduction. The number of I errors per unit time is A,(1 - Q,)xi, and
Some dynamics of living systems
333
these errors must go to some other unit, so (8.14.1)
Y_ Wk, = Al(1 - Ql)
k'I relates the Qk with the WkI.
Putting these ideas together yields the linear rate equations xk = (AkQk - Dk)xk + Y_ Wklxl l#k
(8.14.2)
where all xk >, 0. The quantity (8.14.3)
Wkk = (AkQk - Dk)
is called the metabolic rate by Eigen and Schuster. With this notation, (8.14.2) becomes
xkWk,x,.
(8.14.4)
The eigenstates of (8.14.4) can be written Yk = Y_ «kIXI;
> Yk =
xk,
(8.14.5)
1
which satisfy Yk ='k.Yk-
(8.14.6)
The distribution, yk, of the populations {x,} is called the kth quasi-species. Of course if A k is complex, the quasi-species are the real parts of these expressions. Having obtained the linear interaction dynamics, they next model `natural selection', which is some constraint imposed by the environment. This can be modeled in several ways, but the most common method is to require that the total population is constant d C y xk(t) I = 0; dt
Y_
Xk = c (constant).
(8.14.7)
This is called the constant organization constraint. The rationale for this condition comes largely from a simplification in the following analysis. While some experimental situations may approximate (8.14.7), it is less obvious that nature would ever impose this condition, to produce `natural selection'. The condition (8.14.7) can be insured by adding a flux term, Fk, to (8.14.4), so that the dynamics is now given by xk =
Wk,xl + Dk.
(8.14.8)
'Moderate-order' systems
334
(8.14.7) is satisfied provided that
I Ok = - E Wklxl.
(8.14.9)
k,I
k
Taking Dk to be proportional to the fraction of k units (Dk = (xk/C)Xf;
E 1 k (total flux),
(D,
(8.14.10)
k
then fi, can be determined from (8.14.9), I Wk,xl = - Y_ Wkkxk - Y_ Wklxl
(D[
k,I
kQI
k
= - Y`AkQk - Dk)xk -E AI(I -Q1)x,, k
I
where we used (8.14.1) and (8.14.3). Thus D(, - - > (Ak - Dk)xk = - (.. Ekxk = - E(t) Y_ xk. k
(8.14.11)
k
The last two identities of (8.14.11) simply define the excess productivity, Ek, and the average excess productivity, E(t). Eliminating 1 k from (8.14.8) with help of (8.14.10) and (8.14.11), we obtain xk = E Wklxl - E(t)xk.
(8.14.12a)
I
Because of (8.14.7) we know that E(t) is given by
E(t) = c-' I WkIxl.
(8.14.12b)
kJ
Now, if we reintroduce the quasi-species (8.14.5), and recall the relationship between (8.14.4) and (8.14.6), then it is easy to see that (8.14.12) yields .
k = [2k - E(t)]Yk,
(8.14.13)
and
E(t) = c
Y Akyk.
(8.14.14)
k
While these equations are nonlinear, they can be explicity integrated. We note first that (8.14.13) can be formally integrated E(t') dt' 11
A = Yk exp r 2kt J0
(8.14.15)
Some dynamics of living systems
335
and (8.14.14) can be written E(t) = (Yk/c) Y_ 4Y,/Yk), I
which is true for any k. Using (8.14.15), we see that the ratio within the summation is independent of E(t), and E(t) = (Yk/c) Y_ ),(Y,°/Yk) e(AI -
Aku.
When this is substituted into (8.14.13), we obtain a Bernoulli differential equation (Chapter 3) Yk ='kYk
(A,y /cy) e(z - Ak)rYk
which has the solution (yr/c) eA'`.
Yk(t) = Yk elk`/
(8.14.16)
I
Thus we see that, under the constant organization constraint (8.14.7), the quasi-species
which has the largest `uncoupled' growth rate ' k' (8.14.6), continues to win out in this process of `natural selection'. This is illustrated schematically in Fig. 8.71, for three quasi-species.
Fig. 8.71
Yk
A basic problem in understanding the process of evolution is to explain how increasingly complex systems can evolve. Such systems necessarily must transmit increasing amounts of genetic information in the process of reproduction. In the present model, genetic information is represented by the number, v(k), of digits of information in the kth polynucleotide. Given a certain accuracy of reproduction of any digit, 1 > q > 0, the probability of accurately reproducing the polynucleotide,
336
'Moderate-order' systems
Q = q decreases exponentially as the number of bits of information increases. At some point, when v becomes too large, the reproduction equations (8.14.2) and (8.14.6)
will only yield quasi-species which decay in time. At this stage of the information reproduction a so-called `error catastrophe' occurs, and this reproductive process cannot yield a further evolution in complexity. Very roughly, from (8.14.2), if A and Q are the common reproduction rates and quality factors, we expect survival only if AQ > 1 (neglecting help from `relatives'!). Therefore, we require that Aq"> 1. So
v... z-, - log A/log q
is the maximum number of information digits (molecular symbols) which can be reproduced. If v > vmax, then the `error catastrophe' occurs.
To account for the reproduction of systems with more information, Eigen and Schuster (1979) suggested that there may be catalytic interactions between groups of self-replicating molecules (e.g., the above quasi-species). These groups individually
contain genetic information and are represented by a cycle, Ik (Fig. 8.72). The Fig. 8.72
concentration of this cycle will be represented by yk(t) >, 0. The concept of a hypercycle is one of a supportive (catalytic) interaction between these units 1k, which is moreover
cyclic in character (each unit is catalytic to at least one other unit). The idea is that a hypercycle may be able to reproduce more information than is possible by the
above simple quasi-species dynamics. This is frequently illustrated by various sentences, which contain more information than their individual words (Fig. 8.73). Fig. 8.73
Some dynamics of living systems
337
Each word catalyzes the succeeding word, to overcome errors in the reproduction of the words. The pure hypercycle with N units, yi (i = 1, ... , N), and interactions of degree p, has the dynamic equations N
Yi = Ti(Y) - (Yi/co)
r j(Y)
(8.14.17a)
j=1
where N
I Yi
co
i=1
F1(Y) = Kiyiy1
1
... y[,-P+ 1]
i - JA _ (i - J) mod N and yER". Notice that (8.14.17) again satisfies the condition therefore lie on a so-called simplex in R+,
(8.14.17b)
co = 0. The {yi}
SN={YIEYi=co, yER+}. The simplex S3 is illustrated in Fig. 8.74. Fig. 8.74 y3
'of
The so-called elementary hypercycle has interactions of degree two (p = 2) hi (Y) = Kiyiy1i _ 1)
(8.14.18)
as illustrated in Fig.8.75, for N = 4. Note that the interaction is directional, unless p=N. Eigen and Schuster (1979) have discussed a number of numerical solutions of the elementary hypercycle for various values of N. Briefly, they find that for N = 3, the motion on the simplex, S31 spirals into the fixed point. When N = 4, the motion again spirals toward a fixed point, but much more slowly. Finally, when N > 5 it appears that the system tends toward a limit cycle. These are illustrated schematically in Fig.8.76, where only the (y1, y2) projections are shown.
'Moderate-order' systems
338
Fig. 8.75
Fig. 8.76 Y2
Y2
Y2
yl
yt
An elementary dynamics of the self-replicating units, which is more general than (8.14.17), might presumably be of the form xi = aixi + a*xix[i
- (xi/c0) Z (akxk + ak xkx[k- 1]). k
The present hypercycle approximation, (8.14.17), views the dynamics in the Ik units as following equations of the form (8.14.12), which are less nonlinear (lower degree) than the hypercycle equations (8.14.17). At low concentrations, the dynamics might
therefore be governed by (8.14.12), whereas (8.14.17) might dominate at larger concentrations. This separation of dynamics presumably is similar to solutions of the last equation. See Eigen and Schuster (1979), Eigen, Gardiner and Schuster (1980), Niesert, Harnasch and Bresch (1980), and Eigen (1985) for differing views on the success of this separation of the dynamics.
We will avoid this important issue, and simply proceed to consider the more general, so-called replicator equations m-1
Xi=xi j=o
aijxj - c),
(8.14.19a)
where (D = E a jkx jxk. j,k
(8.14.19b)
Some dynamics of living systems
339
The xi are normalized so that the dynamics is confined to a simplex, Sm, M-1
1
xi = 1, xER+
(8.14.20)
i=o
The elementary hypercycle, (8.14.17)-(8.14.18), corresponds to the special case aij = Ki, if i = j + 1 mod N, and aij = 0 otherwise. Equation (8.14.19) was also proposed in 1930 by Fisher, for the case aij = aji, whereas Maynard Smith (1982) has proposed it to represent an evolutionary game, with (aij) as the payoff matrix. Hence the system (8.14.19) is of widespread interest.
Note that, if we alter the columns of the matrix (aij) by an arbitrary amount, bj, so that
aij=aij+bj the replicator equation, (8.14.19), is invariant
.zi=Xi Yaijxj - Eajkxjxk)
j
(8.14.21)
j,k
because of the normalization (8.14.20). Hence we can make any row of (aij) vanish, say ao j = 0. This will be useful shortly.
Exercise 8.16 Show that, in Fisher's model (aij = aji), (8.14.19) yields a `Theorem of Natural Selection', namely the mean fitness function '(x) - Y_i,jaijxixj, satisfies d(D/dt > 0. In other words, the mean fitness tends to a maximum value. This exercise is but one of many examples of the association of evolution with the optimization of some `fitness' quality of a system. In other words, evolutionary processes are frequently equated with some extremization process. Another example is the maximization of the average excess productivity, E(t), (8.14.12b), as illustrated there. Gradient systems are extreme (nonoscillatory) examples of systems whose flow tends to fixed points, which are extrema of some potential. Sigmund (1983) has reviewed some aspects of the application of gradient systems to various evolutionary processes, involving the use of the nonEuclidean Shahshahani gradient. These interesting topics are outside the scope of this introductory study. Another classic equation in ecology (e.g., predator-prey) is the general Lotka- Volterra equation n
Yi = Yi(aio \\
+ E aijyj
(yER+).
(8.14.22)
j=1
Here the aij may be positive, negative, or zero. The special case n = 2 was discussed in Chapter 5.
'Moderate-order' systems
340
As a particular example of (8.14.22), consider n
E ai jN;Nj
N; = kiN1 +
(aii < 0, /3i > 0).
(8.14.23)
j=1
Assume that the loss of one species in an interaction produces a gain in the other, so that aij = - aji, and the different efficiencies of the species are accounted for by the factors /3i. If we set Vi = In (Ni/qi), where the qi > 0 are arbitrary, then Vi=ki+/3i 11 aijgje1'i. We next assume that the qj > 0 can be selected so that ki/3i = - Y ai ja j
(assume det ( ai j I : 0)
is satisfied. Then /3i Vi =
aijgj(e1' - 1),
(8.14.24)
and Vj = 0 (j = 1, ... , n) is the only fixed point. Following Volterra, we multiply (8.14.24) by (e'i - 1) q1, and sum on i to obtain d
dt E
/3igi(eri - Vi) _ Y aijgigj(exp (Vj) - 1)(exp (Vi) - 1) i.;
_ Yaijg2(exp(Vi)- 1)2,
where the last equality follows from ai j = - a ji. Thus, since all aii < 0, d
dt E
f
igi(exp (Vi) - Vi) < 0.
(8.14.25)
We also note that (exp (Vi) - Vi) >, 1 for all values of Vi. Therefore, if we set L(V) = E /3igi(exp (Vi) - Vi - 1)
(8.14.26)
then L(0) = 0 and L(V) > 0 for all I V I > 0. Moreover, (8.14.25) shows that dL/dt < 0. Hence L(V) is a (global) Lyapunov function. These results show that no Vi can become arbitrarily large, so the system has bounded stability (Laplace stability). Moreover, if aii < 0 (does not vanish) then the associated manifold, Vi = 0, is asymptotically stable. Such boundedness can be used, as in the virial theorem, to make statements about the time average of the Ni, Ni
lim 1 f Ni(t') dt'.
(_OD t
J0
Some dynamics of living systems
341
Since V; is bounded it follows that the time average, (d V;/dt) = 0. So, taking the time average of (8.14.24), and replacing N;, we obtain Y- aij(Nj - qj) = 0.
j
Since det I a; j I :0, the only solution of this is the trivial solution
Nj=qj. Such information, coupled with the fact that the constant of motion (8.14.25) is additive, leads naturally to statistical mechanical considerations (see Goel, Maitra and Montroll, 1971).
Hofbauer (1981) has shown that the solutions of the Lotka-Volterra equation, (8.14.22), and the replicator equation, (8.14.20), can be topologically related, despite
their different degrees of nonlinearity. To show this nice result, first introduce a homogeneous coordinate, yo = 1, to the space (y1,.. . , yn), and define
z;=yr'
(i=0,...,n).
Y Yr
(8.14.27)
j=0
Then the z vector lies on the simplex l
Sn+ 1 =
{zeR++' Y- z; = 1 }.
(8.14.28)
r=o
By setting i = 0 in (8.14.27), it can be seen that the inverse transformation is (i = 0, ... , n).
Yr = zr/zo
(8.14.29)
Differentiating (8.14.27), and using (8.14.22) 2
it = ( .Yr'Y- Yj) -YiY-YJI (
Yj
)
n
a;iYj zo - Yr
= Yr aro + j=1
j
n
= Yr
Yj ajo +
n
ajkyk
z0z
k=1
n
ariYj zo - Yr I Y- aikYiYk j=0 j=lk=1
zo2
If we define a0j = 0, which is consistent with (8.14.22), and use (8.14.29) this becomes n
i; = z;
j=0
auzi -
ajkzjzk zo 1.
(8.14.30)
j,k=0
It can be seen that, aside from the factor zo '(t), this is identical in form to the
'Moderate-order' systems
342
replicator equations, (8.14.20), with m = n + 1. Of course, a0 = 0 in (8.14.30), but we saw in (8.14.21) that this could also always be arranged for the replicator equation. Thus, if we alter the time scale in (8.14.30), by setting zo(t) dz1;/dt = dzi/dT
T = f dt'/zo(t') 0 o
then the solutions zi(T) of (8.14.30) are the same as xi(t) of (8.14.20). This means that the x phase portrait of (8.14.20) and the z-phase portrait of (8.14.30) are identical. It is only the temporal scaling which differs. Note that the scaling differs for different solutions. This nice result shows that the replicator equation and the Lotka-Volterra equation have flows which are topological orbital equivalent (TOE). This is illustrated in Fig. 8.77. Fig. 8.77 Yo
Z3
Lotka-Volterra
X3
/Replicator I
S3
R+
1
S3
Y2
(8.14.29)
(8.14.27)
The replicator and Lotka-Volterra equations are special examples of the following system of ODE xi = xiMi(x)
(i = 1, ... , n)
(8.14.31)
where Mi(x) have the properties: (1) Mi(x) is a real function for all XER+(Mi:R" -+R), and has continuous derivatives of all orders (Cm). (2) For all pairs i and j between 1 and n, if xi > 0, 8Mi/8x; < 0. This is the classic 'crowding-inhibits-growth' condition. (3) There is a constant K > 0 such that, for each i = 1, ... , n, Mi(x) < 0 if II x II > K. This is the 'planet-is-finite' observation. Smale (1976) has pointed out that, these conditions are too general to allow us to conclude much about the specific dynamics, at least if n is large. Indeed, if n >, 5, the equations (8.14.31) are compatible with essentially any dynamic behavior, including the presence of strange attractors. Thus, the Lotka-Volterra and replicator equations, which yield topologically equivalent flows (by Hofbauer's theorem), do not lead to
Epilogue
343
specific behavioral conclusions (if n >, 5), unless further conditions are imposed on
the matrix (a, ). It is, of course, known that periodic solutions can occur if n = 2 (Chapter 5), and it has been proved (Hofbauer, 1981) that limit cycles can occur in the Lotka-Volterra equation if and only if n > 3. For larger n, if we impose the conditions a,3 = a3,, then we can obtain the mean fitness theorem (Exercise 8.16), whereas, if a, j = - a;;, we obtain the Lyapunov function, (8.14.26), and the resulting `virial theorem'. Yet another condition, [a,3 = K;, if i = j + 1 mod N; a,j = 0 otherwise] yields the elementary hypercycle dynamics (8.14.17), (8.14.18). Smale's theorem shows that such restrictions on the matrix (a;;) are generally necessary to draw such specific conclusions.
8.15 Epilogue: open systems; open sesame! This chapter contains a very limited number of topics concerning the dynamics of higher-order systems. A book such as this cannot, of course, do justice to many fascinating ideas concerning dynamics, which are still barely in their formative stages.
It might be useful, however, to at least sketch a more general perspective of these categories of dynamic systems than has been presented above. Probably the most important distinction to be made between models of dynamic systems is whether they deal with closed (isolated) systems, or with open systems. This distinction involves the explicit recognition that all systems exist in some environment,
whose influence on the system's dynamics we must either explicitly or implicitly assess (e.g., by ignoring the environment). For many processes it may be quite permissible to treat the system as if it is isolated from the environment. This depends on what processes are being studied, and for what duration of time. In point of fact no system is closed to the environment, for there are always gravitational, thermal, sonic, or radiation perturbations acting on systems. Borel pointed out in 1914 (see Brillouin, 1964) that the displacement of I gram by 1 centimeter on the star Sirius would result in a variation of the Earth's gravitational field whose perturbation on the motion of gas molecules would lead to our ignorance about the molecular collisions in roughly 10-6 s. With our present appreciation of the generic character of the sensitivity-to-initial-conditions, we must conclude that the above perturbations, which cannot be eliminated, play an ever-present role in such detailed properties of dynamics. Whether this is a more profound influence on higher-order processes than
we presently appreciate, remains to be determined. (As an `outrageous' example, perhaps such perturbations induce sensitive mental-dynamic processes to carry out search-exploratory cognitive processes which are involved in discoveries, inventiveness,
and our sense of free-will.) Whatever may be the outcome of such considerations, it
is clear that such unavoidable perturbations remove strict determinism from all systems, even within the context of classical mechanics (and, of course, quantum
344
'Moderate-order' systems
mechanics, which is based on the Hamiltonian formalism, which in turn is based on closed systems, is likewise approximate, at best).
This, of course, does not mean that many experimental phenomena cannot be treated as deterministic over `reasonable' intervals of time, and to `reasonable' precision. It does mean however that such discussions as the Poincare recurrence `paradox' of irreversibility, based on closed-system dynamics, are quite meaningless;
that we should not ignore realistic (open-system) dynamics when coming to any `understanding' of our observations. Open-system dynamics, however, holds many other exciting prospects, some of which we have discussed in this chapter. Once the system is acknowledged to be open, the nature of the environment needs to be specified. If it reacts to the system and is deterministic (e.g., competing species), then we have simply enlarged the closed system. To avoid this, an open system must interact with an environment which has some specific or statistically defined features. Thus, in chemical reactions, the reagents may be taken to be constant and homogeneous at all times, or, in mechanical systems
we may consider frictional or periodic forces acting on the system. In the study of
thermal conduction, the statistical conditions of the thermal reservoirs at the boundaries are specified. More sophisticated environments, involving inhomogeneous elements which are introduced in some specified fashion (e.g., food, chemicals, or any
resources, introduced in different regions at different times), and which require a mobile system (possibly also induced by a Brownian-motion action) offer a vast realm for exploration - essentially none of which has been discussed in this chapter. Finally, we should note that such open systems may be either passive or adaptive,
depending on whether their dynamical `rules' (abilities) are modified by past interactions with the environment. That is, in adaptive systems, the change of the system in the next time step is not simply a function of its present state, and the present state of the environment, but depends on its past history. This adaption may result from a selection process ('natural selection') from a pool of statistical mutations of dynamic systems (e.g., genetic mutations), or possibly from the reinforcement of certain dynamic channels, such as in neutral networks. The former adaptation may occur over many generations, involving many systems (units), whereas the latter may
occur in one system ('lifetime'). This, of course, leads into the area of cognitive processes, such as memory, learning, and possible inventiveness. Other, quite different types of systems, which at least should be adaptive, are beginning to be explored in
the field of economics (Anderson, Arrow, and Pines, 1988). There is clearly a wonderous world of possibilities, once we recognize the opportunities which are afforded by such open-system dynamics. Introductory, wide-ranging discussions, and other references, can be found in Davis (1988), Davidson (1983), Laslow (1987), and Pines (1985), to name a few. Figure 8.78 attempts to capture some of these perspectives of dynamic systems.
Epilogue
345
Fig. 8.78
Environment
00.
Elements which are precisely or statistically defined in space-time
Open, adaptive systems (dynamic properties modified by interactions)
Hamiltonian dynamics; quantum mechanics; many classic systems
Friction; applied forces; energy exchange; chemical reactions; reaction - diffusion; metabolic dynamics; reproductive dynamics;
Adaptive organization; `natural selection' from genetic mutation pool; speciation; memory, learning
Comments on exercises (8.1) Let L(x) - Yaijxj, and assume that L(x1) = 0 and L(x2) = 0. Let x(s) = sx1 + (1 - s)x2, so that x -+ x 1 as s -+ 1 and x --+ x2 as s - 0. Then L(x(s)) = sL(x l) + (1- s)L(x2) = 0 for all s. Hence the two fixed points are not isolated. (8.2) No comment. (8.3) Use (8.2.7) and (8.2.6) to obtain a., given by &10a = 0. Conditions which must be satisfied are t 0 v and be < 0. Then some algebra yields am = [a - d + (p + v)(- be/µv)1 2]/(µ - v), from which we obtain A... = [dµ - av 2(- ivbc)112]/(µ - v). (8.4) Your intuition may work better for these systems. (8.5) The functions are independent if (a, b + c, d) (0, 0, 0). The proof that the Gk are all in involution, {Gk, G;} = 0, is straightforward. Because of the form of H, it
follows that {Gk, H} = 0, so they are constants of the motion. Hence the conditions for integrability are satisfied. For further details see. J. Moser (1979)
Geometry of Quadrics and Spectral Theory, pgs. 147-88 in The Chern Symposium ed. Hsiang, et al., Berlin: Springer-Verlag. (8.6) For the lattice P. = 0'(Qk+ 1 - Qk) - D'(Qk - Qk- 1) With periodic initial conditions, P1 = b'(Q2 - Q1) - (D'(Q1 - Q3), P2 = (D'(Q3 - Q2) - 1'(Q2 - Q1), and P3=4'(Q1 -Q3)-D'(Q3-Q2). For the molecule (8.7.2), P1=Vi(Q2-Q1)+
V3(Q3-Q1),P2= - V1(Q2-Q1)+ V2(Q3 - Q2), and P3 = -V2(Q3-Q2) V3(Q3 - Q1). The equations are the same if b(x) = V1(x) = V2(x) = V3(- x).
'Moderate-order' systems
346
(8.7) Only the 3 term is changed to 5ac3/(3 x 21/2), so H = + p2 + q2 + q2) + Z(pi q q2 + (5/9)g2. The potential energy has a saddlepoint at q1 = 0, q2 i where V = 3/50. Therefore, if H > 3/50 the motion is unbounded. In the unbounded motion q2 -> - oo, which implies that Q3 - Q 1 - - co, so x3 - xl
- oo. This is unrealistic since it implies that particle 1 has passed through particle 3. (8.8) H=Ek=1(2Pk+2Qk+212/3Qk), where Q1=(q1+g2)21/2, and Q2=(q2-ql)2-1/2 The potentials are related by /3 = a/2. (8.9) 72 is arbitrary. See Toda (1981) for details. (8.10) The expression yields 2 + 4/32 + exp (2y) + exp (- 2y) __ t + ak, Xk - xk+ l = In Y 2 + exp (2Y) + exp (- 2y) C provided that /32 = sinh2 (a), (8.8.10). c = 2a, so vo = sinh (a)/a = 2 sinh (c/2)/c. (8.11) The figure is the same as in the text, except that the slow pulse has a negative velocity. The magnitude of I An I depends on B, (8.8.15). Since x >,U, sinh (p/2) > cosh (K/2)/cosh (p/2), so the shift is greatest in case (a). mf = (K + µ)m and ms = (K - µ)m. (8.12) C is not an additive function of the ps and qs, C 0 F(p) + G(q), as is H. Whether that has some significance (statistically, or otherwise) is not clear. Moreover,
it is unclear how the knowledge of such a constant sheds any light on the behavior of this system. The same is true of the constants, (8.9.6). (8.13) (A) The solution is an elliptic function and hence it is P-type. As in Example 3,
the leading term is x -T-', and the resonant term is unchanged. The Laurent series is, of course, changed. (B) Here the leading term is resonant, for x - T' yields the dominant terms a(a - 1)ATa-2 - 2AT°(aAT
=
1)2/(AT°)2
2Aa2T°-2
so a = - 1 and A is arbitrary. The solution is in fact x = tanh (Cr + D), which has no movable critical points. (C) Again the leading term is resonant, because the dominant terms are a(a - 1)ATa-2 - (aAT°- 1)22/(ATa) or
a(a - 1) - 2a2, so a = - 1, and A is arbitrary. Nevertheless this ODE is not P-type since its solution involves a movable essential singularity, x = tan [In (CT + D)]. This illustrates the fact that the present test is a necessary, but not sufficient
test for an ODE to be P-type.
Epilogue
347
(8.14) The analysis is the same as in Section 8.2, once (A, B, C, D, E, D, D3,) are related to (a, b, c) ICU, v).
(8.15) The x-map is everywhere two-to-one, except at the crossing and `end' points.
It is not a diffeomorphism in the neighborhood of t = 0,1 (end points). The y-map is an immersion; two-to-one at six points and four-to-one at one point. The z-map is six-to-one everywhere except the end points. Is it an immersion? (8.16) Multiply (8.14.19a) by aijxk, sum on (i, k), and use aij = aji to obtain Z aikxixk i,k
ai jx j -
402
j 2
= Exi > aikxk _ i
where we use Y_xi = 1.
k
0,
41
Solitaires: solitons and nonsolitons
In this chapter we will consider a limited, but important class of nonlinear wave phenomena which are described, of course, by partial differential equations (PDE). The phenomenon of interest is the occurrence of localized, traveling disturbances, which retain their structure due to nonlinear effects. These solitary disturbances ('solitaires') may occur in a variety of systems; solids, hydrodynamics, plasmas, biological molecules, optical systems, various field theories, etc., and are therefore of
great interest in many forms of `transport' or `communication' phenomena. Of particular interest is that, in many systems, these solitaires can retain their identity (shape and velocity) even after they collide with each other - which gave rise to the term `soliton'. In other systems, different solitaires can annihilate each other, or interact in more complicated two-dimensional processes. We will examine some of these features, beginning with the historical approach of
Zabusky and Kruskal (1964), which was stimulated by the Fermi-Pasta-Ulam recurrence phenomenon, discussed in the last chapter. This historical approach is by no means the most sophisticated mathematical introduction to this subject, but instead retains some of the flavor of the original discovery, including some analysis with obscure motivation (until, of course, everything fell into place!). Hopefully the reader
will find some heuristic value and stimulation in following this bumpy road to discovery - if not, some of the following sections may be omitted (as noted below).
The study of nonlinear, solitary, propagating disturbances has a long history, particularly in the field of fluid dynamics. Much of this history is reviewed by Scott, Chu, and McLaughlin (1973), with additional details concerning the research of D.J. Korteweg and his student, G. deVries, given by van der Blij (1978). While we will be primarily concerned with the developments in this area since the modern (-. 1960) rediscovery of Korteweg and deVries equation (historical reviews: Miura (1976); Zabusky (1981)), it is enjoyable to pause and read the oft-quoted description by J.S. Russell (1844), concerning his encounter with a solitary wave:
I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass
Solitaires: solitons and nonsolitons
349
of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished,
and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon... Moreover he recognized the importance of this observation, stating: The result here alluded to are those which concern especially the velocity and characteristic properties of the solitary wave, that class of wave which the writer has called the great wave of translation and which he regards as the primary wave of the first order.
Much later, Korteweg and deVries (1895) developed an equation for shallow water waves which provided the basis for an analytic study of solitary waves. In Fig. 9.1
particular, Russell's `heap of water' (Fig. 9.1) can be represented by the particular solution u(x,t)=h+asech2rx-
`\
b
ctl J
where the velocity, c, ambient depth, h, and the amplitude, a, and width, b, of the `heap' are interrelated. However these early studies of solitary waves did not discover the important fact that some solitary waves (in some systems) can retain their identity after they interact ('collide'). This relatively recent discovery, which was due to the
modern computer, is an outstanding example of what Ulam referred to as `synergetics'; the intelligent, selective use of computers for exploring ideas (Zabusky,
1981). One can, however, also observe this phenomena in much more relaxed surroundings, while dawdling in a swimming pool, simply by `launching' two pulses with your hands in the shallow drainage channel at the edge of the pool (Fig. 9.2).
Solitaires: solitons and nonsolitons
350 Fig. 9.2
9.1 The continuum limit of lattices and `solitaire' solutions We will begin this chapter by considering some of the physical consequences which result from a competition between the nonlinear amplitude-dependent velocity of a wave, and either conservative dispersive effects or nonconservative dissipative effects which occur in various systems. The former may lead to `solitons'; that is, localized nonlinear disturbances which retain their identity after interacting ('colliding' and `passing through' each other). The latter systems may produce quite different nonlinear dissipative pulsed structures ('shocks'). Following this, other nonlinear conservative structures that are not solitons in the sense just described, will also be considered
briefly in this limited study. Fortunately, there are a number of authoritative and readable books concerning solitons and nonlinear waves, which may be consulted for more complete discussions.
Since the discovery of the soliton effect was largely a consequence of Kruskal and Zabusky's study of nonlinear lattice dynamics, which in turn was stimulated by the recurrence phenomena found by Fermi, Pasta and Ulam (FPU; see Chapter 8, Section 6), we will approach the subject from this point of view. We will begin with a reduction of lattice dynamic equations to nonlinear wave equations (PDE), followed by a further reduction to the famous Korteweg-deVries equation. It is instructive to critically appraise these various reduction schemes as we go along, and some of these issues will be pointed out.
We consider here one-dimensional lattices consisting of particles that all have the same mass, m, and that only have nearest-neighbor interactions. If x is the position of the nth particle, the equations of motion are of the form 1).
(9.1.1)
Let h be the free-lattice constant, F(h) = 0, and y the displacement from equilibrium, so (9.1.2)
Setting f (z) - F(z + h), so f (0) = 0, (9.1.1) becomes
mjn=f(Yn+1 -yn)-f(yn-yn-1). The most common forces to be studied are: Polynomial:
f (z) = µ(z + K2Z2 + K3z3)
(9.1.3)
Continuum limit of lattices and `solitaire' solutions
351
(For example, to approximate the Lennard-Jones 6-12 potential, (8.7.18), K2 = - 10.5, K3 = 371/16.) Harmonic-plus-hard-core:
f(z) = µz,
z>-b
=-oo, z=-b Exponential (Toda lattice):
f (z) = a(l -
e-b,).
In contrast to either the polynomial or harmonic-plus-hard-core lattices, the Toda lattice is an integrable system, which is discussed in Chapter 8. Another way to write the equations of motion (9.1.3) is in terms of (9.1.4)
un=Yn+1 -Yn,
in which case (9.1.3) becomes (9.1.5)
This gives a particularly nice form for the continuum limit, to be considered shortly (Exercise 9.3).
Historically, two approaches were taken to explain the FPU results: (1) Formal perturbation methods (nonconvergent) were proposed by Ford (1961) and by Jackson (1963), which reproduced some of the FPU results. More recent perturbation methods for the special case of periodic solutions, that can be proved to be convergent, have been developed by Eminhizer, Helleman, and Montroll (1976). (2) The continuum approximation of the discrete lattice was developed by Kruskal and Zabusky (1964), using Riemann invariants to reduce the wave equation to a first-order differential equation in time (but not in space). Their computer
studies of this equation (the Korteweg-deVries equation) was in turn the impetus for the development of analytic theories of solitons and the inverse scattering method. We now follow Kruskal and Zabusky, and consider the continuum limit, to obtain nonlinear wave equations. To obtain a continuum description we replace the functions
of one independent variable by functions y(x, t) (or u(x, t)) of two independent variables, through the following substitution (h is again the lattice constant, (9.1.2), which is also used as an expansion parameter) (or
Yn(t)=Yfx,t);
Yn±1(t)=Y(x,t)±hYx+2h2Y2X+...
(9.1.6)
where ynx =- a"y(x, t)/ax". In other words, (9.1.6) represents a `transcription' of the lattice to Eulerian variables y(x, t). Similarly, for dynamics from Lagrangian variables, the variables (9.1.4) u(x, t);
u,,
1(t) = u(x, t) ± hux + 2h2u2x +
(9.1.7)
Solitaires: solitons and nonsolitons
352
This substitution is done for some arbitrary n, that effectively fixes the origin of x coordinate. For example, we might pick the origin such that y(x = 5, t) = y5(t). This method of introducing a continuous variable x is not entirely satisfactory, because the coordinate system is associated with a moving (accelerating) particle. This is a consideration that rarely concerns people, but will show up below. If we consider the polynomial force f (z) = p Y Knfn
(K 1 = 1)
(9.1.8)
n=1
and use the y variables (9.1.6) in (9.1.3), then 02
Y= P E
ate
Mn
[K.(hyx + Zh2Y2x + 3- h3Y3x...)n 1
- Kn(hYx - 2 h2Yzx + 3! h3Y3x + jj (9.1.9)
Keeping terms through 0(h4) gives a2
Y
ate
2
(2 h2Yzx + I h4Y4x) + m K2(2h3YxY2x) +3 m Ksh4(Yx)2Y2x.
(9.1.10)
The first two terms give the linear wave equation, with the inclusion of the dispersive effect, 02
2
4
ate = 2 ax + h(p/m)1/2 is the sound speed in the lattice, and A2 = h2/12 measures the discreteness of the lattice. It is this discreteness of the lattice that produces dispersion. The significance of this is that an initially localized disturbance (wave packet) will spread out (disperse) as time proceeds. This effect can be seen from the plane wave c=
solution of (9.1.11), y = A(k) cos (kx - (ot),
which must satisfy a dispersion relation w2 = c2k2(1 - k2A,2).
Hence the group velocity, aw/ak, depends on k, if ) # 0 (i.e., if the discreteness of the lattice is taken into account). This means that the different k components, which are required to construct a wave packet, will travel with different velocities, and hence disperse. This is a basic feature of harmonic lattices, which is worth reviewing.
Exercise 9.1 The dispersive effect in a harmonic lattice can also be demonstrated by using the following beautiful solution of the harmonic system (9.1.3), myk = U(Yk + l - Yk) - p(Yk - Yk - 1)
Continuum limit of lattices and `solitaire' solutions
353
The most recent discovery of this solution is due to Schrodinger (1914) (see Maradudin, Montroll, Weiss, and Ipatova (1971), for a history of this solution). First, we introduce the dimensionless time T = 2(µ/m)1/2t, and set
Z-Yk+1) _1
Z2k+1-2(Yk
2k=dYk/dT>
Then, from above, d
dYk
4(Yk+I-Yk)-4,(Yk-Yk-1)=
,
dT(dT
2(-Z2k+1+Z2k-1), 1
so
dZ2k/dT = i(Z2k- 1 -Z2k+ 1). Also dZ2k+ 1 /dT = 2 (dYk /dT - dYk + 1 /dT) _' (Z2k - Z2k + 2Y
Therefore, for all k,
dZk/dT=i(Zk-1 -Zk+1), and we note in passing that K, are all constants of the motion. The nice observation is that this equation is simply the recurrence formula for cylindrical functions, and those functions which are finite at r = 0 are the Bessel functions dfk /dT = 2 (Jk - 1
Jk+1)
We therefore can write the general solution Zn(T) _
Zk(0)J,,-k(T),
Y_
k=
o
where we have used the fact that Jk(0) = (Sk0. A nice feature of this solution is that it is easy to obtain the solution for an initially localized disturbance, and see how it disperses. Obtain the solution of the `plucked lattice' Yk(0) = ak0,
Yk = 0,
that is, only particle zero is initially displaced (Fig. 9.3); e.g., determine yn(T), and sketch this solution for several values of n, to see the dispersion of the initial pulse. Returning to the nonlinear effects, let us write (9.1.10) in the form +c2)2aa4y
ate
=c2aX
x4+I 2ax(ax)2+K3ax(ax)3.
(9.1.12)
While the general solution of such an equation is formidable (and unknown!), there is an interesting (two-parameter) family of special solutions that can be easily obtained,
Solitaires: solitons and nonsolitons
354 Fig. 9.3
by requiring that y(x, t) = Y(x - vot) - Y(z).
(9.1.13)
In this case, (9.1.12) reduces to an ordinary differential equation in z, which has the first integral (VI
- c2)Yz = c2 t2Y3z + K2(Y2)2 + K3(YZ)3 + b1.
Multiplying this by Y2z, we can again integrate to obtain 2(v0
- c2)(Yz)2 =
2c222(Y2z)2 + 3K2(Yz)3 + 4(YZ)4K3 + b1 YZ + b2.
This is a second order equation in Y, or a first order in w(z) - Y, (wz)2 =
-K3 w
z
z
2c 22
AZ
(9.1.14)
Since the right side is a quartic in w, the general solution can be expressed in terms of elliptic functions, with two arbitrary parameters (b1,b2). However the most commonly considered example (simply because it is easiest) is when K3 = 0, in which case (9.1.14) can be written
(wz)2 =c23(a 2 - w)(fl - w)(Y -w), where, because of the w2 term in (9.1.14),
(a+ 1+y)=3(vo-c2)/(2K2), and we can order the constants so that a
(9.1.16)
y. If we want a localized (w(z) - YZ) solution, then we require that w -> 0 as z -> ± oo. If K2 > 0, the right side of (9.1.15) is nonnegative only if a > w >, /3, or if w<- y. However wZ must vanish at some finite z, as well as when z --> ± oo. Hence wZ must have two distinct zeros, so a - w > P. If we require that wZZ - 0 as z --' ± oo, Since this must span w = 0, a >, 0 and 0 then both /3 = 0 and y = 0.
Continuum limit of lattices and `solitaire' solutions
355
Therefore (9.1.15) becomes
wz=
±G3-,
2
1
1/2
)
w(a-w)1/2.
Note again that we are considering the case of a lattice with a 'hard-stretch' (and ,soft-compression') nonlinearity, K2 > 0 (see Exercise 9.5). The solution of physical interest (note that the Lipschitz condition is not satisfied at w = a) is 1/2
2a12tanh '(((X -w)/a)1"2=(
Zz
21k2 )
which then yields a 112
1
w=asech2(2 z I=YZ.
(9.1.17)
(9.1.17) integrates to yield
)1/ 2zI. Y(z)=(6ac2,2/K2)1/2tanh((_I(22cc2
Introducing an amplitude A to eliminate a, and using K2/6c212 = 2K2/h, this can be written as y(x, t) = A tanh ((2AK2/h)(x - vot)).
(9.1.18)
The velocity vo depends on the amplitude, and is obtained by using (9.1.16), or a = 3(v02 - c2)/(2K2), so
vo = c2[1 + 2ahK2] = c2[1 + 3K2A2].
(9.1.19)
This solution is a displacement shockwave, illustrated in Fig. 9.4. Fig. 9.4
0
VO < 2
= vpt
h 2AK2
1401 Notice that in this case the lattice does not oscillate after the shock wave has passed by any point, indicating the very restricted character of the solution, implicit in the assumption (9.1.13). As x - ± oo all lattice particles are displaced +A from
Solitaires: solitons and nonsolitons
356
their original equilibrium positions (recall the meaning of y(x, t), (9.1.2), (9.1.6)), and the transition region is smaller for larger values of A (a `steeper shock'). Moreover the velocity of this shock wave increases with A, as given by (9.1.19). The width of the shock becomes infinite if K2 goes to zero, whereas it goes to zero if h -+ 0 (i.e., when the dispersion tends to zero), so this localization is a true competition between dispersive and nonlinear effects. The shock wave (9.1.18) has a somewhat different structure than the shock wave in the discrete Toda lattice, (8.8.11). Also the relationship between the velocity, vo, and amplitude, A, given by (9.1.19), differs from the vo(A) in the Toda lattice. However
these two shock waves are qualitatively similar, and depend on the same physical effects.
Exercise 9.2 If J3 :A y in (9.1.15) then we can obtain a variety of nonlinear periodic waves for the function w. Show that this is true, with the help of Appendix F, and interpret the physical significance of some of these solutions in terms of an expansion of the lattice (what is required for no net expansion?). In contrast to the displacement y(x, t), the continuum equation for the function u(x, t), defined in (9.1.7), is obtained from the equations of motion (9.1.5).
Exercise 9.3 Show that (9.1.7) substituted into (9.1.5) yields, for any f(u) = µY-K"u", the rather pretty result 2
m at2 = 4p sinh2
(2
ax) f[u(x, t)]
Here, sinh((h/2)(a/ax)) is the operator defined by the usual power series expansion of sinh (z).
To O(h2c2) and O(u3) the last equation reduces to
a2 / h2 02U =c2axel u+l2
alt! _ ate
2
h2K2 a2u2 axe
axe+K2u +-12
3
+K3u +
h2K3 32u3 12
axe
Using the dimensionless space variable (12)"2 x/h - x, and time t(µ/ 12m)112 -> t, this becomes utt = (u + K2 U2 + uxx + K2(u2)xx + K3U3 +K3 (U3)xx)xx .
(9.1.20)
Frequently the last three terms are simply ignored (if K2:0), yielding the Boussinesq equation utt = [u + u2 + uxx]xx,
(9.1.21)
Continuum limit of lattices and `solitaire' solutions
357
where u replaces K2u of (9.1.20). Notice that, to this order, the hard-stretch (K2 > 0)
and soft-stretch (K2 < 0) nonlinear dynamics are simply related. A direct attack on the general solutions of this equation, using the inverse scattering method (see Section 8.6), has been described by Zakharov (1974). We could, of course, again look for traveling disturbances, similar to (9.1.13) for
y(x, t), and again obtain and analogous result to the shock wave (9.1.18). Now, however, since y(x, t) and u(x, t) are related by (9.1.4), together with (9.1.6) and (9.1.7), to lowest order u hyx. If we differentiate the shock wave solution, (9.1.18), we obtain a solitary pulse for u(x - vo t). This `solitaire' (a nonlinear `gem', set alone) travels at
a constant speed, and with a constant shape (Fig. 9.5).
Fig. 9.5
it A
x
Exercise 9.4 Obtain the particular solution of (9.1.21) illustrated in Fig. 9.5, for arbitrary amplitudes, A.
Exercise 9.5 Obtain another particular solitaire solution u(x, t) = g(x - vo t) (scaled variables) for the Boussinesq equation, (9.1.21), which satisfies the boundary condition limx- Y,,,, u(x, t) = u,,. This is physically similar to Russell's `heap of water', discussed
in the introduction, rather than the displaced lattice.
In contrast with `solitons', to be discussed shortly, we do not make any study here concerning how one solitaire may interact with another solitaire. They are, at present, simply isolated solutions. Exercise 9.6 If K2 = 0 in (9.1.20), and we retain only the `lowest order' nonlinearity, the scaled equation becomes utt = uxx + uxxxx + K3 (u3)xx
If K3 > 0 (hard nonlinearity), obtain the solution u(x, t) = w(x - vt), with arbitrary max u = A, satisfying u( ± oo, t) = 0. Show that there is no such solitaire solution if K3 <0.
358
Solitaires: solitons and nonsolitons
9.2 Riemann invariants, and the Korteweg-deVries equation We now consider a special nonlinear wave equation of the form (9.2.1)
Y:t - [F(Yx)]2Yxx = 0
and will first reduce this to a pair of first order equations in terms of the functions u = Yx,
namely
(9.2.2)
V = Yt
Ut-Vx=0 v, - F2(u)ux = 0.
To make these more symmetric, we multiply the first equation by F, then add or subtract the second equation, thereby obtaining
v,+Fu,-Fvx-Flux=0 -v,+Fu,-Fvx+F2ux=0. We can write these in the neater forms
(V+ f F(f1)d?1)-F at
(-v+
(v+
JF(7)di)=0
F(n)dn)+Fax(-v+
J0
Jo
F(n)dn)=0.
(9.2.3)
We now clearly should introduce the functions (the Riemann Invariants) r(x, t)
-v+
F(n) dn;
s(x, t) - V +
fo
F(n) dn,
(9.2.4)
fo
a nd note that r + s = 2 I F(n) do - 2G(u).
(9.2.5)
We can then write
u=G-'(2(r+s)),
G-'(G(u))-u
(9.2.6)
where G-' (`arc G') is the inverse function of G in (9.2.5). Such a function exists and is differentiable if, F(n) > 0. Hence (9.2.3) can be written in the form at+F{G-'[i(r+s)]}
8s
at
=0
- F G-'[i(r+s)]} as = 0 {
which now only involve the functions r and s.
(9.2.7)
8x
ax
,
(9 . 2. 8)
Riemann invariants and the KdV equation
359
The function r does not change along the trajectory dx/dt = F, whereas s is invariant
along dx/dt = - F. Moreover, from (9.2.8) we see that if s(x, 0) = 0 is the initial condition for s, then s(x, t) - 0, and
r,+F(G-'('r))rx=0.
(9.2.9)
This is a right-traveling disturbance, provided that F > 0. It has the (implicit) general solution r(x, t) = H(x - F {G -' [zr(x, t)] } t)
(9.2.10)
with an arbitrary function H(z). Thus, by introducing the Riemann invariants, (9.2.4), we have obtained a general family of right-traveling solutions, (9.2.10), of (9.2.1). Now, returning to lattice systems, to 0(h4), we found (9.1.10) 02
at2
-c2[1 +2K2hax+3K3h2(ax)Z]axZ+c212az4 4 c21+k2ax+k3(ax)2Jaxz+c2A2ax4. 2
(9.2.11)
Except for the last (linear, dispersive) term this is in the form (9.2.1), which is needed for the introduction of Riemann invariants. For present purposes we will keep only the lowest order nonlinearity, and set k3 = 0, so that F(u) = c(1 + k2 u)112,
(9.2.12)
where u is given by (9.2.2). Then, from (9.2.5), G(u) =
JF(n)dii=3k2 [(1 + k2u)3/2 - 1]
and 2/3
k2
G 1(u)
(9.2.13)
so that
F(G-1(u))=c ( 1 + 3k2 2c u ) 1/3
(9.2.14)
Therefore, except for the last term in (9.2.11), we could decouple the r and s Riemann invariants by taking s(x, t) - 0, and obtain 1/3
at+c(1+ 4c r)
ax=0
as a particular class of solutions to (9.2.11). However, retaining this last term in
Solitaires: solitons and nonsolitons
360
(9.2.11), we readily find that the equations for r and s become ar+F{G[2(r+s)]}az=-c2),2uxxX
at
-
as
F{G-' [2(r + s)] }
= + c2),2uxxx
(9.2.15)
(9.2.16)
and, by (9.2.6), (9.2.17)
-1(
Now it no longer follows from (9.1.17) that s(x, t) - 0 for all t, if s(x, 0) = 0. At this point it is frequently simply assumed that s(x, t) remains small if s(x, 0) = 0, and one then proceeds to obtain the equation for r from (9.2.15). This is further simplified by retaining only the linear approximation of (9.2.17), i.e.
1 G-' (r)= 2
2/3
3k
k2(4c r+
1
-k2
2c r.
In that case (9.2.15) becomes, after expanding the coefficient of rx, rt + crx + (k2/4)r rx + (ch2/24)rxxx = 0.
(9.2.18)
This is the so-called Korteweg-deVries equation (1895), frequently denoted as the KdV equation. It contains the lowest order nonlinear and dispersive effects on a conservative system (very conservative, as we will see!). Usually the transformation is made to the moving coordinate system
x'=x-ct,
t'=t
and r is rescaled in some manner to remove one coefficient in (9.2.18). For example, set r = (3k2c)u, so that
U,+uux+62u3x=0
(9.2.19)
where the primes are dropped from (x', t'). By scaling x = ox', t = 8t', and again dropping primes this becomes ut + uux + u3x = 0.
(9.2.20)
If u = - 6u' this becomes yet another favorite form (dropping primes) U, - 6uux + u3x = 0.
(9.2.21)
These are all common forms of the KdV equation. As in the case of the equation (9.1.12) and (9.1.20), that involve second order time derivatives, the KdV equation can easily be shown to possess traveling `solitaire' solutions.
Comparison of the Burgers and Kd V equations
361
Exercise 9.7 Obtain the solution u(x, t) = g(x - ct) of (9.2.20) subject to the boundary condition u( ± oo, t) = u,,,,. Obtain the relationship between the amplitude (A = uo - u., uo = max g) and the velocity, c, of this solitaire solution. Also determine the `width' of the pulse in terms of A. Compare this with the solitaire solution (9.1.17) of the nonlinear wave equation (9.1.12), with K3 = 0-
9.3 A comparison of the Burgers and KdV equations In the previous sections we have considered a conservative system (a lattice), and have seen that the dispersive effects can be balanced against suitable nonlinearities to produce solitary traveling solutions. In this section we consider another famous equation, the Burgers equation (1948), that involves a nonconservative, dissipative ('viscous') effect, rather than a conservative, dispersive effect, in competition with the same nonlinear effect that occurs in the KdV equation. To illustrate some physical systems in which these two equations arise, we consider a fairly general system of `fluid' systems. Let (n, u, f) represent state variables, that are functions of (x, t), and which are governed by n, + (nu)x = 0
(9.3.1)
(nu), + (nut + P)x = 0
(9.3.2)
F(n,u,f,nX,ux,fx,nxx,...)0
(9.3.3)
where (9.3.4)
P = P(f, fX, U, uX, uXX, uXI ).
Specific physical examples of such equations (Su and Gardner, 1969) are given in Table 9.1. Table 9.1
Gas dynamics Water waves
P
F
m- '(p - µux)
p - Any
f
0
a
b
c
YKT/m
0
2v
0
gho
0
0
3gho
density
B0
1
0
1
density
1
0
0
1
p (pressure) density
Zgh2-3h3
(uX, + uuxx - uX)
vo
n
0
n-h depth of water
Hydromagnetic waves
Ionacoustic waves
B-n -(Bx/n)x
ZB2
exp (- 0) - 02
i
n - exp + ixx
B field elec-
trostatic potential
362
Solitaires: solitons and nonsolitons
= ea(x - vot), r = Ea+ 1 t, where vo is the linear wave velocity, u « - llo uxx = 0, and consider perturbations n = no + En(" + ; u=O+Eu()+ we can obtain f=
If we introduce the variables
n(i)n(1)+ea-1bn(i)+E2«-1_C nt') +(_+_ Ao no)
q(1)
0.
(9.3.5)
If b < 0 then the system is dissipative, in which case we set a = 1, and the last term of (9.3.5) is negligible. This gives the Burgers equation
rlt+rp
-zrU44=0.
(9.3.6)
If b = 0, then set a = Z. This case corresponds to no dissipation, but it does have dispersive effects, and yields the Korteweg-deVries equation
nt + qi + 62,444 = O.
(9.3.7)
Ut + UUx = !Uxx
(9.3.8)
Thus the Burgers equation
and the Korteweg-deVries equation Ut +UUx = -
62Uxxx
(9.3.9)
represent dynamics in quite different types of systems. The Burgers systems dissipate u(x, t) through viscous-like effects (the uuxx term), whereas the Korteweg-deVries system disperses u(x, t) (through the uxxx term). Both have the same nonlinear term, that tends to produce a `discontinuity', or shock structure in u(x, t). Let us first consider the shock-producing effect of the nonlinear term. The general solution of Ut + c(u)ux = 0
(9.3.10)
u(x, t) = F(x - c(u)t)
(9.3.11)
is
where F(z) is an arbitrary function of z, and c(u) is any (given) differentiable function of u. This may be proved directly by differentiation of (9.3.11), Ut + c(u)ux =- - F'c't(ut + c(u)ux), which is satisfied by any solution of (9.3.10). The solution (9.3.11) is an implicit solution
for u, but it is not difficult to see that it generates an infinite value for ux after a finite time. What happens is that the regions with larger values of c(u)(= u in the KdV or Burgers equations), catch up with the slower regions, causing a `shock' to develop (i.e., a point x at which I ux I = oo). This is illustrated in Fig. 9.6.
Comparison of the Burgers and Kd V equations
363
Fig. 9.6
Indeed we obtain from (9.3.11)
ux = F'(1 - c'uxt) so
ux =
F'
1 + F'c't
and this becomes infinite at the space-time point where
1+F'c't=0. Therefore the solution becomes singular for dF
dc
tits
L(du)(
dZ )]max
(9.3.12)
In the present case c'=- 1, so that the time to the singularity depends on the maximum negative slope of the initial condition u(z, 0) - F(z). Counteracting this tendency to produce a singularity are the linear dissipative (or dispersive) terms in the Burgers (or KdV) equation. To illustrate the dispersive effect, consider the linear KdV equation (again transforming to remove the uu, term) ut + uxxx = 0.
(9.3.13)
The general solution of this equation can be expressed in terms of the Airy function / 3
00
Ai (z) =
It
1/2
f0
cos
(3 + vz ) dv
(9.3.14)
which gives the bounded solution of J
w"(z)-zw=0. Note the Airy function has the property li
m
X1/2 (3t11/3 Ai I (3i)1/3) = 6(x).
Exercise 9.8 Prove (9.3.15) using (9.3.14).
(9.3.15)
Solitaires: solitons and nonsolitons
364
The general solution of (9.3.13) can be shown to be u(x, t)
Ir1/2 (3t)1/3 J 0000
Ai [(31)1 3] u(x', 0) dx'.
(9.3.16)
To appreciate how the initial state, u(x, 0), becomes dispersed, consider the sketch of Ai (z) (Fig. 9.7). Ai(z)
Fig. 9.7 IzI-1/4cos(2
1z13/2- 1)
r0.6
zz-1/4 exp(-3z3/2)
4
= x/(3 1) 1/3
2
For example, if initially u(x, 0) = 8(x) then (9.3.16) or (9.3.15) -yields U(x, t) _ X1/2 (3t) - 1/3 Ai
\(3 )1/3/
At different values of t, this behaves approximately as illustrated in Fig. 9.8. Recall Fig. 9.8
U (x, 0)
t=0 x
x
t>>0 x
again that x in (9.3.16) refers to the linear sound wave frame of reference traveling in the positive direction. Thus the above figure indicates oscillations after the wave
arrives and also an exponential precursor to the main disturbance. The spatial asymmetry comes from the uxxx term. The linear Burgers equation Ut = fIUxx
Hopf-Cole transformation
365
is symmetric in space, diffusing in both directions. However the solution of the nonlinear equation (9.3.8) is as simple as that of the diffusion equation, thanks to the Hopf-Cole transformation, which we now consider.
9.4 The exact solution of Burgers equation - the Hopf-Cole transformation In 1950 E. Hopf presented (The Partial Differential Equation u, + uux = µuxX, Comm.
Pure Appl. Math. 3, 201-30) a remarkable transformation of the nonlinear Burgers equation (also given by J. D. Cole, Quart. Appl. Math. 9, 225 (1951)) U, + uux - µu.. = 0
(9.4.1)
into the linear diffusion equation. He introduced a new function F(x, t) through the relation [also known in 1906 as a Backlund Transformation - A. R. Forsyth, Theory of Differential Equations, Vol. 6, Chap. 21 (Dover, 1959); also see the section on Backlund transformations] u=-2p10 -
-
inF(x,r).
(9.4.2)
Then, u,= -2µF-'F,x+2µF-2F,Fx and ux= -2µF-'Fxx+2µF-2(Fx)2, and finally, uxx = - 2µF -'Fxxx + 6µF -2 Fx Fxx - 4µF --3 (Fx )3. Forming the combination (9.4.1) yields
- 2µF-' F,x + 2µF- 2 F, Fx + (- 2µF -' Fx)(- 2µF -'Fxx + 2µF- 2(Fx)2) +2 µ2F -' Fxxx
- 6µ2F -2 Fx Fxx + 4µ2F -3 (Fx )3.
The underlined terms cancel, and two others simplify. If we multiply by - F2/2µ, we obtain FF,x - F, Fx - µFFxxx + pFx Fxx = 0, or
F ax (F, - jFxx) - Fx(F, - pFxx) = 0. a
Hence axln[(F`-pFxx)IF]=0
or
F, - pFxx = g(t)F
where g(t) is arbitrary. However this can be written
[Fexp(_Jtgdt')]_P_c[Fexp(_J1gdt')]=o.
Solitaires: solitons and nonsolitons
366
Therefore, if g # 0, it simply gives a multiplicative function of t to the function F(x, t). But such a multiplicative function of time has no influence on u(x, t), because of the
logarithmic derivative in (9.4.2). Hence g(t) can be taken to be zero, and F then satisfies the linear diffusion equation
F, -,uFxx = 0
(9.4.3)
with the general solution F(x, t) = (4nµ t)
_' I2
dzex p
(x 4µt )2 } F(z , 0)
(9 . 4 . 4)
and the multiplicative factor again has no influence on u(x, t). We can obtain a special (traveling wave) solution of (9.4.1) by setting u(x, t) = u(x - ct) - u(z), then from (9.4.1) dz(ZU2-cu-pUZ)=0
or 2µ dz
u2 - 2cu - a.
This has the bounded solution - 2,u tanh - i (a+c2)1/2
(a+c2)'/2-z.
Set A - (a + c2)'"2, then u(z) = c - A tanh
A
(z = x - CO.
(9.4.5)
The thickness of this shock wave (Fig. 9.9) decreases with increasing amplitude, A,
but the velocity is independent of the amplitude, in contrast with the KdV shock, (see Exercise 9.7). Fig. 9.9
History of the inverse scattering transform
367
Exercise 9.9 How is it possible for the solution of a diffusion equation (9.4.3) to give rise to the traveling shock solution (9.4.5) - that is, why doesn't the shock also diffuse?
9.5 A brief history leading to the inverse scattering transform (IST) In 1965 N.J. Zabusky and M.D. Kruskal reported on their numerical studies of the dynamical behaviour of the Korteweg-de Vries (KdV) equation (6 = 0.022).
u, + uuX + 62UXXX = 0
(9.5.1)
First of all, they discovered that the KdV equation not only sustains solitary propagating disturbances (which had been known since the turn of the century), but
that large amplitude waves tend to break up into a spatial series of pulses, with different amplitudes and velocities. Figures 9.10 to 9.12 illustrate the temporal Fig. 9.10
3.0
t=0
-1.0
0
0.50
1.0
1.5
2.0
1.5
2.0
Normalized distance
Fig. 9.11
t=tS
0
-1.0
0
0.50
1.0
Normalized distance
Solitaires: solitons and nonsolitons
368 Fig. 9.12
3.0
t=3.6t,
-1.01
1
1
0.50
0
1.0
2.0
Normalized distance
sequence of a wave shape produced by (9.5.1), with periodic boundary conditions, beginning with the initial wave from u(x, 0) = cos (irx). Figure 9.11 shows the formation of a strong shock, due to the second term in (9.5.1), as discussed in Section 9.3. The time
at which the solution of u, + uux = 0 becomes singular, is given by (9.3.12), where F(z) = cos (itz) in the present case. Since - dF/dz = it sin (nz), which is maximum at z = z, the solution is singular at t, = 1/n. Figure 9.11, at t = ts, shows that the dispersive
term in (9.5.1) not only controls this singularity effect, but gives birth to a smaller scale wave formation. Figure 9.12 shows eight recognizable pulses at t = 3.6tS. These pulses move in the direction indicated in the figure, in the present frame of reference (recall the transformation used in going from (9.2.18) to (9.2.19)). While the behavior of these interacting pulses is not at all obvious at this point, it is possible to make a rough estimate of their motion while they are `separate and distinct', by using the solitary solutions of (9.5.1). From Exercise 9.7, we know that these solutions are u=
(uo - u,,) sech2 [y(x - Ct + ¢)],
(9.5.2)
where 0 is an arbitrary space-time shift, and y2 = (u. - uo)/1252;
C = U. + 3(uo - u.).
(9.5.3)
The value of uo for one of the above pulses can be simply estimated from the maximum of u(x, t = 3.6ts), however the determination of u. for each pulse, from this figure, is more subtle.
Exercise 9.10 Given the solution (9.5.2), and the data illustrated in the figure (at t = 3.6ts), how would you estimate u., for each pulse, and thus predict its width, '/2y , and velocity, c, from (9.5.3) and the value of uo?
History of the inverse scattering transform Fig. 9.13
tR
1
1
Qt
t .
6
I
t
t
1
\
ll©.
3
OO
;\
1
n0
`®
1
Q7
0.6 tR
3 \'0C
1
tR/3
\1
1
(s)
6I
t
tR /4
.Q . 1
z
tR/5 `
tR /6
0.ItR
3
2
\
7
9
8
:,
0
0.5
1.0
t
14
7
IN
1.5
Normalized distance (x)
2.0
369
Solitaires: solitons and nonsolitons
370
A few of their results, obtained this way, are shown in the table. Width (y-1)
Velocity (c)
P u lse
number 1
3
4 8
observed
calculated
0.0455 0.0492 0.0522 0.099
0.0456 0.0493 0.0516 0.109
observed 227 0
- 99
-443
calculated 254 4
-105 -353
The second discovery of Zabusky and Kruskal was that these pulses retain their identity after they interact. This is illustrated in the space-time diagram (Fig. 9.13) (Toda, 1975), showing the motion of the maximum of each pulse (when it can be identified). We see that, while the diagram is complicated, the pulses seem to be `resurrected' after a complicated interaction. The only effect of their interaction is a shift in their space-time lines, corresponding to a temporary `acceleration'. During their interaction their `combined' amplitude decreases, rather than increasing, as we find in the case of the linear superposition of disturbances. Zabusky and Kruskal coined the term `soliton' to describe a solitary, uniformly propagating disturbance, which preserves its structure and velocity after an interaction with another soliton (Fig. 9.14). Other definitions of `solitons' have subsequently been Fig. 9.14
t
occasionally introduced, which allows for the production or destruction of solitary disturbances upon interactions, but we will use the original meaning unless otherwise specified. In particular, somewhat prior to Zabusky and Kruskal's discovery, Perring
and Skyrme (1962) had obtained numerical solutions of another equation (the `Sine-Gordon' equation), 0,t - Oxx + sin 0 = 0
(9.5.4)
in which they found similar (and different) `soliton' properties. They even managed to
guess a number of analytic solutions with two such interacting disturbances, such
History of the inverse scattering transform
371
as the two solitons
sinh [x(1- c2) lie] \4 JJ = cosh [ct(1-c2)
tan (
c
(9.5.5)
Exercise 9.11 Sketch ¢(x, t), from (9.5.5), in the two limits t - ± oo. Determine the influence of the interaction between the two disturbances. These solitons have quite different physical origin from the above considerations, and
will be discussed in Section 9.9. Moreover, it was the study of the KdV equation which led to the general method of solving 'integrable' PDEs, as we will see. Kruskal and Zabusky, and then Miura considered the question of the existence of conservation laws of the KdV equation, namely equations of the form aN
aJ
Wt +ax =0
(9.5.6)
where N = N(u, ux, u2x, ...) is a `density', and J = J(u, ux,...) is the associated `flux'. The most common such conservation equation is the continuity equation an
0
T + ax (nu) = 0 of gas and fluid systems. If J -p 0 as x - ± oo (or if it is periodic in space) then the total `number', f N dx, is conserved, because d
°°
dt
N(u, ux, ...) dx = 0.
(9.5.7)
Ten such conserved quantities for the KdV equation (9.2.21) u, - 6uux + uxxx = 0
(9.5.8)
were found (see Miura (1976) for a history of this development). The first few are N1 = u;
N2=U2 ; N3 = U3 +
J1 = - 3u2 + uxx (the KdV equation) J2 = - 4u3 + 2uuxx - uX 2UX;
(9.5.9)
J3 = -2U4 + 3u2uxx - 6uuX + Uxuxxx - 2Uxx
Clearly these conserved quantities have none of the immediate physical significance that we associate with the continuity equation.
It was conjectured that the KdV equation had an infinite number of such conservation laws, and this was later proved by Kruskal and Miura, and simultaneously by Gardner. If a PDE has an infinite number of polynomial
372
Solitaires: solitons and nonsolitons
conservation laws, it certainly suggests that there might be something `special' about it (may be 'integrable'?). The next question was: How unique is the KdV equation? This led them to consider the number of conservation laws for modified forms of the KdV Wt - 6wPwx + wxzx = 0
(9.5.10)
where p is some integer. For p > 3 they could only find three conservation laws, but for p = 2 vt - 6v2vx + Uxxx = 0
(9.5.11)
they again found a large number of conservation laws. The equation (9.5.11) is called the modified KdV equation. Miura conjectured that (9.5.11) has an infinite number of conservation laws, and that this might imply that (9.5.8) and (9.5.11) are related. He proceeded to obtain the transformation between them, namely, if v satisfies (9.5.11) Qv = vt - 6v2vz + vxxx = 0, and
u=V2+Vx
(9.5.12)
then u(x, t) satisfies Pu = ut - 6uux + vxxx = 0.
Hence, u(x, t) satisfies the KdV equation (9.5.8).
Exercise 9.12 To prove this, show that Pu = (2v + 8/0x)Qv.
The Miura transformation, (9.5.12), is somewhat analogous to the Hopf-Cole transformation (9.4.2), except it connects the solutions of two nonlinear equations - neither of whose solutions are known! Nonetheless, now the plot thickens! View (9.5.12) as a differential equation to be solved for v, where u(x, t) satisfies (9.5.8). This is a Riccati equation (an ODE)
Ux= -v2+u(x,t). Note that t enters the equation simply as a parameter. As was shown in Chapter 3, a Riccati equation is always related to a second order linear equation. Specifically, setting v = 3 In i/i/8x
(9.5.13)
we obtain the linear equation for the function O(x, t) Y'xx-U/
=0.
(9.5.14)
General solution of the Kd V equation
373
Moreover, this can be brought into the form of the 'time-independent' Schr5dinger equation,
0xx - (u - A)o = 0,
(9.5.15)
where u(x, t) is still a function of the `parameter' t. To show this, we note that the KdV equation is invariant under the transformation u(x, t) = u(x + 62t, t) + A,
(9.5.16)
provided that J is a constant (the converse is discussed below). That is, if u(x, t) satisfies (9.5.8), so does u(x, t), defined by (9.5.16). Thus the constant A can be introduced into
(9.5.15) by using the solution (9.5.16). Hence we now have obtained an association with the standard time-independent Schrodinger equation (9.5.15). We are now in the strange position of knowing neither the `wave function' 0, nor the `potential' u(x, t), nor the eigenvalues A in (9.5.15)! One really would be satisfied with determining only u(x, t). The beautiful fact, discussed in the next section, is that u(x, t) can be determined without knowing O(x, t) for all x, but only its scattering behavior, given by the asymptotic values limx +±a, '(x, t)! This is so provided that u(x, t) indeed behaves as a localized potential lim u(x, t) = 0.
x-.±oo
(9.5.17)
Thus (9.5.17) is of basic importance, because it means we already know the answer somewhere!
In any case, the above observations concerning a `Schrodinger equation' and the possible importance of the fact that the KdV equation has an infinite number of conservation laws, apparently advanced the solution of the KdV equation very little. However the presence of a Schrodinger-like equation puts a new face on the problem; a new perspective from which to attack the problem. This led to the collaboration of some ingenious researchers, each adding his individual insight, to discover the beautiful and highly original method of inverse scattering transforms.
9.6 The general solution of the KdV equation We will now discuss a unique and beautiful method for obtaining the general solution of the KdV equation ut - 6uux + uxxx = 0,
(9.6.1)
which was discovered by Gardner, Greene, Kruskal, and Miura (1967,1974). It is an outstanding example of an imaginative approach to a nonlinear equation, and again illustrates the fact that obtaining solutions of nonlinear equations is much more of an art than a science. Their approach, of course, has subsequently been `refined', and
374
Solitaires: solitons and nonsolitons
made more `elegant', as well as generalized (e.g., Ablowitz and Segur (1981), Newell (1985)), but the imagination came first! In this section we will follow this twisted but
highly innovative path. For those who do not like twisting paths, go to the end review - but you will miss the fun!
The approach is entirely unique and, at first, entirely obscure. We begin by considering an arbitrary solution of (9.6.1), and substitute this solution into the `Schrodinger equation' (9.5.15) ixx(x, t) + (A(t) - u(x, t))iJi(x, t) = 0,
(9.6.2)
so that u(x, t) acts as a potential energy, and t enters simply as a parameter in this 'time-independent' Schrodinger equation. The importance of the introduction of this equation is the following: It had been known for some time that if the asymptotic form of O(x, t) (as x -> oo, for all t) can be determined for any incoming plane wave exp (- ikx) (as x oo) - thus obtaining the scattering (i.e., reflection and transmission) generated by the localized `potential' u(x, t) - then it is possible to invert this information to determine the potential u(x, t). In this scattering process, the time t simply enters as a parameter and has nothing to do with the dynamics of the scattering (i.e., t is fixed for the scattering problem). Now the new feature is that the function
u(x, t) is to be a solution of (9.6.1), so by inverting this scattering information we would be obtaining the desired solution! However, we will need to obtain an equation
for the t dependence of fi(x, t), in order to determine the t dependence of u(x, t). Moreover, since u(x, t) is not known, it is clear that the scattering problem will have to lead to new unknown quantities (certain reflection coefficients and other constants). However, the space-time behavior of u(x, t) will thereby be represented in a new form, from which its asymptotic temporal properties (t - oo) can be exhibited.
We next want to obtain an equation involving 0 which moreover makes use of the fact that u(x, t) in (9.6.2) is indeed a solution of (9.6.1). However we need to first determine the behavior of ),(t) (do the unknowns never end?!). For this purpose we introduce the function
Q= r,+`Vxxx-3( + u)Y'x
(9.6.3)
and consider (for entirely unknown reasons!) (Y'Qx-Y'xQ)x=OQxx-'YxxQ = 0Wtxx + 05x - 3(L + u)/sx - 3uxxOx - 6uxoxx)
-'Vxx(Y't - '3x - 3(), + u)tx)
(9.6.4)
Now solving (9.6.2) for u and taking its tim'e'' derivative we obtain 1 2U,
_ 02A, + 'YWxxr - Y'IY'xx. Subtracting this from (9.6.4), removes the mixed space-time derivatives, and
Genera! solution of the Kd V equation
rearranging terms yields, 022'
,/,2U,
+ (YiQx - 0xQ)x = 'V
+ Y'(4'
375
- 3(), + u)03x - 3uxx - 6ux'i)
- Oxx(03x - 3(A + u)Px)
(9.6.5)
We now proceed to simplify the right side by eliminating the higher order derivatives in 0, with the help of (9.6.2). We have 03x =,UxI// + (u - .)Ox
02x = (U - 2(t))j;
''
04x =U2xt/+2U2'Yx+(U-)')'V2x=U2xll/+2uxti
= u3xO + 3u2xfx + 4(u - ))u' + (u
+(u-A)20
-),)2 fix.
Thus all derivatives are reduced to terms proportional to either i or Ox. We now substitute these into the right side of (9.6.5) and collect the coefficient of Ox:
3u2x' + (u - ),)20 - 3(A + u)(u-).)o -3u2xO -(u - ),)(u -).)o +3(u-2)O +u)i/i =0. Similarly, the coefficient of i/i is
i/iu,+u3xO+4(u-,)uxo -3(a.+u)uxo -6ux(u-))O-(u-))uxO =t/i(u,+u3x-6uux). We therefore have from (9.6.5) and (9.6.2) fr2A,
+ (OQx - /)Q) = / 2(u, - 6uux + u3x) = 0.
(9.6.6)
It is only in the last step that we have made use of the fact that u satisfies (9.6.1). Now from
02)"+('Qx-'1' Q)x=0
(9.6.6)
we can conclude: Theorem 1 If u(x, t) is a solution of the KdV equation (9.6.1) which vanishes sufficiently rapidly as IxI -+ oo, then each discrete eigenvalue of (9.6.2), A,,,, [corresponding to bound states I1r, f x /i,2 dx = 1] is constant in time d).m/dt = 0.
(9.6.7)
This follows simply by integrating (9.6.6) over all space.
Having established (9.6.7), the equation for i/i, will be the next problem to be considered. (9.6.6) together with (9.6.2) yields OQxx - QOxx =
IQ.. + Q(A - u)O = 0
or
Qxx + (, - u)Q = 0.
(9.6.8)
Solitaires: solitons and nonsolitons
376
Hence Q satisfies the same equation as does >/i, namely (9.6.2). Therefore Q must be a linear combination of the independent solutions of (9.6.2), say some >/i(x, t) and ¢(x, t)
OX, t) J dx'/>/2(x', t).
(Check that this is a solution - its independence comes from the indefinite integral.) Then, using (9.6.3), Q = r + Oxxx - 3(2 + u)Ox = C(t)t// + D(t)o J
x
dx'/ 2(x', t).
However, if 0 is a normalizable function (spatial bounded), then 4(x) is exponentially unbounded, so we must take D(t) = 0, leaving >/i, + Oxxx
- 3(2 + u)Iix = C(t)o.
Using the derivative of (9.6.2) we can eliminate
Y'xxx
(9.6.9)
from (9.6.9) to obtain
0, + >/ix>/i - (42 + 2u)>/ix = C(t)>/i.
Multiply by//,,I/'i, add - 22(I/i2)x''+I 22(02)x, and using (9..6,.12,) in'thel'second term gives (1/
2)-22(c'2)x+ux>,2-2uY"Vx+(,2) t-4),102)x+4(u/
-Y'xx)Y'x+ux>//2-2u>//I//x
_ (2 02 ), - 42(02 )x + (utr2 )x - 2(02 )x = C(t)>/i2.
Integrating this yields
df
L02 dx - f
dx[41(02)x
- u02 + 2(x)2]" = C(t) f
dx 02.
Since 0 is normalized, the first term vanishes as does the second, for bounded solutions. Hence C(t) = 0, and we finally have from (9.6.9) that the bound `wave functions' satisfy
0, + qtxxx - 3(2 + u)' = 0
(bound t/i)
(9.6.10)
whereas the unbounded solutions satisfy 0, + Oxxx - 3(2 + u)Y'x = C(t)o + D(t)o J x dx'/02(x').
(9.6.11)
The eigenvalues of the bound states are discrete, while the unbound state has a continuous spectrum. Respectively
2m= -kn,,
2=k2.
(9.6.12)
The km are constant, according to (9.6.7), and the ks can be taken to also be constant (because it is the entire infinite k spectrum which is significant - not any particular k).
377
General solution of the Kd V equation
This establishes the preliminary results that will be needed shortly. At this point it is not clear that we are any closer to the solution of the KdV equation. What is required is to consider the inverse scattering problem, in order to deduce the scattering potential u(x, t) from the asymptotic plane wave solutions of the Schrodinger equation (for any k)
exp (-ikx) + b(k) exp (ikx)
X- + oo
(9.6.13)
a(k) exp (- ikx)
x -* - oo
(9.6.14)
with (9.6.15)
Ia12+Ib12=1.
This solution corresponds to a plane wave coming in from x = + oo and being partially reflected and partially transmitted. b(k) is then the reflection coefficient and a(k) the transmission coefficient. In addition to these plane wave asymptotic solutions,
the Schrodinger equation also has bound solutions which decay as e - km I X I (where Am = -k 2M). If the 0m are normalized, then we can define a measure of their amplitude by the definition
c,
lim
X-00
(x) exp (+ kx)
(f \J
dx = 1 0x
/
(9.6.16)
The basic theorem, upon which the inverse scattering method rests, is due to Gel'fand and Levitan (1955), Kay and Moses (1956), Agranovich and Marchenko (1963), and Faddeyev (1963); for a history of these papers, see Kay and Moses (1982).
Theorem (Gel'fand-Levitan) Define a function
B() _ E1
exp (-
2n f "0
b(k) exp (ikx) dk
(9.6.17)
where the c,,,, km, b(k), and k are related to the solution of 0XX + (1, - u)O = 0
(9.6.2)
through the relationships (9.6.12)-(9.6.16). Let K(x, t) be the solution of the integral equation K(x, y) + B(x + y) + fx"O B(y + z)K(x, z) dz = 0
(9.6.18)
Then the potential u(x) in (9.6.2) is given by
u(x) = - 2 dK(x, x)/dx.
(9.6.19)
This theorem is not obvious by any means, but its proof will not be discussed here - only its application.
Solitaires: solitons and nonsolitons
378
The new feature that enters in the KdV case is that the potential in (9.6.2) is not just a function of x but also of the `parameter' t. This means that coefficients cm and b(k) in the function equation (9.6.17), are now functions of t (the km and k are however independent of t, (9.6.12)). If we can determine their t-dependence, then solve (9.6.18) for K (x, y: t), this will give the t-dependent u(x, t) = - 2aK(x, x; t)/ax! Thus `all' that remains to determine u(x, t) (one still has to solve (9.6.18)!), is to obtain the t-dependence of cm and b(k). This can be obtained from the asymptotic solutions (IxI - oo) of the 'time'-dependent equations for t/i (9.6.9) and (9.6.11). Note again that the solution for i/i is required only in the region where the solution for u(x, t) is already u(x, t) = 0! What is needed is: known - namely limlx,
-,
Theorem 2 Under the conditions of Theorem 1 cm(t) = cm(0) exp (4kn, t)
(9.6.20)
b(k, t) = b(k, 0) exp [8ik3 t]
(9.6.21)
a(k, t) = a(k, 0).
(9.6.22)
The result (9.6.20) follows from substituting the asymptotic form of the bound solution V'm ^-'
cme-'"X(x-* + oo), given by (9.6.16), into the t-dependent equation (9.6.10)
yielding (dcm/dt)e-kmx - k,3ncme-k-'+ 3Amkme-kmx
= 0.
Using (9.6.12) (i.e., A. k2 ), this gives (9.6.20). Similarly the unbounded asymptotic solution (9.6.14) substituted into (9.6.11) yields
at + ik3a + 3Aika = C(t)a + (D(t)/a) f x exp (2ikx') dx'.
Clearly this is only possible if D(t) = 0. Using (9.6.12) this gives
a, + [4ik3 - C(t)]a = 0.
(9.6.23)
Similarly, substituting the other asymptotic limit (x -). + oo), given by (9.6.13), into (9.6.11) (now with D = 0), the coefficients of a - ikx and a+ikx are respectively ik3 + 3iAk = C(t);
b, - (ik3 + 3iAk)b = Cb.
Hence C = 4ik3 (constant), so b, = 8ik3b, which establishes (9.6.21). Since C = 4ik3, equation (9.6.23) establishes (9.6.22), and that completes the proof of Theorem 2.
To review briefly: To solve the KdV equation
u,-6uux+uxxx=0
(9.6.1)
General solution of the Kd V equation
379
for the initial condition u(x, 0) (- oo < x s oo) which satisfies lim u(x, 0) = 0, IxI -goo
we first solve the equation (for all x)
lxx + (2(t) - u(x, t))0 = 0
(9.6.2)
at t = 0, for both the bound and unbound solutions. From these we obtain the eigenvalues kand the coefficients b(k, 0) from (9.6.16) and (9.6.13). We next establish that dal/dt = 0, provided u(x, t) satisfies (9.6.1). The t-dependence of b(k, t) are then given by (9.6.20) and (9.6.2 1), which were obtained from 0,+Oxxx-3(2+u)lIix=0
and
(9.6.10)
and
, + Oxxx - 3(2 + u)2x = C(t)o + D(t)q, Jx dx'/O2(x').
(9.6.11)
The c(t) and b(k,t) are now used in the Gel'fand-Levitan theorem (9.6.17), from which we must then obtain the solution K(x, y; t) of (9.6.18). Finally we use this function to obtain the desired solution of the KdV equation u(x, t) = - 20K(x, x; t)/ax.
Since the general solution of u(x, t) is obtained by (a) considering both a scattering (and bound) solution of i/i(x, 0); (b) obtaining its time evolution; (c) then the inverse solution of Gel'fand-Levitan-Marchenko equation, the general scheme is called the Inverse Scattering Transform (1ST) (see Flow chart 9.1). The Flow chart for IST is a sophisticated generalization of the Fourier transform scheme shown in Flow chart 9.2 for solving the linear equation. It will be noted that several of the above steps still require solutions of equations
(9.6.2) and (9.6.18). It is sometimes noted that these are `only' linear equations.
However, as pointed out in Section 3 of Chapter 1, linear equations are not intrinsically simpler than many nonlinear equations - only special linear equations are simple. Thus (9.6.2) may not be easy to solve, to say nothing of the integral
equation (9.6.18). Indeed (9.6.18) is usually solved only for a very special (but interesting!) class of initial conditions that yield b(k, 0) = 0 ('no reflection' potential), and for which K(x, y; t) can also be easily obtained. In general, however, the only assured simplification of this inverse scattering method is that the time-independence has been reduced to known functional forms (9.6.20)-(9.6.22).
Solitaires: solitons and nonsolitons
380
Flow chart 9.1. IST Want u(x, t) satisfying
Solve (for all x)
ut-6uux+uxxx0
Oxx+(A-u(x,0)0=0.
and lim u(x, t) = 0 1xI
Obtain all scattering coefficients, b(k, 0),
- °°
and bound coefficients,
The IST
Time evolution equation for O(x, t), with d ./dt = 0
Solve the integral equation K(x, y) + B(x + y)
+J
B(y + z)K(x, z) dz = 0 x
From b(k, t) and cm(t)
to obtain K(x, y; t). The desired solution is
obtain
t), (9.6.17).
u(x,t)= -2xa K(x,x;t)!! Flow chart 9.2. Fourier transform scheme Want u(x, t) satisfying
Let b(k, t) = 2_
ut + uxxx = 0
lim U=0
_
u(x, t) exp (ikx) dx
Obtain b(k, 0) from u(x, 0)
(*)
IxI- 00
Time evolution:
transform
u(x, t) = J _
b(k, t) exp (ikx) dx
b, = ik3b(k, t), if (*) is satisfied.
l
obtain b(k, t) = exp (- ik3t)b(k, 0).
Pure soliton solutions
381
9.7 Pure soliton solutions The most widely studied case, both because of its intrinsic interest and (fortunately) relative ease, is when the solution of the KdV equation represents N solitons; that is, as t --> ± oo, u(x, t) breaks up into N localized disturbances, each with its own velocity, v; (see Fig. 9.15). The general solution of the KdV equation will contain Fig. 9.15
v2t+62
v4t+64
dispersive oscillations, like those found for the linear equation u, + uxxx = 0. Since the solitons are discrete quantities, whereas dispersion is continuous in character, it would be a reasonable guess (better yet, a correct guess!) to associate the bound states (the cm) with the soliton solutions, and the continuous spectrum (b(k)) with the dispersive portion. The pure soliton solutions are thus associated with the non-reflecting potential u(x, 0) in the Schrodinger equation b(k) = 0
(pure soliton solution)
in which case N
E cm exp (- km b ). zz
(9.7.1)
M=1
This case was studied in detail by Kay and Moses. If this is substituted into the Gel'fand-Levitan integral equation K(x, y) + B(x + y) + fx'O B(y + z)K(x, z) dz = 0 it
(9.7.2)
is natural to look for a solution of the form N
K(x, Y) _
Om(x) exp (- kmY)
(9.7.3)
M=1
where 4m(x) are unknown functions. Substituting this and (9.7.1) into (9.7.2), we obtain eXP (- kmY) 4m(x) + cm exp (- kmx) + cm J M=1
{
eXP C - (km + X m=1
dz } = 0. )
Solitaires: solitons and nonsolitons
382
For this to be satisfied for all y requires that 4'm +
C2
exp (- kmx) + cm2
+"
exp [ - (km + kn)x] = 0
(9.7.4)
n=1KnKm for all m = 1, ... , N. This determines the functions Om(x) in terms of the unknown constants cm, km, which must be consistently related to the bound eigenfunctions and eigenvalues. Hence the function K(x, y) is determined through (9.7.3), and finally the potential u(x) through the Gel'fand-Levitan expression u(x) = - 2(d/dx)K(x, x).
(9.7.5)
The general algebra required to unravel this has been given by Kay and Moses, and also by Gardner, Greene, Kruskal and Miura in their 1974 paper. The general result is: Every reflectionless potential (i.e., a pure soliton solution) can be written u(x, 0) = - 2(d2/dx2) [ln det (M(x))]
(9.7.6)
M = [Smn + CmCn exp [ - (km + kn)x]/(km + kn)]
(9.7.7)
where the matrix
with arbitrary cm > 0 and distinct km. Moreover the time-dependent solution is obtained by substituting Cm = Cm(0) exp (40 t)
into (9.7.7). This shows that x is replaced by terms of the form (x - 40,,t), and the factors vm = 4k,"
(9.7.8)
are the asymptotic velocities of the solitons. These N-soliton solutions were first obtained by Hirota (1971) and by Wadati and Toda (1972). Hirota employed his direct method to obtain this general solution, which will be discussed in Section 9.10.
To obtain a better appreciation of these soliton solutions, we consider now some simple cases.
One-soliton solution Substituting N = 1 into (9.7.4) yields
41 +C2exp(-klx)+C (41/2k1)exp(-2k1x)=0. So, dropping the subscript 1, k(x, x) = _C2 /exp (2kx) + (c2/2k)
Pure soliton solutions
383
and
4k [exp
u(x)
c2 exp (2kx)
(2kx) + (c2/2k)]2
2k2
[cosh (kx + a)]2'
_ c/(2k) i/2 =exp (- 8).
The time-dependent solution (see (9.7.8)) is
u(x, t) = - 2k2 sech2 [k(x - 4k2t) + 6]
(9.7.9)
where k is arbitrary. In this case the velocity of the soliton is twice its amplitude. In the present case S is a trivial coordinate translation (or temporal translation).
Two-soliton solution This case requires a considerable amount of algebra to obtain in explicit form, so we will simply give the two-soliton solution of
u1-6uux+uxxx=0 which was obtained by Miura u(x, t)
12
3 + 4 cosh (2x - 8t) + cosh (4x - 64t) [3 cosh (x - 28t) + cosh (3x - 36t)]2
(9.7.10)
Since the KdV equation is invariant under the group transformation u = a-2U', t = a3t', x = ax', the function u'(x, t) = a2u(ax, a3t) is also a one-parameter family of solitons. We note that at t = 0 u(x, 0)
12
3 + 4[2 cosh2 (x) - 1] + 2 cosh2 (2x) - 1
[3cosh x+4cosh' x-3cosh x]2
12{8cosh2x-2+2[2cosh 2x- 1]2} 16 cosh6 x
that is u(x, 0) = - 6 sech2 x
which corresponds to the nonreflective potential noted by Landau and Lifshitz (for N = 2, see below). To establish the long-time behavior of this solution, showing that it splits into two
solitons, consider the moving point x = + 4t (fixed ). Then (9.7.10) becomes u( + 4t, t)
12
3 + 4 cosh
cosh (4 - 48t)
[3 cosh ( - 24t) + cosh (3
- 24t)]2
Solitaires: solitons and nonsolitons
384
In the large-t limit,
tli mu( +4t,t)= -12
f exp [ ± (- 4 + 48t)]
(2exp[±( -6 b)]2.
[Z exp (± ) + eXp (+ Z
Now set 3 = exp (20) (note 0 > 0), which yields
Jim u( + 4t t) =
t-t0
-6
exp(20)[2exp(± +0)+Zexp(+-0)]2
= - 2 sech2 ( + 0). (I)
Similarly, by setting x = + 16t, we find
lim u( + 16t, t)
(II)
8 sech2 (2 + 0).
t-± 00
Thus we have two solitons traveling with the velocities 4 and 16 (or more generally 4a2, 16x2; vs = - 4, = 4s2a2; s = 1, 2) as t -> ± oo, and which suffer a phase shift of 20 due to their interaction. This is illustrated in Fig. 9.16. It is instructive to compare Fig. 9.16
AA O later O x
x
x=16a2t
x
01=--In /k2+kll k2 k1 1
-+
k1
1
DZ = k2 In
1k2 + k2
k1= 1,k2=2
U
_ kl/ k1
Pure soliton solutions
385
this with the space-time figure for the interacting Toda solitons in Chapter 8, Section 8.
Also compare this with the result in Exercise 8.11. Why isn't there such a solution here?
In the general case, (9.7.4),
Jim u( + 4k2 t, t)
2k,2n sech2 (km ±
+ 00
f
Ar)
Note that the faster soliton is advanced due to the interaction, whereas the slower soliton is retarded in its motion (the rich get richer...). The net x shift, Axn, of the soliton n (0 < k, < k2 ... < kn) over - oo < t <, + oo, is
Axn=k n
ln(kn+km I
n
m
m)
-j
m+kn)1.
lnOm
m=n+1
,
JJ
(9.7.11)
Also the `center of mass' X (t) = $ °° xudx, has a constant velocity. Its velocity is 00
d
d_
xu dx = Jxutdx =
x [3u2 - u]dx = (- 3u2 + u) dx
However,
dt
f u2 dx = 2 J u[6uux - uxx.,] dx = 2JI[(2u)x - uxxxu] dx
=0+2 f uxuxx dx = J(Ux)x dx = 0, so the acceleration of X (t) is
,J
d2 d2 dt2 X(t) = dt2
-, xu dx = 0.
(9.7.12)
f "0
A special group of pure soliton solutions is the Landau-Lifshitz subset (see, Quantum Mechanics, Nonrelativistic Theory, Problem 4, p. 72 (Pergamon, 1958; also see Morse
and Feshbach (1953), p. 768)). This subset of the general N-soliton case is one in which all solitons cross (i.e., meet) at some time. This set of reflectionless potentials is
u(x) = - uo sech2 (ax),
(9.7.13)
so the related Schrodinger equation is 0.,., + [ - k2 + uo sech2 (ax)] fr = 0.
(9.7.14)
Here uo and a are free parameters, governing the width and depth of the potential
Solitaires: solitons and nonsolitons
386
`well' (see Fig. 9.17). If we set then (9.7.14) becomes
= tanh (ax), so 8/ax = a sech2 (ax)a/a = a(l -
((1 _2)4
+
(k/)2
uo-
/
0.
(9.7.15)
I
We now introduce (N, s) by the relations
uo = N(N + 1)a2,
k = sa > 0;
so that (9.7.15) becomes
I (1- 2)
) +(N(N
+ 'H s2 2)
=0.
(9.7.16)
The bound solutions, which are zero at
= ± 1 (x = ± oo), require that s = 1, 2, ... , N, and are the associated Legendre polynomials
(s = 1,...,N)
'YNs=NN51 N(S)
(9.7.17)
where fNS is a normalization constant, satisfying 02 J
dx = I'NsJ
dS
la(1-
(PN(S))2 = 1.
2
)
The number of solitons is given by N = Z(- 1 + [1 + (4u0/a2)]"2)
(9.7.18)
with the eigenvalues ks = as. This is the number of bound solutions in the potential Fig. 9.17
well (depending on uo and a); see Fig. 9.17. The velocities of the corresponding solitons are given by (9.7.8), vs = 4a2s2
(s = 1,..., N).
(9.7.19)
A few examples are: 2a2 sech2 (a(x - v1 t)); while it is only of `academic interest', we note that cu 11= P11(1 - 22)112 = f11 sech(ax), and #212/a = 1.
N = 1: uo = 2a2, v1= 4a2, u(x, t)
Lax formulation
387
N = 2: uo = 60(2, v1 = 40(2, v2 = 160(2; u(x, 0) _ - 60(2 sech2 (ax). Also
2)1i2 = - 3/321 tanh(ax)sech(ax), and 9#321(4x) = 1; Y'21 = -3#21 (1 022 = 3/322(1 - 2) = 3 sech2 (ax), and 9/322(2/0()(3) = 1. N = 3: uo = 120(2, v1 = 40(2, v2 = 16x2, v3 = 36x2; u(x, 0) _ - 12012 sech2 (ax); 031 = (151; 2
-3)(1 - 2)112 = (5 tanh2 (ax) - 1)sech (ax) 032 = (1 - (;2) = tanh (ax) sech2 (ax) 033 = sech3 (ax).
Review and comments ('seventh-section rest') Before proceeding with the many mysterious methods of soliton dynamics, let us pause very briefly to review the larger picture! (1) The FPU results demonstrated that some systems do not readily approach thermal equilibrium, yielding Fermi's `little discovery'. On the other hand some lattices, notably the harmonic-plus-hard-core system, may approach equilibrium rapidly. (2) The KAM theorem gave theoretical reasons why this is to be expected for weak nonlinearities - how weak is not known.
(3) The soliton solutions, from the continuous lattice limit (KdV equation), illustrated another facet of this nonstochastic motion. The soliton motion, however, is not the general motion (only for `reflectionless' initial conditions u(x,O));
the remaining dispersive part may be of basic importance in
understanding how `irreversibility' comes about in such systems (if it does). (4) The Toda lattice illustrates the fact that solitons are not restricted to continuous systems, but are a signature of integrable systems. (5) Other interests in solitary disturbances, particularly in particle physics have centered on solitaires which are not indestructable, but which break up into several disturbances. We will consider such effects in a later section. Now, on to further perspectives of solitaires!
9.8 The Lax formulation Another perspective of soliton dynamics is afforded by a formulation due to Lax (1968). We consider the problem of obtaining a solution, u(x, t), of u, = K(u, ux, uxx, ... ).
(9.8.1)
Lax's viewpoint is that the general inverse scattering method is based on the possibility of finding two operators, L(8/8x, u, ux.... ) and A(8/8x, u.... ), which are related in such
Solitaires: solitons and nonsolitons
388
a way that the equation Lt = [A, L]
(9.8.2)
implies ut = K(u, u, ux,...). Here [A, L] - AL - LA is the commutator of the two operators. Having obtained such operators, known as the Lax-pair operators, we next consider the associated linear equations, Lei = - 2(t)>/i
(9.8.3)
of = Ai/i.
(9.8.4)
and
In (9.8.3) t only appears as a parameter in u(x, t), and possibly 2(t). The first thing to show is that (9.8.2), (9.8.3), and (9.8.4) insure that 2(t) is independent of t,
At=0.
(9.8.5)
To prove this we differentiate (9.8.3) with respect to t
Lt0 +Lit=-2tO-AO, Using (9.8.4), and then (9.8.3), this yields
Ltd+LAi/r= -2ti/i-2At/i=-AtVf +ALi/r, or
[Lt-(AL-LA)]Vf = -.ltz/. The left side vanishes because of (9.8.2), showing that (9.8.5) holds. Once the Lax-pair operators, (9.8.2), are obtained, the procedure is then to: (1) Solve for the bound states, and the reflection coefficient of (9.8.3), b(k,0), using the known initial u(x, 0) contained in L; (2) Solve (9.8.4) in the asymptotic region where u(x, t) __ 0, thereby obtaining c(t), b(k, t). Here (9.8.5) is of basic importance;
(3) Obtain an inverse scattering theorem for L, which is analogous to the Gel'fand-Levitan theorem for the Schrodinger operator. This, of course, may be a very nontrivial step, depending on the operator L; (4) Solve the required integral equation of (3) for the kernel K(x, y), and thereby obtain u(x, t), the solution of (9.8.1). To illustrate this procedure, we consider the KdV example. Knowing that we want (9.8.3) to be the Schrodinger equation, we take z
L
axe - u(x, t) _= D2 - U.
(9.8.6)
Lax formulation
389
Next, we take (experience helps!) 3
A=
OX3
+ a(x, t) CY + a a(x, t) + b(t) - D3 + 2aD + ax + b(t).
(9.8.7)
[A, L] = (D3 + 2aD + ax + b)(D2 - u) - (D2 - u)(D3 - 2aD + ax + b). Now we have D3u = D2(ux + uD) = D(uxx + 2uxD + uD2) = u3x + 3uxxD + 3uxD2 + uD3
so, canceling uD3 terms, [A, L] = -(u3x+ 3uxxD+ 3uxD 2) + (2aD + ax + b)(D 2 -u)-(D2 - u)(2aD + ax + b).
Similarly D2(2aD) = D(2axD + 2aD2) = 4axD2 + 2axxD + 2aD3, so
[A, L] _ -(u3x+3uxxD+3uxD2)-4axD2-2axxD-2aux+(ax+b)(D2-u) -(D2 - u)(ax + b)
-u3x-3uxxD-3uxD2-4axD2-2axxD-2aux -a3x-2axxD.
(9.8.8)
In order for L, = [A, L] to imply that u, = K(u), (9.8.2), we must have [A, L] independent of D - a/ax (i.e., since, by (9.8.6), Lt = - ut is simply a scalar operator, [A, L] must likewise be a scalar operator). We have from (9.8.8) [A, L]
so we must take a
u3x - a3x - 2aux - (3uxx + 4axx)D - (3ux + 4ax)D2,
3u/4, in which case [A, L]
4u3x + 33uux.
(9.8.9)
Hence, if (9.8.7) becomes
A= -4[D 3 - 3uD - tux + b(t)], then L, = [A, L] implies - u, = u3x - 6uux. Therefore (9.8.1) in this case is
u1-6uux+U3x=0 which is the desired KdV equation. Then, if LO = - ),t/i, the equation for 0, (9.8.4), is presently
a = [ - 4D3 + 6uD + 3ux -
(9.8.10)
The form used by Gardner, Green, Krushal and Miura (GGKM) was 0,+D30-3(u+1.)D/i-CO = 0.
(9.8.11)
390
Solitaires: solitons and nonsolitons
The two are related by the fact that
D(L+A)o =(D3-ux-uD+2D)o =0, so 3uxt/i = 3(D3 - uD + AD)O. Using this to eliminate uX reduces (9.8.10) to the GGKM
equation (9.8.11). In the present KdV example the remainder of the program, (1)-(4) above, is of course the one already described in previous sections. Its implementation in other cases is by no means a trivial exercise. Also the search for methods to discover Lax-pair operators continues (e.g., Ichikawa and Ino, 1985). In 1972, Zakharov and Shabat obtained the Lax-pair operators for the nonlinear Schrodinger equation iiui, + 0xx ± 20*02 = 0,
(9.8.12)
where 0 is a complex function of the real variables (x, t). The inverse transform now involves a pair of real functions, comprising 0, the details of which can be found in the references. Historically, Zakharov and Shabat's result was the first example of an integrable nonlinear PDE, besides the original KdV equation. It established the
fact that the KdV system was not an isolated case, and it thereby stimulated the successful search for other integrable PDE (see Section 9.11).
An idea which can be used once is a trick. If it can be used more than once it becomes a method. (G. Polya and S. Szego).
9.9 The sine-Gordon equation In this section we will consider some particular solutions of the so-called sine-Gordon equation
0.. - off =sin 0.
(9.9.1)
The name is, of course, a play on words (not universally appreciated) based on its `nonlinearization' of the Klein-Gordon equation
0..-C-24 =(mclh)2(p. The relationship between the general solutions of (9.9.1) and the IST will be deferred to Section 9.11. The nonlinear equation (9.9.1) occurs in a number of physical systems (or reasonable model approximations). Several nice introductions to these applications are given by Barsone, Esposito, Magee, and Scott (1971) and Scott, Chu, and McLaughlin (1973). One physical model which leads to (9.9.1) is the Frankel-Kontrova model, discussed in Chapter 6, Section 5. In this system a harmonic lattice of particles lie on a periodic potential. The jth particle experiences the force (6.5.4) F j = k(s; + 1 + s _ 1 - 2s;) + A sin (2ns;/ t),
Sine-Gorden equation
0
Fig. 9.18
0
si(t)
391
---sr+i(t)
m
where s, is the location of the ith particle (Fig. 9.18). If m is the mass of each particle, and we set Oi = 2irsi/),, Newton's equations are m
0=k(0j+1 + b _1 -2¢;)-Tsin(¢;)
(9.9.2)
where T = - 2nA/1. Now we are interested in the dynamics of ¢;(t), whereas only the equilibrium configuration was considered in Chapter 6. As in Section 9.1, we make the transition from the Lagrangian variables O;(t) to the Eulerian functions O(x, t). Using the scaled time t' = (T/m)1j2t and distance, so that (k/T)(¢j+ 1 + 20i) - 0xX, (9.9.2) reduces to (9.9.1). Another mechanical system which yields (9.9.2) is a line of pendula (e.g., large
O;-1
nails - see Fig. 9.19) attached to a coiled spacing. See Barone et al. (1971) for a detailed Fig. 9.19
model. In this case (9.9.2) describes the time-dependence of the angles of the pendula
from the vertical, provided that M is the moment of inertia of a pendulum about its end, k is the (harmonic) torque constant of a spring section between two pendula, and T sin 0 is the gravitational restoring torque. The nice feature of this system is
that it can be constructed, and certain topological features (to be described) are particularly obvious. Indeed, even a simpler `pocket model' can be constructed by using a cut rubber band and some straight pins - see Fig. 9.20.
Solitaires: solitons and nonsolitons
392
Lay the rubber band flat, and stick in the pins (with large heads, if possible) at close, regular intervals. Pick up the rubber band at the ends, stretch it stiff, then countertwist the ends in any fashion. This is a very finite, and nonautonomous version of the above pendula system. Following Lamb (1971), we consider first the class of separable solutions of (9.9. 1)
¢ = 4 tan - ' [X(x)T(t)].
(9.9.3)
Note that T and X do not have to be real, only XT. Differentiations yield 0" _
4
7XI
1 +(XT)2
9xx
47X" 1 +(XT)2
8(7X,)2 XT
[1 +(XT)2]2
where X' = dX/dx. A similar expression holds for ¢«. We next note that sin [4 tan -' (f)] = 2 sin [2 tan -'
= 4 sin [tan-'
cos [2 tan-'(g)]
cos [tan-'
11 - 2 sine [tan'
2) Substituting these into (9.9.1), and multiplying by
4[1
4(1+ 02.
+(XT)2]2, (9.9.1) becomes
TX"[1 + (XT)2] - 2(TX')2XT- T"X[I + (XT)2] + 2(T'X)2XT =XT[1 - (XT)2]. (9.9.4)
In a classic example of hindsight, we look for a solution of the form (X')2 = pX4 + mX2 + q
(T')2 = -qT4 + (m
- 1)T2 - p,
(9.9.5)
where (p, m, q) are constants. Using (9.9.5) in (9.9.4), it becomes
T(2pX3 + mX)[1 + (XT)2] =2T 3X(pX4 + mX2 + q) - (- 2gT3 + (m - 1)T)X[1 + (XT)2]
+2X3T(-qT4+(m- 1)T2- p) =XT(1 -(XT)2].
(9.9.6)
It is easy to check that this is satisfied for all (p, m, q), because their coefficients vanish,
as well as the p = m = q = 0 factor. Therefore (9.9.5) is a solution of (9.9.4) for any (p, m, q).
While the general solution of (9.9.5) can be expressed in terms of elliptic functions, we will consider the following special solutions.
Sine-Gorden equation
393
Single-soliton solution: p = q = 0 From (9.9.5)
X'= ±m1'2X;
T'= ±(m- 1)1"2T (m> 1).
If we set c2 = 1 - M- 1, then tan
(/4)=et(X-ct)
-C2)-1/2
(9.9.7)
where c can be positive or negative. As a function of _ (x - ct)(1 - c2)-1/2, the solution tan (0/4) = e+4 is called a soliton, S, and is also called a kink. The solution tan (0/4) = e-4 is called an anti-soliton, A, or an anti-kink. Thus d4s/dl; > 0, and d4)A/dg < 0. Since 0 is determined by (9.9.7) only mod (2n), 4s either goes to zero or - 2ir as -+ - oc (x -> - oc, for finite t) whereas 4)A goes either to zero or ± 2n in the same limit. Thus the possible solutions are as illustrated in Fig. 9.21. Fig. 9.21
We see that in all cases 0 changes by 2ir as x goes from - oo to + oo, hence these are called 2ir pulses. For the pendula system these solutions are particularly obvious,
the solitons (anti-solitons) correspond to right-handed (left-handed, respectively) corkscrews (Fig. 9.22): Fig. 9.22
The boundary conditions are not zero (a `vacuum state') at both x = ± oo, so these are called topological solitons. That is, 4)(+ oo) = 4)(- oc)±2n, so they are topologically
distinct, since they cannot be continuously deformed into one another. On the other
hand any solitons such that 0 - 0 as x - ± oo are nontopological solitons. This topological characterization is particularly obvious physically for the above pendula systems. Holding the ends fixed, we cannot change the number of rotations of the pendula, between x = ± oo and x = - oo, by deformations.
Solitaires: solitons and nonsolitons
394
Next, consider the case p = 0, q 0 0. (9.9.5) then gives (if m > 0)
-1/2sinh-1CmXifq>0 ±x=
dX J [mX2+q]1/2
9
-1/2 cosh - 1 (__x)
ifq < 0,
q and
dT
T[(m- 1)-gT2]1/2 - (m - 1)- 112 sech- 1 C
-m)-1/2cos 1I
\m
q
1
)112T
ifq>0,m> 1,
],
- /1/211
-g
ifq<0,m<1,
1/2
T
(m-1)-1/2csch-1
[(rn_i)
ifq 1.
,
The last two expressions give a singular T(t) at t = 0 or (1 - m)"t = ± n/2 (etc.). Hence the nonsingular solution is
XT= +
m - 1)1/2 M
sinh (m1/2x)
cosh [(m - 1)"2t]'
which is independent of q. We will again use c2 and (9.9.3) are:
(9.9.8)
1 - m- The two solutions (9.9.8)
Soliton-soliton (+) or antisoliton-antisoliton (-) collisions First we consider the (+) case in (9.9.8), so tan
C0)
sinh [x/(1- c2)1/2]
4 J = + ccosh [ct/(1 - c2)1i2]
(c > 0).
(9.9.9)
As t-+ + oo, tan (¢/4) - c{exp [(x - ct)/(1- c2)1i2] -exp [ - (x + ct)/(1 -c2)'/2]j. As t -> - oo, tan (0/4) - c{exp [(x + ct)/(1 - c2)112] - exp [- (x - ct)/(1 - c2)1i2] }.
As Fig. 9.23 illustrates, this is a soliton-soliton collision. The antisoliton-antisoliton collision is obtained by taking the (-) case in (9.9.8). An interesting aspect of the soliton-soliton collision, (9.9.9), is to determine the motion of the `centers' of the solitions, which we will define to be the points where
Sine-Gorden equation
395
Fig. 9.23
t<0
c> t=0
c<
2-1/2
2-/2 .
t>0
x
x
1
a20/ax2 = 0. From (9.9.9),
0 = 4 tan - ' {c sinh [x/(1 - c2)1/2]/cosh [ct/(1 - c2)'/2] } so 820/8x2 = 0 where cosh [x/(1 - c2)1/2]
8
8x {cosh [ct/(1 - C2)112] +2c sink2 [x/(1 - c2)1/2]
0.
2
Dividing out sinh [x/(1 - c2)1/2] (i.e., aside from the root x = 0), this yields cosh2 [ct/(1 - c2)1/2] + c2 sinh2 [x/(1
- c2)1/2] - cosh2 [x/(1 - c2)1/2] = 0.
Therefore the centers of the solitons are given by the roots of (9.9.10)
cosh2 [ct/(1 - c2)1/2] - c2 = c2 cosh2 [x/(1 -C 2)1/2],
or, if (9.9.10) has no solution, then their centers are at x = 0. Their closest point of approach (smallest x) occurs at t = 0. The distance of closest approach is Ox = 2(1 - c2)1/2 cosh(1/c2 - 1)1/2
Ax=0
(c < 2-1/2)
(9.9.11)
(c>2-1/2).
So if the solitons approach each other with too large a velocity (c > 0.707) they lose their identity for a period of time. These two possibilities for the motion of the `centers' is illustrated in Fig. 9.24. The next particular solutions of (9.9.5) to be considered are the cases q = 0, p 0.
Solitaires: solitons and nonsolitons
396
Fig. 9.24
ct/(1 - c2)1/2
ct/(1 - c2)112
//
c=0.6 L
//
/ `Coalesce'
\1
-2
2
1\'\ x/(1 - c2)1/2
Then
±t=
+Jf
dT 1)T2-p]1/2' [(m-
dX X[pX2+m]1/2
(9.9.12)
This gives two types of solutions, depending on whether m > 1 or m < 1.
Soliton-antisoliton solution (m > 1) 1)_ 1/2
If m> 1, then (e.g.) ±t=(m-
±x=
-m-1/2sech((-p/m)112X) so
/
XT= ± 1
sink-1
[(m - 1/- p)1/2T] and, if p < 0, then
1/2 sinh [(m - 1)1/2t]
m
1-m
cosh2 (m1/2x)
With the substitution c2 = 1 - m -1, (9.9.3) then yields
tan
C4J = 1 sinh [ct/(1 - c2)1/2] 4
(9.9.13)
2 1/2 ] c cosh [x/(1 - c)
where the positive sign has been picked (the other case is the time reversal of (9.9.13),
t - - t). This has the asymptotic forms
- {exp[(x+ct)/(1 tan
4
=
-c2)1/2]+exp[-(x-ct)/(1-c)1/2]}-1
c
+1{exp[(x-ct)/(1-c2)1/2]+exp[-(x+ct)/(1-c)112]}-1
(t - - oo (t
+00)
C
Note now that 0 - 0 as x -+ ± oo (a `vacuum state'), which is a nontopological soliton.
Sine-Gorden equation Fig. 9.25
0
t<0.
r"o
o'-l
-2a
+2a
397
X
0
O
x
The behavior of this soliton-antisoliton pair is illustrated in Fig. 9.25. The `centers' of these solitons (where 820/8x2 = 0) are given by (if t 0, see below) 1 + 1/c2 sinh2 [ct/(1 - c2)"2] = sinh2 [x/(1 - c2)1/2].
(9.9.14)
For example, as t - + oo, (4c2)exp [2ct/(1 - c2)1/2] - sinh2 [x/(1- c2)1/2]
so, for x > 0, x - Ct - (1- c2)1/2 In c and if x < 0, x -+ - Ct + (1 - c2)1/2 In c. Note also that In c < 0. Similarly, for t - - 00, x -i ± [Ct + (1 - c2)1/2 In c]. Therefore, these solitons pass through each other and their interaction is attractive. However, as t - 0, their `centers', as defined by (9.9.14), remain at a finite distance apart. At t = 0 there is a discontinuous exchange of these centers, involving a `jump' over a distance Ox = In (1 + 21/2)(1- c2)1/2 --0.88(t - c2)1/2.
(9.9.15)
(See Fig. 9.26). This is, of course, only an `interpretive discontinuity', not a physical discontinuity. Fig. 9.26
'ump'-Ax=1n(1 +21/2). (I -c2)1/2
398
Solitaires: solitons and nonsolitons
Breather mode (1 > m > 0 The dynamic mode can be obtained from (9.9.12), tan
(0)=( m 4/
1-m)
112sin[(1-m)"'t] cosh(m'/2x)
9.9.16)
This is a mode which is localized in space, but oscillating in time. It is therefore commonly referred to as a breather mode. Schematically ¢(x, t) looks like Fig. 9.27. which is sometimes viewed as a oscillating pair of solitons (S) and antisolitons (A) (Fig. 9.28) but of course, 0 is not a linear combination of S and A.
etc.
Fig. 9.28
fQ,
0 0.
9.10 Hirota's `direct method' in soliton theory Hirota (1971) has developed a method for obtaining special solutions of nonlinear PDE which are independent of any explicit dependence on (x, t), N(u, ut, uX, uXX.... ) = 0.
(9.10.1)
This method, like much in soliton dynamics, involves several `artful' steps of unspecified inspiration, but which introduces a new `structure', and hence perspective of nonlinear
dynamics. When you first encounter such a novel method, it helps not to ask too many `Why?'s, but instead repeat `Let's see where this leads'. That may help! Later you may return to the `Why?'s. We begin by introducing Hirota's binary operator D sa(x)b(x) =
a -a f a(x)b(x') as' /
(9.10.2)
(as
where s is a space or time variable, (x, y, z, t). To illustrate, we will set s = x. Thus
Drab = arb - abX = - Drba
(9.10.3)
Hirota's `direct method' in soliton theory
399
Further applications of this operator yield Dxab = axxb - 2axbx + abxx = Dxba Dzab = axxxb - 3axxbx + 3axbxx - abxxx = - D3ba.
(9.10.4)
This may be compared with the usual operator ax(ab) = axb + abx = (ax a2
+ ax
(ab) = axxb + 2axbx + abxx,
)a(x)b(x')Jx,=x,
etc.
(9.10.5)
axe
Moreover, if we exponentiate these operators (defined by the usual power series), we find exp (EDx)a(x)b(x) = a(x + E)b(x - E)
(9.10.6)
exp (Ea/ax)a(x)b(x) = a(x + E)b(x + E).
(9.10.7)
whereas
A generalized binary operator, D..O,xa(x)b(x) _ (a
a -1 ax, )a(x)b(x')Jx,=x
has also been introduced by Hirota (1985), for which (9.10.2) and (9.10.4) are special cases. We, however, will limit our considerations to (9.10.2). Having introduced this binary operator, Hirota's method proceeds by several artful steps:
(1) First introduce one or two functions, f (x, t) and g(x, t), in place of u(x, t), through
some nonlinear transformation. The primary examples of these transformations are u = g/f (type A);
u = (log f )xx (type B);
and
u = log (g/ f) (type Q.
(9.10.8)
That is, this method depends on the fact that the special solutions are the ratio of two functions, which moreover will consist of a finite number of terms (as we will see in step 2).
The object of this transformation is to put the equation (9.10.1) into a form of a homogeneous equation, involving only quadratic terms (f 2, fg, g2) acted on by the binary operators D", and D".
Solitaires: solitons and nonsolitons
400
These operators arise naturally from the proposed transformations, which involve ratios. Thus, for type A in (9.10.8), the derivatives of u are ux = (g/f )x = (Dxgf )lf 2
uxx = (Dxgf )lf 2 - (g/f)(DX ff )lf 2
uxxx = (Dxgf)If 2 - 3(Dxgf/f 2)(D2ff/f 2).
(9.10.9)
In general the direct computation of a"u/ax" = (a/ax)"(g/f) would be quite involved. However a simple relationship can be found by noting first that
g(x+E) _ g(x+E)f(x-E)
f(x+E) f(x+E)f(x-E) holds trivially. Now use the properties (9.10.6) and (9.10.7) to write this as eEa/ax
( g) = exp (EDx )g' f f J cosh (EDx) f f
and from the coefficient of E", (OIx)"
(f)
aE"(cosh(EDx)f f/E=O-
(9.10.10)
The cases n = 1, 2,3 of (9.10.10) are explicitly exhibited in (9.10.9). The objective is to put (9.10.1) into the homogeneous form
f 2 x F1(Dc, Dx, Dxx.... )(g-f) + FO(D,, Dx, Dxx, ...)(g-f) x F2(D,, Dx,...)(f f) = 0 (9.10.11)
by some transformation, such as those in (9.10.8).
(2) The second step is to introduce another unknown function, ), in order to decouple (9.10.11) into a bilinear form, of which there are many possibilities. An example might be F1(Dr)Dx, Dxx, ...)(g-f) =),FO(D,, Dx, Dxx, ...)(g-f)
F2(D, Dx.... )(f -f) = - 2f 2.
(9.10.12)
(3) The final step is to make a formal power series expansion of the functions introduced by the transformations (9.10.8), g
E"gn;
n=0
f
1+ E En fn
(9.10.13)
n=1
It is important that f begins with a constant term, for this will lead to linear equations with constant coefficients (note again that (9.10.1) cannot depend explicitly on either x or t). The series (9.10.13) is substituted into the bilinear equations (9.10.12), and the
Hirota's `direct method' in soliton theory
401
coefficients of each power of s is equated to zero. The objective will be to obtain exact solutions when these series are terminated after a few terms. For example, the second equation of (9.10.12) yields
F2(D, Dx,...)(1.1)= -A;
F20 11 +fl-1)= -22f1
F2-(I-f2 +f1'fl +f2' 1) = - 2(f 1 + 2f/2) F2'(1'f3 + fl f2 + f2.f1 + f3.1) = - 2(2f1f2 + 2f3)
(9.10.14)
The system (9.10.14) is a series of linear nonhomogeneous equations, with constant coefficients, and binary operators. Similarly, the first equation of (9.10.12) yields a series of equations, F1(go-1) = 2Fo(go-1)
F1(g1'1 +go-f1)=2Fo(gl'1 +go-f1)
(9.10.15)
of the same structure. The idea is to find solutions of (9.10.14) and (9.10.15) such that ,,(x, t) = t) - 0 for all n > N (where we are to select N). In other words, a finite number of terms in (9.10.13) yield an exact (special) solution of (9.10.1).
To Recapitulate To use Hirota's direct method we need to: (a) find the right transformation, (9.10.8), to put (9.10.1) into the form (9.10.11), involving the binary operators, D,, etc.; (b) Introduce a separation function, A, to put (9.10.11) into some `appropriate' bilinear form such as (9.10.12); (c) Use a formal series (9.10.13), and
look for special solutions which in fact terminate this series-and hence are exact (special) solutions. Rather remarkably, such a program turns out to be very powerful, as we will see. To examine some possible solutions, consider again (9.10.14). We have
A= -F2(0,0,...)
(9.10.16)
and
F2(D,, Dx, ...)(1'f1 + fl' 1) = 2F2(a/at, a/ax, ...)f1
so that F2(a/at, a/ax, a2/ax2,
...)f1 = - 2f1.
9.10.17)
We note that this is a linear equation, a consequence of the series (9.10.13) for f.
We can proceed to obtain a special solution of (9.10.17) which is a linear combination of elementary solutions, such as f1 =exp(el)+exp(112),
(9.10.18)
i1i=urt+k;x+q19)
(9.10.19)
where
Solitaires: solitons and nonsolitons
402
r1° are arbitrary constants, and (w;, k;) satisfy a dispersion relation F2 (w;, k;, k?, ...) = -A
(i = 1, 2).
(9.10.20)
The special solution (9.10.18), (9.10.19), is of course simply two traveling pulses. Next consider the resulting equation for f2, from (9.10.14)
F2(a/at, a/ax, ...)f2 = -) f2
- iA.f 1 - iF2(D,, Dx,.. ) f, f,
.
(9.10.21)
Now the last factor becomes, with the help of (9.10.3) and (9.10.4), - i F2(D DX, ...) [(eXp ('1 i) + exp ('12))(exp ('1,) + exp ('12))]
_ -zF2(tol -w,,k, -k,,ki-2ki+k1,...)exp(2'1,) - ZF2(w1 - w2, k, - k2, ki - 2k,k2 + k2.... )exp('1, +'12) - -F2(w2 - co1,k2 - k,,k2 - 2k,k2 + ki,...)exp(r1i +'12) -'-ZF2(w2 - w2, k2 - k2, k2 - 2k2 + k2,...)exp(2i12) The middle two terms give the symmetric part of F2
Fi(a,f,Y.... )=i[F2(a,f,Y,...)+F2(-a, -/3, -Y,...)],
(9.10.22)
whereas the factors proportional to exp (2r1,) and exp (2'12) cancel with two terms in - iAf i, because of (9.10.16). Hence (9.10.21) becomes F2(a/at, a/ax, a2/ax2, ...)f2 = - ).F2 - .. exp (r1 i +'12) - F2 (w, - w2, k, - k2 i ...)exp (r1, +'12 ). This has the solution (9.10.23)
f2 = A exp ('1 +'12 ),
provided that A satisfies
A- -[A+F2(w, -w2,k, -k2,ki-2k,k2+k2,...)] [A+F2(wi+w2,k,+k2,(k,+k2)2,...)]
( 9.10.24)
This takes care of the solution (9.10.13) up to e2. The next order, involvingf3, is determined by the last equation of (9.10.14), namely
F2(Dr, DX, Dx, ...)(1'f3 + f''fi + f2'fi + fs' 1) = -
2(2flf2
+ 2f3).
(9.10.25)
Consider the inhomogeneous part (independent of f3). Using f, f 2 = A [exp ('1i) + exp (f12) ] exp ('1i +'12 ). The left factor in (9.10.25) is
F2(fif2 + f2fl) = 2F2 f 1 f 2 = 2[F2(a)2, k2, k2,...) + FZ(wi, ki, k2i ,...)Ifif2, (9.10.26)
where F2 is again the symmetric part, (9.10.22). We now choose to set f3 = 0, based on the special solution being sought, namely two solitons, characterized by the form in (9.10.18). We can set f3 = 0 in (9.10.25), provided that (9.10.26) equals - 2af, f2. But to accomplish this, we need to be able to relate F2 (w;, k;, k?,...) to ), whereas we only have the relationship (9.10.20). Thus
Hirota's `direct method' in soliton theory
403
we must require F2 = F2, so (9.10.26) becomes -4Af1 f2. This equals -22f1 f2i the right side of (9.10.25), only if A = 0, so we are restricted to the case F2 (0, 0, ...) = 0
(9.10.27)
F2(c,A,Y,...) = F2(-a, - fi, -y,...), in which case we can set f3 = 0. In that case the equation for f4, from (9.10.11), is simply, (9.10.28)
F2(l f4 + f2 f2 + f4.1) = 0, and since
F2(D,,Dr,...)f2'.f2 =F2(D,,DX,...)A2exp(rl1
+f12)
=F2(co1+w2-co1-(02,k1+k2-k1-k2.... )A2exp[2(111+h2)] =F2(0,0,...)A2exp[2(111+72)]=0 by (9.10.27). Hence (9.10.28) is satisfied by f4 = 0. All higher order functions, f (n > 4),
satisfy homogeneous equations, because of (9.10.11), and hence can be taken to be zero. Thus we have terminated the series (9.10.13), for f, at f2. Therefore, if (9.10.27) holds, this series is, by (9.10.18), (9.10.23), and (9.10.24)
f = 1 +exp(rl1)+exp(g2)+Aexp(rl1 +i i),
(9.10.29)
which is an exact solution of F2(D1, D., Dx, ...) f f = 0.
(9.10.30)
If (9.10.30) can be obtained by a transformation, such as u = (log .f )XX
(type B)
(9.10.31)
from N(u, ur, uX, uXX, ...) = 0,
(9.10.1)
then (9.10.29) gives an exact solution of (9.10.1). In this case we do not need to introduce
a second function, g, as in the type A and C transformations (9.10.8). If these transformations are needed, then we must turn to the first equation of (9.10.12) and the g series, (9.10.13), and (9.10.15).
We will now briefly indicate some explicit Hirota Transformations: (See Solitons,
Direct method references for the Benjamin-Ono, Massive Thirring, Boussinesq, self-induced transparency, and Kadomtsev-Petviashvili equations.)
Type A (1) Nonlinear Schrodinger equation (9.10.32)
Solitaires: solitons and nonsolitons
404
is transformed by i' = g/f (f real) to (a) (iD, + Dx 2)g- f = 0 and D' f f = qgg* (if q > 0), with the boundary condition
limxxx-.1012=0; or (b) (iDe + Dx - A)g f = 0 and (D2 - ,)f f = qgg* (if q < 0), with the boundary condition limxxx 10 I2 =1/I q 1. (2) Derivative nonlinear Schrodinger equation
d iii, + Y'xx + i(0*'Y 2)x = 0
(9.10.33)
is transformed by ili = f *g/f 2 into the three bilinear equations
(iDe+DX)g f=0; and
+ig*g=0. Type B This type, which involves derivatives, is related to Backlund transformations, discussed in the next section.
(1) Korteweg-deVries equation ut + 6uux + uxxx = 0
(9.10.34)
is transformed by u = 2 (log f )xx into
[Dx(D,+Dx)+A.]f f =0 where A=O, if limxxx . co u = 0. Hirota (1971) used this to first obtain the impressive N-soliton solution of the KdV equation {
N
JN = E. exp µ=0.1
N
E, Pi/1iAij + E 'Uif1i i<j i=1
where rli = kix - k3 t + n°; rl0 are arbitrary phase constants, and the Ai; satisfy exp Aij = (ki - kj)2/(ki + k;)2.
The sum Y_µ= 0,1 involves all possible combinations of µ 1 = 0, 1; µ2 = 0, 1; ... ; µN = 0, 1. The solution (9.10.29) is for N = 2. This solution can also be expressed
in the compact, and relatively obscure form fN = det I I + M 1, where I is the unit matrix, and
M
-_ (kikj)1/2 e- 1/2(ni+n;)
`'
ki+k;
Hirota's `direct method' in soliton theory
405
Using the same transform:
(2) Sawada-Kotera equation (9.10.35)
ut + 45u2ux + 15(uuxx)x + u5x = 0
transforms to DX(Dt + DX)f'f = 0.
(3) Lax's equation ut + 30(u3)x + 10(2uxuxx + uuxxx) + u5x = 0
(9.10.36)
transforms into [Dx(D, + DX) -
3Dt(Dt + D3)]f'f = 0
The N-soliton solution of this equation is f = det l 1 + MI;
(1: unit matrix)
and M is the N x N matrix with elements
_ 2(C;Ci)"2 Mi'
C,+C
1
expz(qj+t1i)
(i,1= 1,...,N)
where the C, are arbitrary constants, and
r1;=k;x-kst+ This may be compared with the above solution for the KdV equation. Type C Involves the log of a ratio of functions and possibly their derivatives. (1) Modified KdV equation
ut+6u2ux+uxxx=0
(9.10.37)
is transformed by u = - i [log (f */ f )]x into the pair
=0
and
=0.
(2) Sine-Gordon equation Oxx - 0tt = sin 0
is transformed into (D2
- Di )f *.f = 2(f #2 - f 2)
by
4= -2ilog(f*/f).
(9.10.38)
Solitaires: solitons and nonsolitons
406
There is, of course, no `formula' for obtaining a Hirota transformation for a given nonlinear PDE (if it exists!) except for experience. It is, however, a powerful method which continues to be generalized (Hirota, 1985), and certainly represents
another of the `analytic gems' in the analysis of integrable PDEs.
9.11 The AKNS formulation of the IST (!!) In this section we will briefly examine the general formulation of the inverse scattering transform, (IST) due to Ablowitz, Kaup, Newell and Segur (AKNS, 1973), and others. This method uses Zaharov and Shabat's method for obtaining the inverse transform for a pair of `Schrodinger equations' (the scattering problem)
a01= q(x, t) i2 -'AO1 ax
(9.11.1)
S:
002
ax
=r(x,t)li2+i2,2
where 2 is required to be a constant. We will not treat the details of this generalization of the Gel'fand-Levitan equation (e.g., see Newell, 1985), but instead will concentrate on some practical extensions which the AKNS method provides for us. In (9.11.1), the two functions q(x, t) and r(x, t) are the physical functions of interest, replacing the single function u(x, t), for example, from the KdV equation. In the usual way, we are interested in functions which vanish at x = ± oo.
The temporal equations for ail and 02 are assumed to be given by ao 1 = A(x, t, 2)01 + B(x, t,2)02 at
T:
1.2)
a02 at
C(x,t,2)>/i1+A(x,t,2)Y'2
Note that the `diagonal' function, A(x, t, A), is the same in both equations. The integrability conditions for (9.11.1) and (9.11.2) are at
(ax)
ax (aatk)
(k = 1, 2)
or ,I'
at
(q02 -'AO 1) = ax (A01 + B02)
a (rl/1+'412) =a (CY'1-A02)
AKNS formulation of the IST
407
Requiring that al/at = 0, and using (9.11.1) and (9.11.2), we obtain
grr2 + q(CI1 - Ac2) - i2(Ac1 + B02) =Axi/i1 +A(g0/i2-iAi/i1)+Bxc2+B(ro 1 +i 02) and
rr/1 + r(A , + Btfi2) + i i(Ci/il - A02) = CXc1 + C(g02 -
i2c1)
- A.,,02 - A(ro1 "02)-
Since 01 and 02 are independent functions, their coefficients must vanish. The coefficients of i/i yield 1
Ax=qC - rB
(9.11.3)
Cx = rr + 2rA + 2i2C,
(9.11.4)
where the coefficients of 02 yields (9.11.3) and
Bx = qr - 2qA - 2i2B.
(9.11.5)
The evolution equations for the physical functions of interest (namely q(c, t) and/or
r(x, t)) are obtained by eliminating 2 from (9.11.3)-(9.11.5), where however, the functions A, B, and C depend on A. Specifically AKNS take these functions to be finite power series in 2 N
N
A=
B=
AkAk;
k=M
N
C= = AkCk.
.i,kBk;
(9.11.6)
k=M
k=M
The procedure is to substitute (9.11.6) into (9.11.3)-(9.11.5) and to equate the coefficients of each power of 2 to zero. This will yield some evolutionary equations.
Whether they are of some physical interest remains to be seen. We see that this approach does not start with a given physical problem, but essentially `backs into' a large group of evolutionary equations, some of which (hopefully) are of physical interest. Happily, that turns out to be the case, which is why the AKNS method is of physical interest.
The simplest possibility turns out to be N = M - 1 in (9.11.6). Then (9.11.3) and (9.11.4) generate the system
B-1=-21gr; C-1=Zirr; (B_1)x= -2gA_1; (C_1)x=2rA_1.
(9.11.7)
Thus B and C are now related to the `physical' functions, q and r. Introducing the notation A _ 1 - a(x, t), (9.11.3) and (9.11.7) require that ax =
iigrr +
iigr =
ii(gr)r
(9.11.8)
408
Solitaires: solitons and nonsolitons
whereas (9.11.7) also requires that
qxt = - 4iqa;
rxt - 4ira.
(9.11.9)
At this point there are still a variety of possible evolution equations which can be obtained for q(x, t) and r(x, t), but they must all be special cases of the two third-order PDE obtained by eliminating a(x, t) from (9.11.9) using (9.11.8). We note that (9.11.9) allows the possible relationship q(x, t) = ocr(x, t), for any constant a. Two cases of particular interest are q r. If q = - r, this leaves the equations ax = - 2i(g2)t;
qxt = - 4iqa.
(9.11.10)
We could proceed, in an uninspired fashion, and operate on the second equation with q8/8x, and use the first equation to obtain the evolution equation for q(x, t) ggxxt = - g3gt + gxgxtg
which is of obscure physical interest. The inspired procedure (a la AKNS) is to introduce yet a new function, u(x, t), related to q and a(x, t) by
q = - ?ux;
a = 4i cos (u).
(9.11.11)
This, of course, must represent a particular solution of (9.11.8) and (9.11.9), so we must have sin (u)ux = uxuxt;
Uxxt = Cos (U)ux.
Both of these are satisfied if u(x, t) satisfies the sine-Gordon equation uxt = sin (u).
(9.11.12)
Thus, this evolution equation is related to q =- - r by (9.11.11), which in turn appears in the Schrodinger equations (9.11.1). Hence we see that the sine-Gordon equation can also be solved (in general) by the IST. Special solutions of (9.11.12), of course, can be obtained relatively easily, as we saw in Section 9.9. We noted above that (9.11.9) is satisfied by q = ± r. If we now take q = ± r, and replace (9.11.11) by q = tux;
a = 4i cosh (u)
(9.11.13)
we find that (9.11.8) and (9.11.9) are satisfied, provided that u(x, t) satisfies the evolution
equation uxt = sinh (u).
(9.11.14)
This is obviously known as the sinh-Gordon equation. Numerous physical applications of this equation can be found in the references.
AKNS formulation of the IST
409
To illustrate another example, return to (9.11.6) and set M = 0. Quite generally, we find from substituting (9.11.6) into (9.11.3)-(9.11.5) AN = aN (constant);
BN = 0,
CN = 0,
(9.11.15)
and for k = 0,1, ... , N (Ak)x = qCk - rBk (Bk)x = bkogt
- 2gAk - 21Bk-1
(Ck)x = bkort + 2rAk + 2i Ck
-1
(9.11.16)
where bko = 0 (if k :0) and 1 (if k = 0). Now we take N = 2. Then using (9.11.15), the terms k = 2 in (9.11.16) give C1 = ira2,
B1 = iga2;
and using them in the k = 1 equations yields so A1= a1 (constant);
(A1). = 0;
(B1)X=ia2q.= -2gA1 - 2iB0 (C1)x= ia2rx = 2rA1 + 2iCo.
(9.11.17)
Finally, for k = 0, we have (A0)x = qC0 - rB0; (Bo)x = qr - 2qA0;
(CO). = rr + 2rA0.
(9.11.18)
Again we are left with a variety of choices, each yielding a different evolution equation. One simple possibility is to take
Al -a1 =0 so
Bo = -
2a2q,;
Co = 2a2rx.
Therefore, from (9.11.18) (Ao)x = ia2(gr)x
or A0 = 2a2gr + ao.
Making the choice ao = 0, (9.11.18) gives the evolution equations - 2a1gxx = qr - a2g2r 2a2rxx = rr +
(9.11.19)
a2qr2.
A special case of physical importance is when r = - q*, which also requires that a2 be imaginary for (9.11.19) to be satisfied. If we take a2 = 2i, (9.11.19) yields the nonlinear
Solitaires: solitons and nonsolitons
410
Schrodinger equation iq, = qxx. + 2gxg2.
(9.11.20)
As mentioned in Section 9.8, this was first solved by this IST method by Zakharov and Shabat (1971). By retaining terms up to A3 in (9.11.6), it is possible to obtain the KdV equation as well as the modified KdV (MKdV) equation qr ± 6g2gx + gxxx = 0.
(9.11.21)
Finally we note that variations of the AKNS formulation have been introduced by Kaup and Newell (1978) and Wadati, Konno, and Ichikawa (1978), which essentially
amounts to replacing
r - 2r,
q -> ),q
in (9.11.1) and (9.11.3)-(9.11.5). This yields new evolution equations when the series (9.11.6) is employed. For example, if r = - 1, q = u - 1 and
A= - 4iu- 1/223 _ u- 3/2ux)2 B = 4u-' /2(U - 1)23 + 2i u - 3/2 ux,.2 - A(u - 3/2ux)x
C= -4u-' /2A3 then (believe it or not!), the evolution equation for u is Dym's equation ut = 2(u- 1/2)xxx.
(9.11.22)
The applications of this equation have been discussed by Kruskal (1975). These examples illustrate quite clearly both the generality and the inventiveness which is required to seek out physically interesting evolution equations which can be solved by the IST method.
9.12 Some Backlund transformations between difference equations Another of the mathematical art forms which has been employed to solve nonlinear PDE is the so-called Backlund transformations. These transformations are not the familiar types, relating changes of coordinates or functional interrelationships, but instead involve various partial derivatives of several functions. There is a long history in the use of such transformations, but a precise general definition is not easy to find. We will give one suitably general definition, but the real understanding will come from examining some special examples. Let u(x, t) be a solution of a partial differential equation D1(u) = 0,
(9.12.1)
Some Backlund transformations
411
and v(x, t) be a solution of another partial differential equation, D2(v) = 0.
(9.12.2)
A Backlund transformation is a set of partial differential relationships (i = 1, ... , n)
B; (u, v) = 0
(9.12.3)
which is such that, if u(x, t) is a solution of (9.12.1), and v(x, t) is a solution of the set (9.12.3), then v(x, t) is also a solution of (9.12.2). Generally it not also required that, if v(x, t) is a solution of (9.12.2) and (9.12.3), then u(x, t) satisfies (9.12.1). This is certainly not your everyday transformation, so let us proceed with some examples. We begin with the Burgers equation z
(9.12.4)
at+uax+ax2=0,
which represents (9.12.1). The claim is that this equation is related to the simpler linear diffusion equation, representing (9.12.2), 2
0
(9.12.5)
(the Hopf-Cole transformation)
(9.12.6)
T + aX2
by the Backlund transformation: ax
? uv
(vax+uax).
at
(9.12.7)
J)
Thus (9.12.6) and (9.12.7) are representative of (9.12.3) for i = 1, 2. We want to show that, if u(x, t) satisfies (9.12.4) and the v(x, t) satisfies (9.12.6), and (9.12.7), then v(x, t) also satisfies (9.12.5).
We first establish the latter fact by differentiating (9.12.6) with respect to x, and use (9.12.7) to obtain 02v
1
ax2-
au
av
av
llax+uax
-Ft
°
Hence (9.12.5) is indeed satisfied.
What we need to show next is that the pair of equations (9.12.6) and (9.12.7) can be integrated to obtain v(x, t), if u(x, t) satisfies (9.12.4). The integrability condition is
a
T(at
(00=
Solitaires: solitons and nonsolitons
412
Hence we need to show that 8
8t
_
8
8x
28t(uV)
8
(3v)
8x
8t
18
au
av
23x CvaX
FX
holds, if u satisfies (9.12.4). This condition is uvi + u,v = - (2vxux + vuxx + uvXX),
from which vx, vt =
(9.12.8)
Zu2v), and viX = i(uXv + Zu2V) can be eliminated, with the help of (9.12.6) and (9.12.7). Dividing out v, (9.12.8) yields 2(vuX +
- i u(ux + u2) + U,
uuX - uXX - 2 u(ux + u2)
2
2
which is indeed satisfied by solutions of (9.12.4). Therefore (9.12.6) and (9.12.7) can be integrated for v(x, t), if u(x, t) satisfies (9.12.4). This illustrates one example of the use of a Backlund transformation in some detail. We now outline a few other examples. A second example is to show that the Liouville equation, ux3, = exp (u)
(9.12.9)
is related to the nearly trivial equation VXY=0
(9.12.10)
by the Backlund transformation vx = ux + a exp [(u + v)/2]
(9.12.11)
vY = - uY - (2/a) exp [ - (v - u)/2].
(9.12.12)
Indeed, differentiating (9.12.11) with respect to y, and using (9.12.12) and (9.12.9), we have vxY = exp (u) + (a/2)(uY + vY) exp [(u + v)/2] = exp (u) + (a/2)(- 2/a) exp (u) = 0,
which satisfies (9.12.10). By similarly showing that (vy)x = 0, we show the integrability condition, vxY = vYX, is satisfied by (9.12.11) and (9.12.12). This completes the proof.
Finally we note that the KdV equation, uY + 6uux + uXXX = 09
(9.12.13)
can be related to the modified KdV equation, vY - 6v2vX + vXXX = 0,
(9.12.14)
by the Backlund transformation VX = + (u + v2)
(Miura's transformation)
Vy = + uxx - 2(vux + uvx).
(9.12.15) (9.12.16)
Invariant Backlund transformations
413
Writing (9.12.15) in the form u = - v2 + v, and substituting into (9.12.13) yields - 2vvy + vxy + 6(- v2 + v.,)(- 2vv + v.,.,) - 2(vv )xx + Uxxxx = 0,
or
which is satisfied if v is a solution of (9.12.14). Note that we have now interchanged the role of u and v in our definition (9.12.1)-(9.12.3). To check the integrability condition, differentiate (9.12.15) with respect to y, and use (9.12.13) and (9.12.16) to eliminate uy and/ vy, (vx)y = + (6uux + uxxx) ± 2v(+ uxx
- 2vux + uvx).
Comparing this with (from (9.12.16)) (vy)x = :" Uxxx - 2(2vxux + vuxx + Uvxx),
we see that they are equal if + 6uux + 4v2ux ± 2vuvx = - 4vxux - 2uvxx = +4(u+v2)u,, + 2u(ux + 2vvx),
where (9.12.15) has been used again twice. These few examples illustrate how Backlund transformations may be employed to associate solutions of a complicated equation with a simpler equation. A number of other examples can be found in the references (under Backlund transformation) but, alas, there is no known systematic method for finding such transformations.
9.13 Invariant Backlund transformations In the last section we saw how Backlund transformations can be used to relate two different partial differential equations D1(u) = 0
and
D2(v) = 0,
where one is complicated, and the other may be simpler. In the present section we will illustrate another use of the Backlund transformation, (9.12.3), for cases where the two PDE are the same (D1 = D2), D(u) = 0
and
D(v) = 0.
In this case we have a so-called invariant Backlund transformation
B.(u,v;c)=0
(i= 1,...,n),
where c is an arbitrary constant in these equations. One may wonder why we want to associate two identical PDEs, since they are obvious equally difficult to integrate.
414
Solitaires: solitons and nonsolitons
However, the `artful device' now comes from the arbitrary parameter, c, which appears
in these Backlund transformations. As we will see, this parameter can be put to good use. We begin with our favorite, the KdV equation uy + 6uux + uxxx = 0
(9.13.1)
Wahlquist and Estabrook (1973) and Lamb (1974) showed that (9.13.1) is invariant under the Backlund transforrmation
ux+vx=m-2(u-v)2
(9.13.2)
u,, + vy = (u - v)(uxx - vxx) - 2(ux)2 + uxvx - 2(vx)2
(9.13.3)
where m is the desired arbitrary constant. By `invariance' we mean that, if the relationships (9.13.2) and (9.13.3) hold between the functions u(x, y) and v(x, y) and if u(x, y) satisfies (9.13.1), then v(x,y) will likewise satisfy a similar equation, vy+6uux+uxxx = 0.
To appreciate the above invariance property, we need to carry out the necessary differentiations of u(x, y) in (9.13.2) and (9.13.3), and with algebraic manipulations, eliminate u(x, y) from (9.13.1) to obtain the identical equation for v(x, y). This process involves some tedium and pain, but should be done once to make it believable; which is a way of saying that it is left as an exercise. Proceeding from there, we note first that (9.13.2) is an ODE, rather than a PDE.
Moreover, with the help of the arbitrary parameter m, we can obtain a nonlinear superposition, relating four solutions of (9.13.1). Let u be a solution and ul another solution related by (9.13.2)
ux+Ulx=m1 -2(U-UI) U.2 1
(9.13.4)
Next, use u1 to generate another solution u12, with m = m2, u12x + ulx = m2 - 2(u12 - u1)2.
(9.13.5)
However, u12 could also have been generated from u by first constructing a solution u2 using m2
u2x+ux=m2-2(u2-u)2,
(9.13.6)
u21x + u2x = m1 - 2(u21 - u2)2.
(9.13.7)
and following this with m1
This process permutes, so that u21 = u12 (proved by Bianchi in 1879; see Fig. 9.29). To eliminate the derivatives, we can use [(9.13.5)-(9.13.7)]/[(9.13.4)-(9.13.5)] = 1 to
Invariant Blicklund transformations
415
Fig. 9.29
2 = U21
obtain
U12=U+
2(m1 - m2)
(9.13.8)
U1 - U2
This nonlinear superposition of four solutions of (9.13.8) is related to the fact that (9.13.2) is a Riccati equation for w - v - u,
wX+2w2+2u,, - m = 0, whose solutions satisfy the 'cross-ratio' relationship (w - w2)(W1 - w3) = k, (W - W1)(w2 - w3)
which is (9.13.8). Actually there must be a superposition relationship involving only three solutions, because the Riccati is related to linear second order equations (see the discussion in Chapter 2). A simpler example involves the sine-Gordon equation, (9.9.1),
a2¢_a20=sin ax2
0
at2
which can be written in the form a Ox
0)(0 ax
at
+
a
at
0= sin 4.
(9.13.9)
Let 4(x, t) = a(r, s), where r = (x + t)/2;
s=(x-t)/2.
Then (9.13.9) becomes a2a Or Os
= sin a.
(9.13.10)
Solitaires: solitons and nonsolitons
416
This equation is invariant under the Backlund transformation ar (a1 2
a)= as in \ a' 2 a)
as
(a1 2
= a sin
( 9 . 13 . 11 )
( 9 . 13 . 12)
(a1 2
which are again ODE, and where a is an arbitrary constant. This is easy to see, for differentiating (9.13.11) with respect to s, and using (9.13.12), yields a2
ai - a
arcs
ai + a
= a cos
2
2
a1 - a
1
a sin
2
= i[sin a1 -sin a]. Since a satisfies (9.13.10) this simplifies to a2 a1
arcs
= sin a1,
which proves the invariance. Because this invariance transformation contains an arbitrary parameter, a, we can
again obtain a nonlinear superposition between solutions, because the Backlund transformation is of the Riccati type. For if u = tan 4(ai + a), then
a
a sin (
i
2 1(0,1+ ur=sec qa)ar[4 (a1 - or + 2a)] = 2
COs2
ai+a 2
(al +a )
+ (1 +
U2)
s
tar
or
u,=au
+i(OQ
(1 +u2),
which is a Riccati equation for u, and a superposition relationship is assured. The nonlinear superposition can be obtained from (9.13.11) and (9.13.12) by using different constants, (a1, a2), and the permutability illustrated in Fig. 9.30, beginning with a solution ao. This yields the superposition tan
(a3-a°)=a1 4
+a2tan(a1 -a2).
a1 - a2
4
(9.13.13)
Invariant Backlund transformations
417
Fig. 9.30
Exercise 9.13 Consider the functions 2(o k ± o'o) o o in (9.13.11) and (9.13.12), and, more generally, o 11 . Use these together with two constants, a1 and a2, to o 1) obtain (9.13.13) according to the scheme in Fig. 9.30.
The Backlund transformation (9.13.11) and (9.13.12) can also be used to generate a solution, starting with the `vacuum' solution, a - 0, so
1361_ - a sin
2 Or
61
2
1361_1 2 3s
a
sin
2.
61
This shows that a 1(r, s) = a 1(ar + a -'s) = a 1(p), and 2361(p)lap = sin (a1/2). 02 ). Let a1/2 = 2 tan -' (i/i). Then sin (61 /2) = 2 sin (tan -' i/) cos (tan -' 0) = Hence d(a1/2)/dp = d(2 tan-' 0)/dp = (2/1 +02 ) do/dp = sin(a1/2) = 21/i/1 + 1/,2, or simply dpi/dp = . Therefore 1/i = eP, ignoring the constant of integration, since this corresponds only to a shift in p. Therefore
a1 = 4 tan -' [exp (p)] = 4 tan-' [exp (ar + a -'s) ], or
a1 =4tan-' [exp(±(x-ct)/(1 -c2)'"2)],
(9.13.14)
where ± = sgn (a), and c = (1 - a2)/(1 + a2) is the velocity of propagation. This is the `kink' (+) or `antikink' (-) solution, discussed in Section 9.9. If a1 and 62 are two solutions of the form (9.13.14), with velocities (cl, c2), (9.13.13) gives a third solution
tan(63/4) = K(c,,c2)tan {tan ' [exp(vl)] - tan where Vk = sgn (ak)(x - ck t)/(1- ck )' 12. Since tan - 1 (a) - tan
tan
[exp(v2)]} (b) = tan
(a - b/1 + ab),
3 ) = K(c, C2) exp (v,) - exp (- v2) - K(c1, c2) sinh (2(vl - v2)) 4
1 + exp (v1 + v2)
cosh (2(v1 + v2))
(9.13.15)
Solitaires: solitons and nonsolitons
418 and
+ a2
= sgn (a,) [(1 - c,)/(1 + c, )] "2 + sgn (a2)[(1 - c2)/(1 + c2)] "2
a, - a2
sgn(a1)[(1-c,)/(1+c,)]'12-sgn(a2)[(1-c2)/(1+CA "2
K(ci, c2) = a'
If a, > 0, a2 < 0, then a3 goes from - 27T to + 2ir as x goes from - oo to + oo, and
hence is called a 4n pulse (Fig. 9.31). The location of a3 = 0 is at v, = v2, or Fig. 9.31
03
2ir
x -21r
(x - c, t)/(1 - c )''2 = - (x - C2010 - c2)1'2, and hence has a velocity velocity
c,(1 - c2)"2 + c2(1 - ci)'12
(1-c2)1"2+(1-c2)''2
If a, > 0, and a2 > 0, then a3 is a On pulse.
9.14 Infinite number of conservation laws We will see in this section that invariant Backlund transformations which have a free parameter, yield an infinite number of conservation laws. Recall from the discussion in Section 9.5, that conservation laws were the `guiding light' which led Miura to his transformation, (9.5.12), and thence to the Schrodinger equation (9.5.14).
We therefore return to this important topic. A differential equation, involving a function O(x, t), is associated with a conservation law if any solution, 4(x, t), satisfies at
D(01 OX, 0r, ...) +
a flo,
Ox,
0.
(9.14.1)
This is called a conservation equation, involving a `density', D, and a `flux', F. More common conservation equations are the continuity equation an/at + a(nu)/ax = 0, expressing the conservation of particles, or aQ/at + aJ/ax = 0, involving the energy density, Q, and heat flux, J (conservation of energy). (9.14.1) is a generalization of this idea. Now consider the sine-Gordon equation, Ox, = sin 0, and its invariant Backlund transformation Ox = Ox + 2a sin (¢ + 0)/2,
(9.14.2)
0, = - 0, + (2/a) sin (0 - 0)/2.
(9.14.3)
Infinite number of conservation laws
419
We now treat a as a small parameter, and consider the expansion (9.14.4)
0=0+aO1+a202+...
Substituting this into (9.14.3) yields I'
01+21
Equating the coefficients of powers of a, 01
=20,
W3-203,+34't,
'tt,
Y'2=24/
05 = 2451 + 30t 03t + 50,0 t
04 = 2041 + 20tott,
+ o¢t
etc. (9.14.5)
These relationships are also consistent with (9.14.2). Because (9.14.2) and (9.14.3) is an invariant Backlund transformation, t/i(x, t) also satisfies
0xt-sin0=0. Now this equation can be written in a conservation form, (9.14.1) namely (102), + (cos 1/i - 1)x = 0.
(9.14.6)
Substituting (9.14.4), and using (9.14.5), we can obtain an infinite number of conservation laws, one for the coefficient of each a". These have `densities', D, and `fluxes', F" which satisfy a
a atD"+axF"=0.
The lowest order Do and F0 are Do
Fo = (cos
=
-1).
The conservation equation in this case is simply the spatial gradient of the sine-Gordon equation, which *(x, t) satisfies, by assumption. The next few densities and fluxes are D2 = 2(ox,)2 + 2oxoxtt, 2
= -2i0i2sin0-i/i;cos
F2= aa2cos0 a=o
44[[ sin 0 - 4(x[)2 cos 0 = 4(cos /)[[
and D4 = 20xox[n[ + 40x[[4'x[[[ + 20x[[Y'x[02 + 20r 0x4'x[[ + 20x[[ + 40xO,Ox,O,,
and so forth for F4. The odd order densities and fluxes are simply the time derivative
420
Solitaires: solitons and nonsolitons
of the even ones. For example
D, =
a
at
F, =
Do,
a
at
F0.
Hence the odd densities and fluxes do not yield independent conservation laws. As in the case of the KdV equation, with its infinite number of conservation laws, we see here another explicit example of this association between that property of the PDE and the existence of soliton solutions.
9.15 Onward Perhaps no place in the study of nonlinear dynamics is there such a varied juxtaposition of amazing analytical methods as there is in the above examples of
integrable PDEs. The inter-relations of these many methods is only partially understood, and continues to be explored (e.g., see Newell, 1985). We must leave these considerations for the future, and instead explore some extensions of the above soliton/nonsoliton concepts. One of the most obvious extensions of interest is the possibility of solitaire solutions
in higher dimensions. It is reasonably clear that solitary disturbances, which are localized in all directions (in two or three dimensions), will not retain their shape in time, if the system is `simple'. If a disturbance is initially localized in space, like a mound of water or an explosion (Fig. 9.32), it will propagate in all directions, and Fig. 9.32
04
decrease in amplitude. Because of the isotropy of most media, it is not generally possible to balance the amplitude-dependent velocity against dispersion in only one (propagating) direction. The introduction of gravity, or magnetic and electric fields breaks this isotropy, and may provide `channels' for traveling localized disturbances (e.g., in plasmas), but these are rather specialized circumstances.
Onward
421
Fig. 9.33
The simplest possibility for a nontrivial two-dimensional semi-localized disturbance in `simple' systems, such as water or plasmas, appears to be interacting solitary ridges. Such an interaction is illustrated schematically in Fig. 9.33 in a cutaway view of two ridges. Of course the ridges are localized only in one direction, but the combination of the two ridges makes the system nontrivial. Such systems have been modeled by several equations. The Kadomtsev-Petviashvili equation (KP) (1980) (9.15.1)
(u, + uuX), + uyy = cuxxxx
is the two-dimensional generalization of the KdV equation (see reviews in Physica 18D, 1986), whereas Yajima, Oikawa, and Satsuma (YOS) (1978) used
8tf
-VZf +«
=0
(9.15.2)
to investigate interacting ion-acoustic solitons in plasmas. A solitary ridge can be characterized in terms of a wave vector, K, and a dispersion relation D(K, f (K)) = 0. Thus, in the case of the YOS equation, (9.15.2), a solitary ridge is
fl(K) t]
(9.15.3)
D(K, n) - S22 - K2 - 4#02K2 = 0.
9.15.4)
p = (6/3/a)K2 sech2
where p,
= -V2f,
and the nonlinear dispersion relation is
The interaction of two of these ridges has been studied both analytically (for which only approximate solutions apparently exist) and numerically, by Kako and Yajima (1980) (also see Yajima (1985) for a survey).
These and other studies indicate the importance of `resonance conditions' S2(K3) = S2(K1) ± f2(K2);
K3 = K1 ± K2
(9.15.5)
in understanding the possible interactions between two and three solitons. When the
422
Solitaires: solitons and nonsolitons
angle between two wave vectors K, and K2 (normal to two asymptotic ridges) is small, it may be possible for these ridges to travel, with a `virtual soliton', connecting the spatially shifted ridges. This is schematically illustrated in Fig. 9.34. Similarly, if
the angle between K, and K2 is very large, two ridges may again persist, with the Fig. 9.34
Fig. 9.35
help of a phase-shifting virtual soliton (Fig. 9.35). However, when the angles approach the resonance condition (9.15.5), the `virtual soliton' becomes longer, and apparently becomes a `real' ridge (i.e., extending to infinity) at the two critical angles satisfying (9.15.5). In each case there is theoretically a `Y'-soliton ridge structure, as illustrated
in Fig. 9.36 (from Yajima, 1985). For wave vectors between these two resonance angles, there is apparently no reference frame in which the soliton is time independent.
This resonance breakdown of Hirota's two-soliton solution of the KP equation, (9.15.11), was discovered by J.W. Miles, and has been related to the breakdown of the Zakharov-Shabat multisoliton solution of the KP equation by Newell and Redekopp (1977; see Kaks and Yajima, 1980). Other interactions of water ridges are illustrated and reviewed by Segur (1986). When the physical system is not `simple', as in the case of many field theories, then there are new opportunities for higher-dimensional solitary disturbances. One type of field-theory solitaire requires that (a) the system have degenerate `vacuum states', and (b) there is more than one field at each point (x, t) (i.e., vector fields). Just as in
Comments
423
the case of the sine-Gordon equation, which has an infinite number of 'zero-energy' states, sin 0 = 0 (¢ = 2nn, n = 0, 1, 2, ... ), and correspondingly yields an infinite number of topological solitons, the higher-dimensional fields can also have solitary topological solitons. However, to be solitary in all directions, they need the vector field character, to distinguish the degenerate ground state at large distances. Localizations need not be limited to three dimensions. Localized solutions in space and time can be obtained in some systems, and are referred to as instantons. These solitons represent events rather than spatially localized disturbances, or particles.
These topics are beyond the scope of our present discussion, but introductory discussions can be found in the references (Rebbi, 1979).
It is important to recognize the fact that once a phenomenon such as soliton interactions is discovered, it changes our perspective on the dynamics of many other systems, which may not have simple solitaire excitations. This is an important benefit of such discoveries. Thus in attempts to understand `turbulence' or space-time patterns
in continuous systems, the knowledge that an unperturbed system may tend to generate localized structures can greatly assist in formulating research about perturbed systems. If an unperturbed system has various soliton solutions, then when this system is perturbed by some force, it may be useful to view the dynamics as making transitions between soliton states. At the very least, this offers a very different perspective from
the one usually based on perturbed linear modes of dynamics. Many of these perspectives can be found in the review references.
Comments on exercises (9.1) (Fig. 9.37) The initial conditions yield ZY1(0) = ± 2, all other ZK(O) = 0, so Zn(T) =
z
[J - 1(T) - J,, + 1(r)] This yields yk(T) = J2k(T). More generally Yk(T) =
Zm(°)'12(k+1)+ 1 -m(T) M
!
Solitaires: solitons and nonsolitons
424 Fig. 9.37
y'o v,
1
(9.2) If /3 0 y, then (9.1.15) yields (see Appendix F) [(a-x)(x-f)(x-y)]-'12dx=2(a-y)1"2sn-1
[k(w _
)112
\w - y
JaW
where k2
y), or
w(z') _ /3 - yk2 sn2 [Z (a - y)'12z', k] 1 - k2 sn2 [2(a - y)1"2z', k]
w varies between a (where sn2 = 1) and
fi
(sn2 = 0). Since
w=yz, if
fOK (k )
w(z') dz' = y[K(k)] - y(O) = 0
there is no expansion of the lattice, otherwise there is a constant expansion each wave length. Note that the 'quarter-period', K(k), is the half-period of y(i) because of the above quadratic dependence, sn2 (2(a - y)1/2z', k). Thus, there is a two-parameter family of periodic solutions, even when the no-expansion is satisfied. (9.3) We have
(U.,.1)2=U2±2huu'+h2[uu"+(u')2]±h3[U'u"+ 2 uU"' - 3! 4 +4h 1(U,,)2 +
-
UrU"
+ - UU nn ] + 4!
3!
=
U2 + h(u2)' + -1h 2(U2)"
+
1 h3(u2)"r 3!
+
1
h4(U2)1"f + ...
4!
so
(unt1)2
=exp(±hax)u2
Similarly, (U.,.1)3 = u3 ± h[U(U2)' + U2U'] + Zh2[U(U2)"/+ 2u' (U2)" + U2U"] + ...
=u3±h(u3)'+2h2(u3)"+
=expl
±h
Iu3.
425
Comments
We conclude that (u,,, 1)' = exp (± h(a/ax))um. Therefore, if f (z) is given by (9.1.8), then (9.1.5) becomes rr
mu(x,t)=µ[exP(h = 4p sinh2
a
a
ax)+exp(-ha)-2]f (u)
(h 8) f(u). 2ax
(9.4) Integrating, and using the boundary conditions yields uxx + u2 + (1 - vo)u = 0 or (ux)2 + 3U3 + (1 - vp)U2 = 0. This leads to u(x - vot) = A sech2 [(A/6) 1/2(X - vot)],
(9.5) g(x - vot) =A sech2 [(A/6) 1/2(X - vot)] + (9.6) Proceeding as in Exercise 9.4, we obtain
where vo = 1 + (2A/3).
where vo = 1 + (2A/3) + 2u,'.
(wx)2 = - (1 - v2)w2 - ZK3w4 PMIf K3 > 0, this yields the desired solution w(x - vot) = A sech [A(K3/2)1"2(x - vt)],
where v2 = 1 + (K3A2/2) which may be compared with Exercise 4. If K3 < 0, then P(w) (above) only has one zero bounding the P(w) 0 region. Hence w cannot be bounded. (9.7) u(x - ct) = u , + (uo - u) sech2 [y(x - Ct + ¢)], when y2 = (u,, - uo)/1262, and
c=u"" +3(uo-u""). (9.8)
lim 7r- li2(3t)- lie
t-0
00
f- F(x) 1 focos (v3 + v x (3t)l/3
J
r-o - F((3t)1j3z)
3
7t 1/2
ll3
1
cos
1im
(- + vzdv dz
fo'o
)
3
= F (0)n112 JT dz Ai(z) = F(0). (9.9) Its motion is due to diffusion, but its shape is retained as it moves because of the special initial spatial arrangement of F(x, 0). Generally, of course, the diffusion leads to a dispersion of the shape. (9.10) From (9.5.2) one readily obtains min (uxx) 2(uo - u"")y2. Thus from the data of on uxx at the maximum of the pulses, and using (9.5.3), the values of u., (etc.) can be determined for each pulse (see Zabusky and Kruskal (1965)). (9.11) See (9.9.9). (9.12) Calculate.
ak' sin o--K0 (K = 1, 2). (9.13) From (9.13.11) ao 0/ar = ak sin TKO and ao 0/as aa = Also ea31/ar = a2 sin Q31; aa31las = a2 1 sin o 31 ; 32/ar = a1 sin a32; &732/as = al 1 sin o-3-2 From o 32 = X31 + oc 0 - o 20 we obtain a1[sin 032 - sin o-+ j =
426
Solitaires: solitons and nonsolitons
-
a2 [sin r 31 - sin 620], or 2a1 sin !(632 - 610) cos 2(632 - 6 ') = 2a2 sin !(a31 -a a) = a2 2sin!(a31 -a'0or a20)cos!(631 + 620). Therefore a1 sin-!(a3'2 2 a1 sin 2(630 - 612) = a2 sin--(630 + 612), or a1 [sin 2630 cos 2612 - cos 2630 x sin 1612 ] = a2 [sin l2 a 30 sin l612 ]. Dividing by cos l630 cos l612 - cos l2a i 2 + cos yields (9.13.13). l630
10
Coupled maps (CM) and cellular automata (CA)
10.1 An overview In the previous chapters we have obtained a number of insights and perspectives concerning nonlinear dynamics with the help of various simplifying models. These models necessarily involve some mathematical paraphernalia, such as differential equations or maps (difference equations) in various dimensions. To put this chapter into context, it is useful to first organize and extend this range of mathematical methods of describing the dynamics of physical systems. We began with models based on ordinary differential equations (ODE), which have one independent variable, which we usually take to be the time, t. Thus these models are characterized by n differentiable functions, x(t)ER", of one continuous ('independent') variable, teR. In the case of models based on difference equations (DE), or maps, we generally still have n real functions, x(t)ER", but the `time' now is taken to be discrete. We denote this by teZ, where Z stands for all real integers. These models, all involving the one independent variable, t, may be usefully referred to as Lagrangian models. We also considered rather briefly another Lagrangian class of models, in the studies
of the influence of finite accuracy of computational and empirical methods. These involved lattice maps (LM), in which both the `time' and the functions x 1(t), ... , x"(t) can only take integer values. Thus, for LM, we have teZ, and x(t)EZ" (Z possibly mod (N)). Z may obviously also refer to any discrete set, not just the integers. Time
X(t)
Models
R
R" R"
ODE DE (maps) LM
Z Z
Z"
These Lagrangian models can be organized in a table. Looking at such a table, there is clearly at least one logical possibility missing, namely tER, x(t)EZ".
Coupled maps and cellular automata
428
There are certain physical situations where such modeling might be useful, as in catastrophe situations. Thus if stresses build up continuously in time, but the location of matter only moves in discontinuous steps (e.g., earthquakes), such modeling might be appropriate. It would appear, however, that the time steps might as well be taken to be discrete, teZ, in which case the LM modeling could be used. Nonetheless, some admixture, teR, x(t)EZ", y(t)ER' might prove to be useful in the future. The most common alternative to the Lagrangian description of physical systems is the Eulerian description, in which both space and time are taken to be independent variables. The classic models of this sort involve fluid dynamics, or field theories, described by partial differential equations (PDE). In this case, both space and time are real and continuous, as are the physical functions, f (x, t). Hence, for PDE, tER, xER", and f (x, t)ERm.
Beyond this point, the mathematical modeling becomes less standard and more imaginative. One of the first examples of this imagination was Turing's cellular model, discussed in Chapter 8. In that case, space was viewed as consisting of discrete cells, k = 1, 2, ... , M, in which `morphogens', Mk(t)c- R', are dynamically governed by
some ODE. From a mathematical viewpoint this is simply a large system of ODE, where the total set of morphogens is M(t) = {Mk(t)ERm, k= 1,.. ., N}ER"`N, and tcR. However, from the physical point of view, the spatial patterns are of primary interest, so the fact that space is being treated discretely is noteworthy. In other words, the physical viewpoint is essentially Eulerian (a space-time description), with a discrete space (Z"; Z' in Turing's model).
Time (t)
Space (x)
f (x, t)
Model
R R
R"
R'" R'"
PDE CDE
Z"
Thus, from the Eulerian viewpoint, we can represent these modelings as shown in the Table. Here CDE refers to cellular differential equations, where each cell has m functions (e.g., morphogens). Once again, this is a physical rather than a mathematical distinction (where CDE = ODE), but we shall adopt the physical viewpoint. Comparing the last table with the previous table, it is clear that there are a number
of additional mathematical methods for modeling physical systems. Indeed one example was introduced in Exercise 6.1, where Turing's idea was modified to incorporate map dynamics for the morphogens in the cells, rather than ODE dynamics. In this case the time steps are discrete, so tEZ. We will refer to such models as coupled (or cellular) maps (CM), and consider further examples of these in this chapter.
429
An overview
A final class of models which we will examine, uses functions, f (x, t), which also take only discrete values, as in the above case of lattice maps. In this case tEZ, xeZ", and also f (x, t)EZm, where Z may also represent a finite set of integers. These models,
invented by von Neumann and Ulam at roughly the same time as Turing's CDE conception, are called cellular automata (CA). They have many novel features, which we will explore shortly. The table completes our `Eulerian table'. It does not exhaust
all of the logical possibilities, but it does represent the principal Eulerian models which have been used to date. Time R R
Z Z
Space
f (x, t)
Model
R"
R'° R'° R'°
PDE CDE CM
Z'°
CA
Z" Z" Z"
The basic problem of Eulerian dynamics is to `relate' the spatially localized dynamic rules and the consequent global space-time dynamic patterns (Fig. 10.1).
Fig. 10.1
We have already seen the surprising example of the Turing instability, in which local
`diffusion' between damped cellular systems can lead to `global' oscillations (a la Smale) or stationary spatial patterns. We are presently interested in simplifying the dynamics from a flow (tER) to a map (teZ), either with continuous cellular functions (f (x, t)ER") or discrete-valued functions (f (x, t)EZ"). As noted above, these dynamic systems will be called coupled (cellular) maps (CM) and cellular automata (CA) respectively.
Exercise 10.1 It will be noted that the term `pattern' is beginning to enter the discussion more frequently. Why did we not consider `patterns' in Lagrangian models? Or did we? What do Eulerian `patterns' mean from a Lagrangian point of view?
430
Coupled maps and cellular automata
It should be noted that the last figure represents an example from a broad program of synthesis within the sciences (and, indeed, in even broader social contexts). It is the 'reverse-direction' of the frequent research programs of analysis, which involves the separation or division of some compound system into constituents which can be studied individually, and in more limited interactions (e.g., two-body interactions). This point of view is associated with the philosophical emphasis on reductions, which has dominated much of science since the time of Newton. Indeed many physicists have managed to convince others that it is only through this process that we obtain `truly fundamental' knowledge about nature. Synthesis, on the other hand, involves
combining separate elements to form a whole which has some coherent structure and behavior; an emergent behavior, which may be quite unanticipated from the
behavior of its constituents. More importantly, not only may the behavior be unanticipated, the behavior usually (generically) cannot be predicted by any computations based on `fundamental theories', because there is no short-cut method to obtain the answers; it is generally necessary to `run the dynamics' to see what happens. With `fundamental theories' this process is crippled and disabled by sensitivities to numerical errors, the unavoidable influences of the environment, and the unknown precise initial
conditions of their immense number of variables. A significant advantage of CA dynamics is that it avoids all numerical errors by dealing only with finite sets of integers. Because of this, these theories are able to explore the emergent behavior of complex systems, while avoiding the difficulties which are inherent in THE INFINITE
of mathematics (see Sections 4.11, 6.14, and 10.16). We will see that, even so, the emergent behaviors frequently cannot be predicted from the local dynamic rules, except by following the dynamics step by step. One of the great interests, of course, is to understand better which emergent behaviors can be predicted from the local rules, and how this may be done. This program has just begun (e.g., Aiazwa and Nishikawa, 1986; Jen, 1986; Langton, 1986; Oono and Yeung, 1987). At the risk of trivializing the distinction between analysis and synthesis, the rather simple-minded Fig. 10.2 is presented, in order to emphasize this important issue; moreover, it is worth a moment's thought to clearly formulate in what sense this example is relatively trivial, compared to other complex systems around us. Even this simple example illustrates an important aspect, already noted in Section 8.14 concerning the dynamics of living systems. Namely, the processes of discovering the emergent behavior of a complex system involve the synthesis of elements which themselves have a considerable degree of complexity. In the case (Fig. 10.2) of an old-fashioned pocket watch, the elements (springs, gears, screws, etc.) are fairly simple,
but they certainly could be still analyzed further in terms of solid structures, atomic
forces, and so forth. To do this, however, would eliminate the possibility of this particular synthesis, which takes the illustrated parts as the basic elements of the dynamics, with known types of interactions. It would be a hopeless exercise to try
Some coupled maps Fig. 10.2
431
SYNTHEs
EMERGENT BEHAVIOR
REDUCED
INTERACTIONS
to determine the behavior of this watch from more `fundamental' theories; the watch would never have been made, based on such theories! Understanding that synthesis generally begins with elements of some complexity, we recognize that a `judicious' amount of analysis of a 'macro-complex' system, to discover some set of complex elements (components) and their interactions, which accounts deterministically for some limited emergent behavior, forms one 'circle-of-research'. When dealing with complex systems, it is important to emphasize that there will never exist one `fundamental', all-encompassing circle-of-research. For, while it may be possible to carry out further reductionistic analyses on a system (e.g., the atomic content of the watch parts), the new synthesis process becomes impossible both empirically and theoretically. Instead, many such 'theory-circles' can be equally fundamental, applying to different complex systems. We have seen a few examples in Section 8.14, and will explore more in this chapter, before returning to this basic point in Section 10.17. Historically, CM have only recently attracted much attention, whereas CA have existed since around 1950. From a physical point of view, however, it is much easier to begin these studies with simple examples of CM and recent applications of CA, rather than the historic CA dynamics. We will therefore proceed in this somewhat reverse historical order.
10.2 Some coupled maps (CM) In this section we will first consider a finite, one-dimensional collection of identical cells, i = 1, ... , N. Each cell has a single dynamic variable, whose dynamics is
432
Coupled maps and cellular automata
given by some map Ci
I,-, N)
(10.2.1)
which depends only on its cell (i) and its two nearest neighbors (i ± 1). Generalizations
to more than one variable per cell, and longer ranges of interactions between cells, are obvious possibilities, but (10.2.1) is already dynamically rich. When F(x, y, z) is a continuous function of its variables, we call (10.2.1) a coupled (or cellular) map (CM) system.
Many models are based on the logistic map dynamics occurring in each cell, which are then coupled together in some fashion to allow the `populations' to migrate from one cell to another. We will therefore introduce the notation for the logistic map
f(x)-cX(1 -X)
(xE[0, 1]),
(10.2.2)
where 0 < c < 4 is the constant control parameter. Note that all cells are assumed to be identical (neither F(x, y, z), (10.2.1), nor c depends on i). Hence we are considering here only homogeneous CM (see below).
There are many ways that the cells can be coupled together in (10.2.1). Early examples (Kaneko, 1984; Waller and Kaprall, 1984) used Turing's diffusive coupling, and took (10.2.1) to be (10.2.3)
where D is the `diffusion' constant. This is a generalization of Exercise 6.1 (for two cells) to the case of many cells (i = 1, ... , N). As in Turing's treatment, periodic boundary conditions are frequently assumed (a cellular `necklace') 1) =
(10.2.4)
Since the last factor in (10.2.3) represents the average of the `population' flux to cell i from the adjoining cells, it is reasonable to restrict D to the range i >, D >, 0. As noted by Yamada and Fujisaka (1983), the map (10.2.3) can give unbounded solutions for i >, D > 0. If I x (i) I > 1 for any n, then f (x,,) is even larger in magnitude, and this is generally not controlled by the linear diffusion term. 1) Exercise 10.2 Show that all solutions of (10.2.3) are bounded from above when D < c'12 - (c/2), where f (x) and c are defined by (10.2.2). Try to find a condition for the lower bound, 0 <
The fact that (10.2.3) yields unbounded solutions for reasonable values of D shows that the modeling did not take proper account of the effect of the diffusing `population' (or `concentration') on the limited resources (food or chemicals in the ecological or
Some coupled maps
433
chemistry contexts). This shortcoming is often ignored, but in so doing we lose touch, and to an unknown degree, with our physical intuition, and possibly with `physical reality'. `Reality' can be recovered by separating the diffusive processes from the reproductive (f-dynamic) process. In the first step (say the reproductive step) the new population in each cell is taken to be (i = 1, ... , N).
x' (i) = f (xn(i))
(10.2.5a)
This is followed by the diffusion between cells, so that the final `next generation' is xn+ I(i) =
D(x;,(i - 1) + x;,(i + 1) -
(10.2.5b)
Combining these, we obtain the CM equations xn+I(i)=.f(xn(i))+D[f(xn(i- 1))+f(xn(i+ 1))-2.f(xn(i))7,
(10.2.5)
where i = 1,..., N. Yamada and Fujisaka (1984) have also considered a long-range coupling N
xn+ I (i) = 9(xn(i)) + D
,
j=1
[9(xn(j)) - 9(xn(I))l,
whose physical basis differs from above.
Exercise 10.3 Assume that 0 < f (x) < 1 for any 0 < x < 1. Show that for this f (x), (10.2.5) yields 0 < xn + 1(i) < 1 for any i >, D 3 0. Hence if initially all 1 > xo(i) >, 0, then all solutions remain bounded if >, D >, 0.
i
Another form of coupling, which is generally not diffusive in character is (Keeler and Farmer, 1986) 1
xn+1(i)=2r+1;
i+r
rf(xn(j)),
(10.2.6)
where r is the range of the `coupling'. In this case the value of xn+1(i) is the average-dynamics of cell i, and the r adjacent cells on each side. Clearly, if 0 < f (x) <, 1,
then the same inequalities hold for all xn + 1(i), and solutions remain bounded, as in Exercise 10.3. Indeed if r = 1 in (10.2.6), it is the same as (10.2.5) for D = 3. However, these models generally differ (r > 1), and the physical rationale for the averaging in (10.2.6) must be based on a very different type of (more direct) coupling than produced by diffusion. All of the couplings (10.2.3), (10.2.5), and (10.2.6), allow for homogeneous dynamics, xn(i) = xn(j) (all i, j, and n), as a special case. This is frequently referred to as in-phase dynamics, which is the same as decoupled maps. Also, all of these coupled maps have
434
Coupled maps and cellular automata
three control parameters or
(c, D, N)
(c, r, N)
and we can anticipate that these permit a rich variety of dynamics and bifurcation properties. Much of this possible wealth of dynamics/bifurcations has not yet been investigated, and the present account is likewise highly abbreviated. Many interesting possibilities exist, some of which we will outline here. Let us first follow Turing and consider the stability of homogeneous states. We can first transcribe the CM equations into maps of normal modes, by setting N
Y
(10.2.7)
M=1
Using the orthogonality relationship N
E exp[27rij(m - k)/N] = Nbkm,
j=1
where bkm is the Kronecker delta function bkm=O (if k
m);
bkk = 1,
the mth normal mode is given by
(m) = N
N
,E1
exp (- 21rijm/N).
(10.2.8)
The map (m)
1(m) of course depends on the CM (10.2.1). The mode map is obtained by substituting (10.2.7) into (10.2.1), multiplying by exp(- 27cijk/N), and summing over j = 1, ... , N. In the case of (10.2.3) this yields the normal mode map (Waller and Kapral, 1984) N-1
(m) = [c - 4D sin2(irm/N) ]
(m) - c E (k) (m - k)
(10.2.9)
k=0
whereas, for (10.2.5) we obtain the map N-1
(m) = c[1 - 4D sin2(nm/N)] l; (m) - Z
k=0
k)
.
(10.2.10)
In (10.2.9) the diffusion does not couple the normal modes, whereas it does couple them in (10.2.10). There are two possible homogeneous solutions of the CM, both of which correspond to fixed points of f (x),
x=x*=1-(1/c) (ifc>1),
or
x=x**=0.
(10.2.11)
435
Some coupled maps
The stability of the homogeneous solution x"(i) = x* can be studied by substituting
x"(i) = x* + bxji)
(10.2.12)
"(m) = X*8mo + 8"(m),
(10.2.13)
into (10.2.8), yielding where N
1N j=1 Y Sx"(j) exp(- 2nijm/N).
(10.2.14)
Substituting (10.2.13) into (10.2.10), for example, and keeping only the linear terms in we obtain (from the terms k = 0 and k = m)
5 ,,+i(m)=c[1 -4Dsin2(nm/N)](1
(10.2.15)
Therefore, if boo(m) is the initial perturbation, the solution is 5
(m) = {c(1 - 2x*)[l - 4D sin2(nm/N)] }50(m).
(10.2.16)
The analogous result for (10.2.9) is (Waller and Kapral, 1984) [c(1 - 2x*) - 4D sin2(nm/N)]" 5 0(m).
(10.2.17)
Identical results are obtained for perturbations about the fixed point x** = 0, when x* is replaced by x** = 0. A distinction in these two cases comes from the fact that (10.2.12) now reads x"(i) = X** + bxji) = (Sx"(i),
and hence all bxji) > 0. This means that it is not possible to initially excite only one normal mode, (10.2.8). However, that does not influence the stability of those modes which are excited. Hence the equations (10.2.16) and (10.2.17) likewise hold with x* +x**. We see, from both (10.2.16) and (10.2.17), that the most unstable mode is m = 0, and that: x"(i) = x* is stable if 1 <, c x"(i) = 0 is stable if c
3 1.
which is identical with the uncoupled maps. In other words, the diffusion does not introduce an instability into the homogeneous system. The only instabilities are those which are present in the uncoupled cell dynamics. This is totally different from Turing's
result, where the cell dynamics is governed by ODE. In that case, cells which only have damped motion to a fixed point, can yield unstable perturbations away from homogeneous states when they are diffusively coupled. We thus have the interesting
Coupled maps and cellular automata
436
result:
The homogeneous states of the CM (10.2.3) and (10.2.5) are stable, if the uncoupled cells are stable, whereas homogeneous states of coupled ODE can be unstable, even if the uncoupled cells damp to a fixed point (two or more morphogens).
The map dynamics is similar to a Poincare map of a damped oscillator. What (10.2.18) indicates is that a diffusive coupling of such damped dynamics, only `at' the surface of sections, is insufficient to destabilize the homogeneous systems. It remains to be determined how general is the result (10.2.18) for other CM, (10.2.1). Hence, for at least the CM (10.2.3) and (10.2.5), it is necessary to consider cells with periodic dynamics (c > 3) to obtain interesting diffusive effects. If x is such a
periodic solution (x+ P = xn), and we consider the perturbation of a periodic homogeneous solution, xn(i) = x* + bx(i)
(xn+p = xn ),
(10.2.19)
then (10.2.15) is replaced by
k n+ 1(M) = c[1 - 4D sin2(nm/N)] (1 - 2xnn*)d n(m).
(10.2.20)
Since (1 - 2x*) returns to the same value after p iterations, we can formally write the solution of (10.2.20) at the iterates n = kp (k = 0, 1,...) as S kp(m) = [1 - 4D sin2(1Cm/N)]'Prc" LL
fl
j=
(1 - 2x7)Jbk o(m).
1
We see again that the mode m = 0 is the most unstable, and it is unstable only when P
cPfl(1-2x*)>1. j=1
According to (4.2.11) this is precisely when the period-p cycle of the uncoupled map
becomes unstable. Hence, diffusion again plays no destabilizing role for the CM (10.2.5).
We see that in the present case of coupled one-dimensional maps, the diffusive coupling appears to have the stabilizing effect, which we might intuitively expect. To examine this further, we will consider several large inhomogeneous initial conditions,
x0(j)=A0 +A1cos(mmcj/N)
(0<(A0±A1)<1).
(10.2.21)
If 1 +6 1/2 > c > 1 the uncoupled cells tend to period-two motion. In this case, if D = 0, we expect that the initial state (10.2.2 1) will asymptotically (n -+ oo) yield a temporal period-two pattern. The spatial structure of this pattern depends on which
Some coupled maps Fig. 10.3
437
Xn (j)
I I
I
I
I
9
10
11
12
I
13
i
14
i
15
initial conditions, (10.2.21), produce motion with a common phase. Thus the space-time
structure is not just a question. of periods, but also of phase. This is illustrated in Fig. 10.3, where adjacent values are connected by straight lines for easy visualization.
Borrowing from Winfree (1987), we will call the initial states of f (x) which are asymptotically in phase, isochronal states. When c is such that f (x) has a period-1 attractor, the isochronal interval of initial states, Ij = {bj_ 1 < x < M, all tend to the same fixed point of f `(x). Here (bj - 1, b) are respectively the (smallest, largest) values of x which contain no other contiguous isochronal states. Thus points in Ii and 1;+ 1
tend to distinct fixed points of f (x). If, for example c yields a period-two attractor for f (x), then I; and Ii 12 would contain isochronal states, even though the intervals are not contiguous. Of course, when D :0, the identification of `isochronal states' of the coupled system involves at least neighborhood considerations, and may be a rather useless concept, particularly if c > c.. However, for small values of D, and c < c., this concept clarifies some aspects of the resulting space-time patterns.
Exercise 10.4 In the period-two region of c, how many isochronal regions, Ij, are there within the interval 0 < x < 1? How are they distributed? Onto what point do all the boundary points, b;, map? The dependence of the asymptotic pattern on the initial distribution is illustrated in Fig. 10.4, when N = 50, c = 3.4 (period-two), and D = 0. The left figures is for a smaller amplitude (A0 = 0.3, A 1 = 0.1) than the right (A0 = 0.25, A 1 = 0.24). Only one iteration pattern is illustrated (see the last figure). We see that more isochronal Fig. 10.4
large t
Coupled maps and cellular automata
438
regions are present in the latter case. The straight line is x = 2, and the smooth curve is the initial configuration (10.2.21).
When D # 0 we might expect that these spatial patterns become smoother. Fig. 10.5 illustrates the change in the larger amplitude (A0 = 0.25, A, = 0.24) when D = 0.1 (left figure) and D = 3 . This is the same as (10.2.6), when r = 1. Fig. 10.5
D=3
D = 0.1
We see that, for small D, the isochronal zones of f (x) continue to determine the spatial pattern, but not for larger D. Increasing to D = 2 does not substantially change this pattern. Exercise 10.5 If the initial amplitude is doubled (A0 = 0.5, A, = 0.48), the same isochronal regions of f (x) occur in two spatial locations (f (x) is symmetric about x= Do you expect the above spatial pattern to be correspondingly reduced in scale by a factor of 2, if D = 0, 0.1, or 3? Try some computations, and determine the pattern for D = 2. 2).
If c = 3.5 (period-four of f (x)), the isochronal regions of f (x) produce patterns at four values, as illustrated in Fig. 10.6 for D = 0. Here four iterations are superimposed, and only the value x = is also represented. 2
Fig. 10.6
D=0
D=0.1
Some coupled maps
439
It is now seen that when D is increased to D = 0.1, not only has the spatial pattern been altered, but the temporal periodicity is no longer clearly (i.e., empirically) the same in all cells. Two groups of cells appear to have period-three (within graphical accuracy), whereas the remaining regions have period-four. When D = (Fig. 10.7), 3
Fig. 10.7
there are clearly period-two and period-four regions. As we will also see in the next section, multiple-period regions are not uncommon in these systems. If we further increase D, do these spatial variations continue to smooth out? If we take D = i we obtain an unexpected spatial feature, as illustrated in Fig. 10.8. The spatial 'saw-tooth' pattern has a period-four behavior. This is therefore the first of Fig. 10.8
our examples which indicates that diffusion can produce a spatial instability. Computations indicate that this instability sets in when D >, 0.492, but we may reasonably suspect that this is initial-state-dependent. Indeed it appears to be a rather singular phenomenon, when c < ca.. You may wish to explore this point along the lines of Exercise 10.5.
In contrast to the smooth initial conditions (10.2.21), Fig. 10.9 illustrates how diffusion (D =
3)
smooths out random initial fluctuations. The case c = 3.4 (left) settles
to two smooth period-two regions, whereas c = 3.5 again tends to period-two and period-four regions.
To remove the strong dependence on the initial state, it appears to be necessary (but perhaps not sufficient) to take c > cam. Consider first the case c = 3.6, so the
uncoupled cells are `chaotic' (they have a small positive Lyapunov exponent; Chapter 4, Section 5). Again take AO = 0.5, A 1 = 0.45 in (10.2.21), and increase D. Fig. 10.10 shows 30 consecutive iterations in the asymptotic (large n) region.
When D = 0.2 (top figure) we find that there are four spatial regions with very
440
Coupled maps and cellular automata
Fig. 10.9
different time/phase regions. There are two period-four regions with different phases, period-eight regions, and several less obvious regions. One region, even if periodic, may be fairly called (empirically) semi-periodic turbulence, since, if it is periodic, the period is beyond observational memory (who can remember more than 32 complicated configurations?). It is not totally `turbulent' because intermediate amplitudes never occur in the region and hence 'semi-periodic turbulence' appears to be a fair description (as an extension of semi-periodic chaos). When D = (middle figure) the periods have changed, but there appears to remain an apparently3semiperiodic turbulent region.
When D = i (bottom figure) we again see a spatial instability, as in a previous example. Clearly, too much `diffusion' in coupled map systems can lead to more, rather than less, spatial irregularity. For the remaining examples, we will take D = 3 (a `moderate' value), and explore a few cases where c > cam, always starting from the initial state A0 = 0.5, A, = 0.45, and N = 50 in (10.2.21). We might expect that the chaotic dynamics of the uncoupled
cells is most difficult to influence when c >, 3.7, where there is no semi-periodic dynamics.
When c = 3.7, however, the asymptotic state (n > 300) is found to be very regular, as Fig. 10.11 indicates (despite its appearance, there are no fixed points). Aside from the two different regions of period-four the pattern is rather uninteresting.
Exercise 10.6 But wait! Doesn't such a `uninteresting' result raise some interesting questions? Even if we keep c = 3.7, D = 3, what does such a figure suggest needs to be investigated?
Some coupled maps Fig. 10.10
D = 0.2
D=2
Fig. 10.11
441
442
Coupled maps and cellular automata
Exercise 10.7 What modification of the c = 3.7 system will certainly make the dynamics more interesting? What practical difficulty will you encounter in computing such cases? These results show that all (c, D, N) systems do not yield interesting spatial-temporal patterns. One of the most interesting of such patterns in hydrodynamic systems (e.g.,
Fig. 10.12
n = 800
n = 950
n = 1400
444
Coupled maps and cellular automata
the Rayleigh-Benard system) involves the phenomenon of spatial-temporal intermittency. In some `turbulent' conditions it is not uncommon to see some local regions of order, surrounded by a `turbulent sea'. These islands of order tend to come and go in various regions of space as time progresses. In other words, the local order is intermittent, as is the turbulence. It is of some interest to see that the present system can also exhibit spatial-temporal intermittency. We take c = 3.83 and D =L3. While c = 3.82 is the period-three window of f (x), the present cells are not uncoupled, so this particular value is not necessarily unique. Each part of the Fig. 10.12 shows 30 iterations, after approximately n iterations. The values of n are selected simply to illustrate interesting differences in patterns; specifically, the intermittent effect. When n = 800, many regions (P) show simple period-four motion, whereas other
(T) regions are very `turbulent' (simply a qualitative characterization - nothing profound). By n = 950, the T-regions have begun to migrate, and by n = 1400, no P-regions are present. A few regions are T-regions but many are s-regions (with semi-periodic cells). By n = 1850 nearly all regions are T-regions, but shortly afterward (n = 1900) most are either P or s-regions. At n = 2400 all have again become T-regions,
only to return to P-, s-, and T-regions at n = 2800. Scanning these figures, it is also clear that the P-regions and T-regions wander through
the necklace of cells, or may simply disappear. Hence this dynamics appears to have some of the features of spatial-temporal intermittency in hydrodynamics. Obviously this simple result gives us no deep understanding of such systems. At this point it is merely an interesting result.
This brief introduction to CM dynamics clearly leaves most of the interesting research for the future. In addition to studying other maps, an important area of research is inhomogeneous coupled maps (1CM). After all, the world is not homogeneous, and many of the important dynamics depend crucially on this inhomogeneity (such as life itself!). Even for only two cells it is interesting to explore how ordered motion in one cell can influence chaos in another (Fig. 10.13). Exercise 10.8 Replace f (x) in (10.2.5) by f;(x (i)), where f;(x) = c;x(1 - x). Take N = 2,
c, = 4, and (for example) c2 = 3.4. As D is increased, who wins the `battle', chaos or
Fig. 10.13
Coupled lattice maps
445
order? That is, does order entrain chaos? What if D - D. (i = 1, 2), and D2 - 0? Does entrainment occur?
One of the drastic restrictions of this section is to CM with only one variable in each cell, xn(i)ER. It is likely that the coupling of cells, each with several variables (xn(i)ERm; m = number of morphogens) will prove to be more important in the future. As we have seen, the CM involving a single variable, does not generate the Turing-like instabilities, which are of great interest. We must, however, move on!
10.3 Coupled lattice maps (CLM = CA) Before considering general cellular automata (CA) in the following sections, we will first explore several special examples, which are perhaps more readily `understandable', based on the discussions in the last section. Recall, from Section 6.1, that the dynamics is now being characterized by discrete time (teZ), discrete space (cells) and, finally, the function values at the cells are also discrete (a 'state-lattice'). In the present section we will consider a few special discrete map functions, which can be used to discrete the CM dynamics discussed in the last section. Since the maps
will now occur on a lattice, it may be useful to refer to them as coupled lattice maps (CLM).
As in (10.2.5), we consider first a coupled linear map (Oono and Kohmoto, 1985) x;,(i) = xn(i) + (D/2)[xn(i - 1) + xn(i + 1) - 2xn(i)]
(10.3.1)
can have any value, because of the multiplicative factor D. To obtain the next step on the lattice we therefore need a map from We write this as where 1 >, D > 0. The variables xn(i) take values on some lattice, but
xn+ 1(i) = F(xn(i)),
(10.3.2)
where F(x) takes only the lattice values. The dynamics on the lattice is given by (10.3.1) combined with (10.3.2), and periodic boundary conditions xn(N + 1) = xn(1), xn(0) = xn(N) (the cells are on a `necklace'). If D = 0, (10.3.1) is the identity map and xn+i(i) = F(xn(i))
(D = 0),
(10.3.3)
so F(x) defines the lattice map of the uncoupled cells. We will now consider several specific examples.
We will take the cell states, xn(i), to occur on a set of k points ('lattice'), which we can take to be positive, xn(i)E{sj};
0<,sj,<M (j=1,...,k)
(10.3.4)
so that F(x') = s,,
if b; _ 1 < x' < b;.
(10.3.5)
446
Coupled maps and cellular automata
The x' boundaries, b;, satisfy bk = 00
bo = 0,
with the remaining b; taking any values, bj _ 1 < b,, greater than zero and less than M. It might be noted that, because bk = oo, this system is defined for any D > 0, even though we will take 1 >, D > 0. Fig. 10.14
F(x') FT-741
M=S3 S4 S1
S2 1
b1
I
b2
I
b3
x'
As can be seen from Fig. 10.14, the variety of possible maps defined by (10.3.5) is too large to be easily surveyed, so we will consider only two examples, by way of illustration. The first example is due to Oono and Kohmoto (1985), who explored the nature of chemical turbulence. The physical interest is to better understand how cells which have an intrinsic oscillatory dynamics (e.g., the Belousov-Zhabotinsky chemical oscillations) will behave when they are diffusively coupled. The specific
interest is to try to characterize the type of space-time patterns which result from such coupling, and in particular to explore the meaning of `turbulence'. The dynamics of the uncoupled cells, (10.3.3), is taken to be a simple three-step oscillator. This is accomplished by taking (10.3.5) to be M
F(x') = 0 1
if if if
0<x'<0.5 0.5<x'<1.5 1.5<x'
(10.3.6)
as illustrated in the Fig. 10.15. They allowed M to have any real value, but the lattice consists of only three states (k = 3). Indeed, if 1.5 > M > 0.5, the uncoupled cell only oscillates between 0 and 1. In general, M indicates the amplitude of the oscillation (e.g., the maximum chemical concentration), which is usually large in experiments. We can see that the coupled dynamics will only change character (bifurcations will occur) when M and D are functionally related. Bifurcations in the dynamics can only occur for those values of M and D which cause some x' in (10.3.1) to lie on one of {s;} = (0,1,M). the nonzero boundaries of (10.3.6), namely x'=0.5 or 1.5, when the
Coupled lattice maps
447
Fig. 10.15 0
M X,
1.0
I
I
I
0.5
x'
1.5
Thus bifurcations can only occur where (M, D) satisfy b = (0.5 or 1.5) = s;(1 - D) + (D/2)(s, + s,).
(10.3.7)
Of these many possibilities, Oono and Kohmoto numerically found the dynamic behavior of the coupled system only changed its character when the parameters (M, D) cross five bifurcation lines, illustrated in Fig. 10.16. The reason for this limited number of `emergent bifurcations' was subsequently established by Oono and Yeung (1987).
Exercise 10.9 Determine the values of Si, sj, s1, and b in (10.3.7) which produce the indicated bifurcation lines, D vs M.
Fig. 10.16
3
D=
D= ZMD= M+ M
M 3
2 1
0
0.2
0.6
0.4
D
1
448
Coupled maps and cellular automata
Exercise 10.10 Given a set of states {s;} and boundaries {bj}, show that the dynamic behavior of (10.3.1), (10.3.2), (10.3.4), and (10.3.5) can only bifurcate at discrete values of D. Determine this set of (possible!) bifurcation values {D,}, (1 = 1,..., L). What is L? The actual bifurcations are less frequent (Fig. 10.16), and difficult to predict. In the region X of Fig. 10.16 the dynamics is complex because M is relatively small, which is not usually of experimental interest. The stronger oscillations exhibit three different types of space-time patterns, in regions (3, T, S, 3'). In regions (3, 3'), all cells develop period-three dynamics. Not surprisingly these regions occur when D is small or very large. These space-time patterns are illustrated in Fig. 10.17 (M = 4, D = 0.2 Fig. 10.17
D=0.2
D=0.8
and 0.8). There are 60 cells horizontally. Each line corresponds to a single iteration, with the base line, x = 0, stepped upward as n is increased. As in Section 10.2, the adjacent values x (i), 1) are connected by a line for easy visualization. Only ten iterations are shown because period-three dynamics has been established in both cases, starting from a random initial state. This type of graphics program is easy to construct and requires no elaborate software. The region S contains traveling solitary disturbances, as illustrated in Fig. 10.18 (D = 0.4). After a complex initial stage, the remaining 'domain-wall' disturbances pass through each other, much like solitons. Note that the coupled logistic map dynamics in the last section never yielded such traveling disturbances. Fig. 10.18 D = 0.4.
Coupled lattice maps
449
Fig. 10.19 D =0.3. -
..-..mot
The region labeled T has a `turbulent' character, as illustrated in the Fig. 10.19
(D=0.3). Any scientist looking at such a dynamic pattern would probably be comfortable with the description `turbulent' system. However, a precise definition of `turbulence', which is sufficiently general to distinguish between those general CA which do or do not generate `turbulence' proves to be a difficult problem (Oono and Yeung, 1987).
Exercise 10.11 All dynamics of a CLM with N cells, each of which can have k values, must be periodic. What is the largest possible period (ergodic motion)? Compute this
when k=3, N=50. We can reasonably require that (Eulerian) turbulent systems have properties similar to (Lagrangian) chaotic systems, such as sensitivity to initial conditions, and some positive (suitable) dynamic entropy. Moreover we can require other conditions, such
as the rapid decay of dynamic correlations in both space and time. That is already adequate to distinguish the above T dynamics from the remaining (S, 3j) dynamics. The difficulty is not due to such reasonable necessary properties of `turbulence', but whether these properties are sufficient to distinguish turbulent systems from other systems which, for other reasons, we do not consider to be `really turbulent'. We will delay a discussion of this problem until we have defined general CA dynamics in the next section. To explore some rather different types of space-time dynamics, we return to (10.3.1) and (10.3.5), and consider the following family of CLM:
O<sj<, 1;
bj=j/k
(j=0,...,k-1),
bk=00.
(10.3.8)
Now the maximum state value is taken to be 1, and the boundaries are uniformly spaced below 1. We will moreover consider only systems satisfying
sl,sk
si>bi
(j=2,...,k-1).
(10.3.9)
450
Coupled maps and cellular automata
This means that the uncoupled cell dynamics, (10.3.3), causes all cells to end up in the state s1, in no more than (k - 1) steps. In other words, the state s1 is a global attractor of the uncoupled cells. This is the opposite extreme from the system (10.3.6), with its periodic cellular dynamics. The present cells `die' if they are not coupled (recall Smale's theorem for coupled ODE). We are interested here in exploring what type of `life patterns' can be brought about by diffusively coupling such cells. Fig. 10.20 Ik 33
I
I
0.5
I
1.0
The first example is illustrated in Fig. 10.20. s1 = 0.3, b1 = 0.33,
s2 = 0.9,
b2=0.66
s3 = 0.2;
(10.3.10)
There are 16 values of 1 > D > 0 where the dynamic rules change (Exercise 10.10), and many of these lead to different space-time patterns (hence are bifurcation points). In what follows, the values of D are rounded off for simplicity. For D < 0. 1, all patterns become homogeneous, at x = s1. For 0.10 < D < 0.33 there are asymptotic patterns consisting of `sawtooth' groups,
all propagating parallel to each other. These groups can be spatially represented as either the left-traveling (s1)(s2,s3)(s1) or right-traveling (s1)(s3,s2)(s1) where the parentheses signifies any number of sequential repeats of the group (consistent, of course, with the necklace size, which is N = 50 in these examples). Thus we have propagating solitary disturbances, but not `solitons'. These features are illustrated in Fig. 10.21. For 0.34 > D > 0.33 we find the first example of many, where the system has spatial
regions with different periods. Fig. 10.22 illustrates regions with period-two and period-four. One or both of these regions are the attractors of all initial states. A
Fig. 10.21
Coupled lattice maps
451
Fig. 10.22
Fig. 10.23
more interesting example of these multiple-periodic patterns is when D = 0.805. As
Fig. 10.23 shows, there are patterns with two different period-one regions, one period-two region, and one period-three region all coexisting in this simple system.
Exercise 10.12 Characterize these local periodic patterns in terms of (s,, s2, s3) of (10.3.10). Write a computer program and verify these results. For all 0.69 > D > 0.34 `turbulence' appears to be the most characteristic dynamics. What is rather surprising is that this system can have both a periodic-pattern basin of attraction, and a 'turbulent'-pattern basin of attraction for the same value of D. Thus, for 0.6 > D > 0.4, different (but `nearby') initial states can yield simple period-four
regions (of several types), or else patterns with very long periods ('turbulence' - see Exercise 10.11). Specifically, the initial states x0(j) = 0.3 for all j, except IA: IB:
x0(5) = x0(25) = x0(32) = 0.9 IA and x0(10) = 0.9.
(10.3.11)
The upper part of the Fig. 10.24 shows that the initial state (IA) has gone to a period-four configuration after only nine iterations. The lower part of the figure shows
how the (only) change from x0(10) = 0.3 to x0(10) = 0.9 causes the system to tend to a `turbulent' state. There is, of course, nothing unique about the initial states (IA) and (IB) in (10.3.11), but the problem of predicting which initial states lead to the different asymptotic states is not easy to solve. Even systems with these simple dynamic rules can have `turbulent' states which have different characteristics, even though it is not simple to quantify these differences.
Fig. 10.25 shows some configurations when D = 0.35 and 0.85. They are clearly `different'.
Coupled maps and cellular automata
452 Fig. 10.24
Fig. 10.25
D = 0.35
D = 0.85
Exercise 10.13 The `turbulence' at D = 0.85 bifurcates from the multiple-periodic regions considered in Exercise 10.12. Determine the bifurcation value of 0.85 >, D 0.805. What `bifurcated'?
There are other lessons to be learned from the simple system (10.3.10), which you
Coupled lattice maps
453
may wish to explore. We will, however, consider a final example which exhibits space-time patterns of an entirely different character. This is the case when each cell has four states (k = 4). Specifically we take
s1=0.2,
s2=0.6,
s3=0.9,
b1 = 0.25,
b2 = 0.50,
b3 = 0.75,
s4= 0.1
(10.3.12)
as illustrated in Fig. 10.26. There are now roughly 35 values of D which change the 0.2. dynamic rules, but for D < 0.14 the system goes to the fixed point Fig. 10.26
F(x') 1.0-
Ik 44
1I
0.5
1.0
x'
Perhaps the most conspicuous effect produced by the added freedom of one more state in the cells, is that it frequently takes many iterations for the system to reach its asymptotic space-time pattern. Thus, for D = 0.15, the asymptotic state apparently only contains regions with period-eight and several with period-four but it may take several hundred iterations (when N = 50), for the system to reach this state. When the space-time patterns become involved, it may take many thousands of iterations to reach `asymptotic states' - which, indeed, may not be simple to recognize, much less to define. Thus the system (10.3.12) already is sufficiently complicated to challenge our inventiveness. We briefly consider some examples.
Fig. 10.27 illustrates a space-time pattern at D = 0.215. The necklace of cells has a large region with period-four. Traveling through this region (at least for the recorded
period) is a complicated nonsymmetric sawtooth pattern, inside of which is a (temporarily) fixed period-five region! The velocity of the moving pattern is i cell per iteration! This is obviously only a temporary configuration, but is typical of the juxtaposition of complex and simple patterns which this system can develop. Even short-period patterns may have a considerable spatial complexity, as illustrated in Fig. 10.28 for D = 0.205. The complexity of such space-time patterns is best appreciated
by carrying out a synergetic analysis with your computer. Exercise 10.14 Not all patterns are necessarily complex. Explore the dynamics near D = 0.26. Are there solitaires or solitons? Are you sure you found the asymptotic behavior?
454
Coupled maps and cellular automata
Fig. 10.27
mow'/`\J,\
"
.MA,
Fig. 10.28 D = 0.205.
/\;\,-\i\ / i\ J\ This introduction to a special class of cellular automata gives some suggestion of
the complexity which such systems can demonstrate. We now turn to the more general, and therefore more abstract, historical development of cellular automata. Exercise 10.15 For those interested in exploring different forms of dynamic patterns (e.g., oscillating traveling solitaires; local turbulence in a periodic sea), modify (10.3.12)
by changing only s2 to s2 = 0.45. Since s2 < b2, uncoupled cells in either state sl or
s2 remain fixed. That is, this system is bistable. For D = 0.42, investigate the configuration (0.9, 0.9, 0.45, 0.2), with all other cells at 0.45. Also explore random initial states when D = 0.35.
10.4 General cellular automata (CA) In this section we will turn from the specific, physically motivated examples of the last section, to a more general (and abstract) formulation of CA dynamics.
The study of cellular automata had its origin in a question which John von Neumann, the renowned mathematician at Princeton's Institute for Advanced Study, posed around 1950, and essentially solved before his death in 1957. The problem was
General cellular automata
455
(briefly) to determine the kind of logical organization that is sufficient for an automaton
to be self-reproducing. (An automaton is a robot which gives specific responses to specific inputs. The actual questions which von Neumann asked are listed in the appendix at the end of this chapter.) His initial approach to this problem, beginning before 1948, involved a rather realistic kinematic automaton, which consisted of about ten primitive elements (e.g., `muscles', switches, delays, sensing elements, rigid girders, welding and soldering units, etc.). This kinematic automaton then moves in a random `sea' of such primitive elements, selecting parts for self-reproduction. The dynamical and sequential complications of this kinematic theory tended to obscure the logical aspects of the problem, so that von Neumann, following a suggestion of S.M. Ulam, then developed the theory on the more abstract basis of cellular automata dynamics - to be described shortly. (See Burks (1970) for a historical account of this development.) This form of dynamics has subsequently been used to `model' many complex physical phenomena, and interest in its potential applications continues to grow. At present, however, there is only a fragmentary understanding of the advantages of this type of dynamic modeling. The following discussion is a simple introduction
to this area of research, and does not attempt to give a comprehensive description of the many diverse details which are presently known about these systems (see e.g. Burks (1970), Toffoli and Margolus (1987), Wolfram (1986)). Instead, a few examples
will be given, which illustrate some of the breadth of phenomena which can be encompassed by cellular automata. As noted in the last section, the dynamics of CA is entirely discrete. The space of the system consists of cells (a tessellated space) which may have one, two, or more dimensions. Moreover the space may be finite (with boundary conditions) or infinite.
In each cell the system can assume a discrete (usually small) number of state values - say k values. The configuration of the entire system at any time is defined by the set of state values, {si}, in all of the cells, {i}. For example, si may have the possible values si = 0, 1, 2, ... , k - 1,
(state space, S)
(10.4.1)
and i = 0, 1, 2,... (over the entire space, finite or infinite). If the cells are twodimensional, one may prefer to give the state in the form si; = 0, 1, 2, ... , k - 1,
(S)
(10.4.2)
using two position indices, but this is a nonessential detail. Finally, the dynamics in `time' is also discrete, as in the case of maps. The dynamics differs, however, from previous maps in two essential ways: (1) The future state of the system can only consist of discrete values, in this cellular space (Fig. 10.29), namely the set {si}, (10.4.1); (2) The value of s, at the next time step can depend not only on the present value of si,
Coupled maps and cellular automata
456 Fig. 10.29
(k = 4) so = I
sl =3
S2 = 0
53=2
s4=3
s5=2
but also on the present values on some set of neighboring cells. In the onedimensional case, this dynamical rule can be written in the form
s,(t+1)=F[s;-r(t),si-r+1(t),...,si(t),...,S1+r(t)]
(10.4.3)
where r is the range of the intersection between the cells. The generalization to higher cellular dimensions is obvious, but more tedious to write.
The collection of cells (i - r,.. . , i, ... , i + r) about the cells i will be called the neighborhood of i, Ni. CA dynamics, as defined by (10.4.3), is a mapping of the set of
all neighborhoods, N, to the set of all states, S. Since there are k states, there are k2'+ 1 possible neighborhoods. Therefore (10.4.3) defines the map
N->S
(SEZ,NEZ2'+'),
(10.4.4)
where Z represents the k-integer state space. Since k2r+ 1 neighborhoods map into k states, the map (10.4.4) is clearly many-to-one. In this abstract formulation, a transition function such as (10.4.5)
s.(t + 1) = F(s; _ 1(t), s.(t), si+ 1(t))
is defined by a table of all possibilities. Thus, if si = 0, 1 are possible states, a dynamic rule (10.4.3) is defined by specifying the central state value in the next time step for all initial neighborhoods. We might write this in the form 000/ca; 100/c4i
001/c1; 101/c5;
010/c2; 110/C6;
011/c3; 111/c7.
(10.4.6)
The dynamic rule is defined by specifying the value of each c;e {0,1 }, j = 0,..., 7. See Section 10.8 for an elaboration of this representation. What is of greatest interest, of course, is the behavior of the entire system. The state
of the entire system is defined by the state values in all of its cells, which will be called the system's configuration, C
C = {s1,s2,s3,...,}.
(10.4.7)
The dynamics (10.4.4) changes the values of the s,, yielding a new configuration for the system
`Legal' cellular automata
457
This map, (parallel map),
C -+ C'
(10.4.8)
which is produced by (10.4.4), is sometimes referred to as the parallel map. It represents the macroscopic dynamics of physical interest, whereas the dynamics (10.4.4) represent
the `microscopic' rules. The basic mystery of CA dynamics, as we will see, is the relationship between (10.4.4) and (10.4.8)
N-+SpC-->C'.
(10.4.9)
It might be thought that we could simply use a computer to study all possible CA rules, (10.4.4), and thereby establish the relationship (10.4.9). However, since there are k2r+1 neighborhoods, each of which can produce k states, this means that kk2r+' functions, F(si ... , si+,), may be studied, which is essentially impossible except for k = 2, r = 1 (256 possible CA dynamic rules). The function F(si _ ... , si+,) in (10.4.3), which is called a transition function, can only yield the discrete values (10.4.1), for all allowed values of its arguments. It is
this restriction of the possible values in the cells to a small set of values, (10.4.1), which distinguishes (10.4.3) from the computational solution of partial differential equations. This distinction is important, because the dynamics of CA is always exact, whereas there are always truncation errors when the transition function F (and the states si) can take on a continuum of values (as in the case of partial differential equations). In other words, because the state variables can only have discrete values, (10.4.1), there are no roundoff errors in the evaluation of the dynamics, (10.4.3), and therefore the dynamics is always exact (e.g., two different computers will not give different results, if they are running properly). The fact that the dynamics is exact, means that any `conserved' property of the system will indeed be exactly conserved, rather than slowly modified. We will return to the feature of conserved quantities when we discuss invertible CA (sometimes, unfortunately, referred to as `reversible' CA) later in this chapter.
10.5 `Legal' cellular automata To limit the number of possible CA dynamics to a manageable size, and yet account for the many `natural' systems, historically it was frequently assumed that the following conditions hold: (1) The quiescent condition: There is some quiescent configuration which remains fixed, say
{si}={0},
and
F(0,...,0)=0.
(10.5.1)
458
Coupled maps and cellular automata
Because of this condition on the transition function, F, once all the cells of the system are in the state s = 0, they remain in that state (no excitation, or `life', is created out of this quiescent configuration). If we consider only finite CA (e.g., by
imposing periodic boundary conditions), then the quiescent condition is mathematically redundant, and therefore can be omitted. More specifically, any
finite CA is equivalent, upon changing the time scale, to another CA with quiescent condition (Toffoli, 1977). This is a consequence of the fact that all finite
CA are periodic. However, the original time step is the one of physical significance, and hence this distinction is of great physical importance. Indeed we
have already examined both a quiescent and nonquiescent CA in the case of (10.3.8) and (10.3.6) respectively.
(2) Systems which obey (10.4.3) have a homogeneous dynamics (i.e., F does not depend on the location of the cell i), and frequently it is also assumed to be symmetric dynamics,
F(.... si_j,...,s;+j,...)=F(.... si+j,...,si_j ....)
(10.5.2)
where the inversion is carried out for all j. Thus, for example, F(11000) = F(00011);
F(10010) = F(01001)
In higher dimensions, this is replaced by invariance under rotations about the central cell. The CA which obey the restrictions (10.5.1) and (10.5.2) are often referred to as legal CA. While the restrictions (10.5.1) and (10.5.2) reduce the number of possible transition kk"'(k'+1)/2-1 (dynamic rules) in a one-dimensional functions, (10.4.3), from kklr+1 to system, this is still too many rules to survey easily, except for small values of k and r.
legal dynamical rules in a Exercise 10.16 (a) Show that there are one-dimensional system. (b) Some dynamic rules are related to other rules simply kk'+I(kr+1)/2-'
by some permutation of the k states (permutation-equivalent dynamics). For the case k = 2, r = 1, determine the four pairs of legal dynamic rules, (c0, ... , c,) - (co, ... , c,) which are equivalent under the interchange of 0 and 1. Thus there are 32 - 4 = 28 independent dynamic rules in this case.
As discussed at the end of Section 10.3, there are examples where it is useful to consider nonquiescent, inhomogeneous, or non-symmetric CA dynamics ('nonlegal'), due to interactions with `outside' agents such as external forces, chemical reactants,
and the like. An interesting example of such models are the `ring models', due to M. Kac (1959; Dresdan (1962)), which shed light on the irreversibility that is introduced into (exactly) reversible dynamics by Boltzmann's stosszahlansatz (see e.g., Kac (1959),
Prigogine (1962)). Other interesting exceptions to (2) are in the study of neural
General association for legal CA
459
networks, specifically, in some models of the mechanism of memory, and in chemically active systems, which also do not satisfy (1).
10.6 A general association for legal CA The cellular automata considered by von Neumann were two-dimensional, with the transition function limited to the five contiguous cells which form an orthogonal cross in space. This abstraction of the problem of self-reproducing machines, while it removes many of the kinematic complications, at the same time removes much of
the realism from the model. This is, of course, true of any application of CA to complex systems. In broad outline, it is perhaps useful to keep in mind the general associations shown in the table which are commonly made for the legal CA models of complex systems:
General associations for legal CAs (A) A `base state' of a system (the `vacuum state', `unexcited state', or unused raw material the primordial soup). (B) The laws of dynamics (e.g., physics, biology, chemistry, ecology, economics, sociology).
(C) The various elementary parts or divisions
The quiescent configuration; all si = 0, (10.5.1).
The transition function, (10.4.3): dynamic rules. The cells, sometimes where
of the system. N*si : 0 (vacuum). (D) The types of components which make up The states of a cell, given by the parts of the system (e.g., molecules and S*the values of si, (10.4.1). their excited states; spin states; nuts, bolts, and rods; production products, ethnic (E)
groups). The total system (e.g., organic molecule, crystal, machine, economic system, city structure).
The spatial configuration of the
E*set
{si:0} in the cellular
space.
This general association is clearly a rather severe abstraction of real systems, at least in many cases. In particular, the spatial arrangement of `excited' cells (si 0 0) may or may not bear a simple (or any) relationship to the actual configuration of a system in physical space. Usually, however, such a simple relationship is assumed to hold,
and these physical considerations are accounted for in the laws of dynamics of the system, (10.4.3). More abstract associations between real systems and CA dynamics have been developed in recent years, but the above simple association gives the
460
Coupled maps and cellular automata
rationale for the development of many CA dynamical models. This was illustrated by the examples in Section 10.3, and will be explored further in Sections 10.11, 10.15, and 10.16.
Exercise 10.17 The association (B) holds, of course, for any CA, legal or not. Obtain the transition function for both the `illegal' CA (10.3.6) (e.g., in the `turbulent' system, D = 0.3, M = 4) and the legal CA (10.3.8) in the periodic/'turbulent' system, D = 0.5. To limit the labor consider only the rules s1, s1s;/cj, where s1 = 0, 0.2 respectively in these C. Determine cj for each s; (i = 1, 2, 3). See (10.4.5) and (10.4.6) for the simpler system with only two states.
10.7 Simple examples To illustrate these dynamical ideas with a simple but interesting example, consider the infinite one-dimensional CA
s;=0,1
(i=0,±1,±2,...),
(10.7.1)
(so that k = 2 in (10.4.1)), with the dynamics defined by
s;(t+ 1)=si_1(t)+s;+1(t)
(mod 2),
(10.7.2)
(so that r = 1 in (10.4.3)). The addition in (10.7.2) is modulo-2 (e.g., 1 + 1 = 0), so that
the values of s;(t + 1) always satisfy (10.7.1), if the initial states have these values. Moreover it is clear that (10.7.2) satisfies both (10.5.1) and (10.5.2), so this is a legal CA. The dynamic rule (10.7.2) could, of course, also be expressed in the tabular form (10.4.6).
At present our understanding of CA dynamics is so rudimentary that, given the dynamical rule, (10.4.3), - or, in the present case, (10.7.2) - there is no known general method of predicting many of the qualitative dynamic properties of the system. The
only general method we know to determine these properties is to simply `run the dynamics' on a computer, for various initial states. To illustrate this for the present CA, (10.7.2), consider the particular example illustrated. (t = 0) (t = 1) 00000101110110000 (t = 2) 00001001010111000 (t = 3) 00010110000101100 (t = 4) 00100111001001110 (t = 5) 00000001011000000 00000010011100000
Simple examples
461
The unspecified cells all have the values s, = 0. From this it is clear that the `excited' state, s, = 1, tends to propagate at a unit speed to the left and right (a `sound' or `light'
speed), except when it encounters another excited state. This speed is, of course, governed by the range r, (10.4.3), which is one in the present case. These results can be represented in a more compact form by replacing the states s = 1 with a black mark, and making no mark where s = 0. The previous results, when extended to t < 20, is shown in the left picture of Fig. 10.30, with time again increasing downward. Fig. 10.30
If we compress the time and space scale by a factor of two, the result (now for t < 40) is illustrated in the middle picture. We have thereby lost some details, but it
is easier to see the global sensitivity of the dynamics to a change in the initial configuration. Thus the picture on the right (t <, 40) follows from the initial configuration 0111000111110, and we clearly see the loss and shrinkage of the triangular quiescent regions. A difference between two configurations, C'(t) _ {s;(t) } and C"(t) _ {s; (t) }, is sometimes measured in terms of the Hamming distance, H(t). This is simply the number of cells which are in different states at time t,
H(t) _
h[s'(t) - s'. (t) ];
h(0) = 0,
h(n) = 1 if n 0 0.
(10.7.3)
Using this `distance', one can define `expansion rates' or `Lyapunov exponents', and the like. However, this `distance' is not very suitable nor revealing. First of all, by trivially
shifting all the cell numbers of some configuration the value of H(t) is changed, without changing the subsequent dynamics of the configuration. We might minimize H(t) subject to such shifts in space, but the minimum is not necessarily unique, and its subsequent behavior is likewise nonunique (or should we continue to minimize?).
Moreover, two configurations of the form C = (...010...n zeros... 010...) are `equidistant' from each other, regardless of the value of n. On the other hand this does not indicate their relative spatial extent, which influences their dynamics because of the limited interaction range, r (which governs the maximum propagation velocity of disturbances). Thus the behavior of H(t), as a function of time, is highly nonunique. There is, of course, the bounds 2r > H(t + 1) - H(t) for sufficiently large t (e.g., 2rt > size
Coupled maps and cellular automata
462
of initial non-quiescent regions), but there is no such bound for small t, which is typically used for defining Lyapunov exponents (e.g., Section 7.10). It is not difficult
to invent other measures for the distance between two configurations which incorporate this spatial extent (in relationship to the range r), but that will not be pursued here.
Exercise 10.18 Show that, if (10.7.2) is replaced by s; (t + 1) = s; _ 1(t) + s.(t) + s; + 1(t) (mod 2), then the above initial state develops groups of five contiguous excited states. Is this a legal CA? To check, consider the initial configuration in which only one cell has s, = 1, all others being zero. Exercise 10.19 Show that the dynamics (10.7.2), and the one in the last example, can be given by a polynomial function F(s; - 1 , ... , Si + 1), without the use of mod 2 notation.
Exercise 10.20 Show by computation that, for the transition function (10.7.2), the initial state 0101110 self-reproduces in such a way that there are only 2"(`) copies of this state (and nothing else) at the times t = 4k, k = 1, 2, .... If this is done for t not greater than 48, it is found that N(t) = 1 for t = 4, 8,16, and 32, whereas N(t) = 2 when t = 12, 20, 24, 36, 40, and 48, and finally, N(t) = 3 when t = 28, and 44. Note the temporal `overlap' in this reproduction process (N(t) is not a monotonic function of t). In this case the `self' reproduction is being strongly modified by a 'group' action. These examples illustrate the much more abstract character of general CA dynamics, compared to the physically oriented examples of Section 3. Frequently a dynamic rule is studied simply by running the CA dynamics on a computer, for a variety of initial configurations; perhaps obtaining statistics concerning its `behavior'. In many cases, the possible relevance of a dynamic rule to a physical system only comes after such studies. This is clearly not the different approach of Section 3, but instead tends to `stumble onto' a physical application. More recently, many studies are turning to the physically motivated dynamic rules (see Sections 10.11, 10.15 and 10.16), but for the present we will continue with these general considerations.
10.8 Neighborhood configurations and dynamic rules It is useful to develop a numbering scheme for identifying the kk2k " different possible CA transition functions (10.4.3), or (10.4.4). We will first illustrate such a scheme for the case k = 2, r = 1, expanding upon the discussion on (10.4.6). In this case, there are 3(= 2r + 1) cells in a neighborhood, so the list of all 8(= 23) possible neighborhoods around any (central) cell is N:
111
110
101
100
011
010
001
000
S:
c7
c6
c5
c4
c3
c2
c1
co
(at time t) (at time t + 1)
(10.8.1)
Neighborhood configurations and dynamic rules
463
Below each neighborhood configuration is a number c; = 0, 1, which represents the state value of the central cell at the next time step. We will refer to these as dynamic conditions (i.e., s1s2s3/c). Thus this tabulation of eight dynamic conditions represents all 28 possible transitions (10.4.3), in other words, the map (10.4.4)
N -S. The legal CA satisfy the dynamic conditions CO = 0;
c, = c4,
c3 = c6
(legal CA)
(10.8.2)
in which case there are only 25 possible distinct transitions. Note that the subscripts on the c; have been chosen to equal the number represented
by the neighbourhood configuration, when it is `read' in binary notation (e.g., 101 = 1 x 2° + 0 x 2' + 1 x 22 = 5). Similarly, any neighborhood configuration (with any k and r) can be associated with a number, when the neighborhood is taken to be the number in `k-nary' notation (e.g., for k = 3, r = 2, the neighborhood 02102 can be denoted by the number 2 x 3° + 0 x 3 + 1 x 32 + 2 x 33+0 x 34 = 65). This is a very neat way of labeling any configuration (of any number of cells), or any collection of state values. In particular we can associate a number with the set of state values (CO, ... , c,) in exactly the same way. Following Wolfram, we introduce the dynamic rule number 7
R = Y c"2"
(10.8.3)
n=0
which uniquely defines the map (10.8.1). For example, the legal CA defined by the rule (in the more concise notation) 111/0, 110/1, 101/0, 100/1, 011/1 010/0, 001/1, 000/0, is rule R = 2 + 8 + 16 + 64 = 90, whereas the `illegal' CA 000/1, 001/1, 010/1, with all other c" = 0, has the rule 7. Note that only the nonquiescent CA have odd rule numbers. Clearly (10.8.3) can be generalized to any CA, by taking the rule number to be k2k +t _1
R=
Y
(10.8.4)
cn k".
n=0
While this rule designation is unique and generally concise (it is not concise for legal CA), it is purely formal, and does not give any indication of the resulting dynamics of the parallel map (10.4.6). For example, the only CA in (10.8.1) which propagate a `signal' through the tessellated space are those for which c, = 1
and/or
c4 = 1
(propagating CA)
(10.8.5)
(try some initial configuration, say 010110, and verify (10.8.5)). However, given R, it generally takes several algebraic steps to determine whether (10.8.5) is satisfied. In
Coupled maps and cellular automata
464
other words, given R, it is not immediately clear whether the very basic physical property of propagation is present or not (e.g., does R = 229 propagate?). Clearly, any more subtle dynamical properties of the parallel map cannot be associated with the value of R in a simple fashion. Indeed, perhaps the first basic question is `What are the possible qualitative dynamical features we can obtain from "simple" CA (small values of k and r)?' We turn to this question next.
Exercise 10.21 Determine R for the case in Exercise 10.18. This is one example of a `totalistic' CA, which are those CA that have the dynamics
j
r
s;(t+1)=f[ 1
=-r
s,+,(t)].
(10.8.6)
In this case s,(t + 1) is only determined by the sum of the values in all cells within the range. What other R values have this totalistic property (with different functions, f, of course) when k = 2, r = 1? If k = 2 and r = 2, how many totalistic CA are there? Compare this with the total number of CA (when k = 2, r = 2).
10.9 Several classifications of CA dynamic properties The qualitative classification of CA dynamics has only begun to be studied. For one-dimensional infinite systems, Wolfram (1983, 1986) found in the course of many
computational studies of CA dynamics that, if the initial state was taken to be `disordered' (so each site value is chosen to be from 0 to (k - 1) with probability 1/k), four qualitative behaviors can be distinguished: D1: Evolves in time to a spatially homogeneous configuration (all cells in one state);
D2: Tends in time to fixed or periodic temporal structures in different regions of space;
D3: Tends to chaotic aperiodic patterns throughout space; D4: Evolves into complicated localized patterns, some of which propagate in an irregular fashion. This classification does not indicate what happens to specific initial configurations, some of which we will consider shortly. Wolfram (1984; also see Martin, Odlyzko, and Wolfram, 1984) made a detailed
investigation of those CA which are restricted to the totalistic rules, (10.8.6), to determine how many rules belong to each of the above four classes. He found that the approximate fraction of CA in each of these classes depends on k and r according to the table opposite. It should be particularly noted that Wolfram found no class 4 behavior in the simplest CA, here k = 2, and r = 1. We will return to this point below.
Several classifications of CA dynamic properties
465
Wolfram also concluded from other studies that the CA with totalistic rules exhibit all of the dynamic features which are found in all types of CA, and therefore represent no special dynamic simplification. Hence he felt that the above classification should represent the possible qualitative behavior of all CA which began from disordered initial states. However, there presently exists no proof of this classification-equivalence between totalistic CA and general CA dynamics.
Class 1
2 3
4
k=2 r=1
k=2
k=2
k=3 r=1
0.50 0.25 0.25 0
0.25 0.16 0.53 0.06
0.09
0.12 0.19 0.60 0.07
r=2
r=3 0.11
0.73 0.06
Wolfram has assigned a `code' number to such totalistic CA, given by (2r+1)(k-1)
C=
E
n=o
Of (n)
(10.9.1)
where f (n) is the function which appears in (10.8.6). Since f (0) = 0 these numbers can all be divided by k, and are therefore unnecessarily large. However, the values of C are much more concise than those of R (many R values do not occur). Using the definition (10.9.1), Wolfram found that, for k = 2, r = 2, the following codes belonged to the above dynamic classes: (1) 0, 4, 16, 32, 36, 48, 54, 60, 62 (2) 8, 24, 40, 56, 58 (3) 2, 6, 10, 12, 14, 18, 22, 26, 28, 30, 34, 38, 42, 44, 46, 50 (4) 20, 52
The `chaotic' patterns generated by the dynamics of class are not all equally chaotic, as is illustrated by Fig. 10.31. The picture on the left is generated by code 10 (only f (1) and f (3) equal 1), and the one on the right by code 12 (only f (2) and f (3) equal 1).
A third example (Fig. 10.32) generated by code 26, falls somewhere between the last two in its chaotic behavior. One can quantify these differences by measuring the various forms of `entropy' of these patterns (Wolfram, 1983), which will be discussed in the next section.
Attention has sometimes been drawn to the similarities of some CA patterns (particularly those containing `triangles') and the patterns found on sea shells; an example of which is illustrated in Fig. 10.33 (Waddington and Cowe, 1969). However,
Coupled maps and cellular automata
466
Fig. 10.31 k = 2, r = 2, totalistic rule, code 10 (001010)
Fig. 10.32
Fig. 10.33
k = 2, r = 2, totalistic rule, code 12 (001 100)
code 26 (01 1010)
Past
`Present'
Several classifications of CA dynamic properties
467
there are obviously important differences, and a broad spectrum of such patterns on
the shells of molluscs apparently can be obtained by PDE reaction-diffusion, activator-inhibitor models (e.g., Meinhardt and Klingler, 1987). The dynamics in the case of Class 4 can take a long time to establish a simple or complex pattern, sometimes involving much `searching' in the course of establishing these patterns. A few examples, for code 52, are illustrated in Fig. 10.34. For many more examples of one dimensional CA dynamics, see Wolfram (1986). Fig. 10.34 code 52 (110100)
k = 2, r = 2, totalistic rule, code 52 (110100)
When we consider more specific initial configurations, rather than the `disorder' configurations considered above, then it is much more difficult to characterize the general asymptotic behavior of the patterns, much less any of the details. Even if we simply restrict the initial configurations (i.e., nonquiescent states) to a finite number of cells, the above 'disorder-excited' classes exhibit some subclasses with distinct qualitative characteristics. The 'finite-excited' classes can be described as follows: Fl: The initial patterns disappear with time (tend to the quiescent configuration). F2: (a) The initial patterns evolve to a fixed finite spatial size. (b) The initial patterns evolve into a uniformly expanding region which is entirely periodic in time. (c) The initial patterns evolve into a uniformly expanding region which is entirely homogeneous, except for a finite spatial region. (d) The initial patterns again have uniformly expanding boundaries, but the
interior pattern may be any combination of chaotic, chaotic/periodic, or periodic. The chaotic/periodic combination does not generally develop uniformly in time. F3: The chaotic patterns grow indefinitely in size, at a fixed rate bounded by r. F4: The patterns grow, contract, and possibly propagate irregularly with time.
Coupled maps and cellular automata
468
Wolfram (1986) pointed out the importance of such finite classes, and identified many of them. We will return to examples of these classes shortly, and discuss the reason that many F2 classes, degenerate into one D2 class. Because the concept of sensitivity to initial conditions is recognized to be of basic importance, another useful characterization (Wolfram, 1986) is to note how the above disordered classes react to a small (localized) change in an initial configuration. The resulting changes he found to be:
Cl: There is no change in the final configuration. C2: The patterns change only in a region of finite size. C3: The chaotic patterns change over a spatial region which increases indefinitely in size.
C4: The patterns experience irregular changes in both space and time. Note that, if the changes had been made to finite-excitations, the change-classes would be more numerous.
A few examples of localized disturbances for the totalistic case, when k = 2 and r = 2, are illustrated in the following exercises. The reader might also explore small perturbations of these initial configurations, and see how they relate to the above discussion.
Exercise 10.22 In the case of code 20, show that the initial states 11101001, and 10111101 have simple periodic dynamics, while 11000011 leads to a complicated periodic pattern; 1001111011 yields a simple drifting motion, and 10111011 produces a complex drifting motion.
Exercise 10.23 Using the initial state 1001111011, which produces a simple drift motion, obtain another initial state that produces a drift motion in the opposite direction. Determine what can happen if there is an initial state in which these two drift motions interact in the future. Is there more than one possibility? Aizawa and Nishikawa (1986) proposed a method to `decompose' the dynamic rules, hopefully to better understand the resulting global qualitative features, such as the above D classes described by Wolfram. They noted that any transition function for the legal, k = 2, r = 1 CA dynamics si(t + l) = F(s;_ 1(t),
si(t),
s;+ 1(t))
(10.9.2)
can be composed in terms of the null function, f = 0, three symmetric functions
fo=si_1+s,+si+1 mod2 f1=si_1si+sisi+l+s,-1s,+1 mod 2 f2 =S,-1SiSi+1,
(10.9.3)
Several classifications of CA dynamic properties
469
and the asymmetric function
A=(s,-si-1)(s,-s,+1).
(10.9.4)
All of the local transition functions can be composed as some sum (mod 2) of the functions from the set (0, fo, f,, f2, A, Af, ). Moreover each of these functions represents one legal transition function, namely (F(0), F(150), F(232), F(128), F(36), F(32)) respectively. Hence this decomposition is minimal. The difficulty in under-
standing the emergent global behavior, even with the help of such a reduced representation of the local dynamics, is illustrated by the following considerations.
Referring to the above finite-dynamic classes, the corresponding 32 transition functions, F(R), for the dynamic rules R, (10.8.3), are found to be composed as follows (all sums are mod 2):
F(128)=f2; F(160) = f2 + Af, F(250)=f0 +fl +f2 + A + Af,; F(254)=f0 +fl +f2 Class 2: (a) F(4) = A + Af,; F(36)=A; F(72) = f, + f2 + Af,; F(76)=f1 + f2 + A; F(104)=f1 + f2; F(108)=f1 + f2 + A + Af,;
Class 1: F(O)=O;
F(32) = Af,;
F(132)=f2+A+Af,; F(164)=.f2+A; F(200)=f,+Af,; F(204) = f, + A; F(232)=fl; F(236) = f, + A + Af, (b) F(50) =.fo +.f2 + A; F(178) = fo + Af,; (c) F(218)=f0 + f, + f2 + A; F(222)=f0 + f, + f2 + Af,; (d) F(94) =.fo +.f i + Af, Class 3: F(18) = fo + f2 + A + Af 1; F(22) = fo + f2; F(54) = fo + f2 + Af,;
F(90)=fo+f, +A; F(122)=fo+f, +A+Af,; F(126)=fo+f,; F(146)=fo+A+Af,; F(150)=fo; F(182)=fo+Af,. To try to understand these results, note that the rules 001/c, and 100/c4 indicate a propagation of a disturbance when c, = c4 = 1. This only occurs if the transition function contains the function fo. Thus it is not surprising that all of the Class 3 systems contain fo in their transition functions. Similarly the appearance of fo in the Class 2 systems yields the expanding subclass F2(b), but it is neither necessary nor sufficient for oscillatory behavior. Thus while F(50) and F(178) tend to oscillating states, F(218) and F(222) do not. On the other hand, F(108) can oscillate even though it does not contain fo. Indeed, F(250) and F(254), which both contain fo, tend to homogeneous states (Class 1). This, however, gives a clue concerning the 'propagation-
impotency' of fo, because the transition functions R = 218, 222, 250, and 254 all contain the combination f o + f , + f2. We can therefore conclude that this combination cannot yield more than a finite inhomogeneous region. What insight that represents remains to be clarified, however. Moreover Aizawa and Nishikawa noted that
470
Coupled maps and cellular automata
F(R) + A + some members of (0, f t, f2, Aft) implies Class 2 but, of course, the converse does not hold. Also, there appears to be no other general statements which can be made in terms of such a composition. We see rather clearly the difficulties involved in predicting the emergent behavior from the local rules, even when 31 nonquiescent transition functions are composed additively from only five functions. One of the particularly interesting rules, noted by Aizawa and Nishikawa, is F(94) = fo + f t + Af 1. As noted under the above F2(d) description, some initial states go to period 2 or 3 in any localized region of space, even though period 1 continues to expand, as illustrated in Fig. 10.35. However, any initial configuration composed Fig. 10.35
........ .. ...... ...................... .......................... ........ ............ .......................... .......................... ........... ............... .......................... ................................ ........... ...............
................................ ................................ .............. .................. ................................ ...................................... .............. .................. ......................................
of even numbers of contiguous ones and zeros yields a space-time fractal structure, as the middle picture illustrates (now in high-density). If one of those even-numbered groups of zeros is replaced by an odd-numbered group, a spreading inhomogeneous region develops inside the fractal region (in space-time). The initial configuration of the bottom picture differs from the middle picture by replacing a ten-zero group with a nine-zero group. It can be seen that the fractal region retreats in an irregular fashion from the center with an average speed, v, which is slower than the propagation speed of the boundaries (one cell per iteration, r = 1). Therefore the asymptotic state will have a fractal region over a fraction (1 - v/r) of its nonquiescent region.
Several classifications of CA dynamic properties
471
Fig. 10.36
The character of another fractal/inhomogeneous boundary is illustrated in Fig. 10.36 (t < 200, using smaller dots, for clarity). The boundary only propagates when a quiescent triangular region occurs, which happens very irregularly. However, the boundary is well-defined, and one can determine the average speed, u, and its dependency on the initial configuration. If we define a velocity as the number of cells, N(t), from the boundary to some fixed cell (e.g., one to the right in the figure) divided by t (> 0), v(t) = N(t)/t, then we obtain the result shown in Fig. 10.37. Of course,
v(t)
Coupled maps and cellular automata
472
Fig. 10.37
VW
1.01
0.5
100
200
Fig. 10.38
300
400
600
t
X
X.
X
500
411:
X
J:
conditions distributes these inhomogeneous regions everywhere, and they ultimately dominate the space-time pattern. The complexity of the patterns which are generated by CA can be very different from that associated with such maps as 1 = 2x mod 1, discussed in Chapter 4, Section 11. If x is written in binary notation, so that
x = a1a2a3 ... , then we see that this map produces xn+1
='a2a3a4....
That is, the `dynamics' amounts to a shift of the numbers ak to the left, and discarding
the first number (a `shift dynamics'). Thus, any `randomness' which arises in the numbers xn+r='a1+ra2+ra3+T'**
Entropies of one-dimensional CA
473
is simply an `unfolding' of the initial information contained in the initial numbers ak. To the contrary, in CA dynamics, simple initial conditions (one integer!) can produce a complicated sequence of numbers at one cell. Fig. 10.39 (Packard and Wolfram, 1985) illustrates the situation when one site is initially excited in the (k = 2, r = 1) totalistic cases C = 102 (left figure), and C = 138. As another example, from many
initial states the simpler (k = 2, r = 1), but 'nonlegal' CA (R = 30) apparently can generate sequences of zeros and ones which contain all possible subsequences with equal frequency (e.g., Jen, 1986).
We will now examine a few ways that have been used to measure several types of randomness, by means of various 'entropies'.
10.10 Entropies of one-dimensional CA One of the ways that the complexity of CA dynamic patterns has been characterized is by various forms of `entropy'; that is, some spatial and/or temporal measure of the uncertainty or information associated with the dynamics. In this section we will briefly
explore some of the dynamic properties which these entropies do, and do not determine.
Historically the statistical concept of entropy was defined as a measure of randomness associated with some ensemble of possible `states' (e.g., configurations in be the distribution of probabilities over the m possible the case of CA). Let (Pi,. .. , configurations (i = 1, ... , m), satisfying m
pi>, 0
(i= 1,...,m);
p;= 1.
(10.10.1)
A total entropy associated with this probability distribution can be defined (following Boltzmann, Shannon, and Kolmogorov) as m
S= - E Pklnpk,
(10.10.2)
k=1
is continuous at the origin. Before an experiment where Oln 0 = 0, so that S(p1,... , is performed, S measures the uncertainty which is associated with the possible outcome
of the experiment. Conversely, after an experiment is performed, S measures the
474
Coupled maps and cellular automata
amount of information obtained by that experiment. In both cases Pk is the a priori probability of the state k, which in physical situations is determined by the ensemble of physical interest. The maximum uncertainty/information occurs when Pk = 1/m, yielding S = ln(m). These ideas have been discussed in other sections (see index). In the case of a CA, it is necessary to use a finite tessellated space (finite m in (10.10.1)) in order to obtain nonzero probabilities for the spatial and/or temporal configurations. One useful finite CA is the spatially periodic CA, si+,v(t) =Si(t)
(i = 1,..., N)
which has only N independent cells. This CA can also be pictured as a `necklace' (Fig. 10.40) with N `beads' (cells) which can acquire `colors' (states). The size of the Fig. 10.40
necklace, N, can of course be taken as large as we like, but it must be finite in order to obtain nonzero probabilities for reasonable ensembles. Wolfram (1984) adapted the entropy (10.10.2) to CA dynamics. There are several
possibilities, depending on whether we wish to measure spatial or temporal, or spatial-temporal forms of randomness. Consider first the spatial form of entropy, and take a block of X cells. We consider an ensemble of random initial conditions, and examine the resulting configurations in the block X after a long period of time. Let p()" be the probability of the configurations (i = 1, ... , kx) for some CA rule of interest, where k is the number of states in each cell (so the number of possible configurations in this block is m = kx). The specific spatial measure entropy is then defined by sM(X) _
X
1 E PI") logk
(10.10.3)
The division by X makes this a specific entropy (entropy per cell). The term `measure'
(corresponding to the subscript µ), distinguishes this entropy from a `topological' entropy which considers only whether a configuration i occurs (regardless of the value of its probability, pi > 0). Let
N(X) = number of configurations which occur in a block of length X.
(10.10.4)
Of course N(X) <, V. Then the specific temporal topological entropy is defined by SL(X) - X logk N(X).
(10.10.5)
Entropies of one-dimensional CA
475
The topological entropy (sometimes called set entropy) is clearly less `detailed' in its characterization of the dynamics than the measure entropy, which depends on the probability distribution. Correspondingly, sz(X) is easier to determine than sµ(X). A second type of entropy measures the temporal randomness which occurs in a specific (arbitrary) cell. We now consider a temporal `block' of T time steps, and write p;`° for the probability of the sequences (i = 1, ... , kT) in this time block (note that this block is taken a long time after the random initial conditions). Then the specific temporal measure entropy is defined by kT
s,,(T) = 7 Y_ j=j
logk p!`).
(10.10.6)
Similarly, if
N(T) = number of sequences which occur in the time interval T
(10.10.7)
then the specific temporal topological entropy is defined by sz(T) = T log, N(T).
(10.10.8)
The spatial entropies are in the original spirit of Boltzmann, whereas the temporal entropies stem from ideas due to Kolmogorov. Generally (for arbitrary CA rules) it is only possible to determine these entropies
by computer calculations. In that case, one is always obliged to estimate the probabilities, or the values of N(X) or N(T), after some finite amount of evolution of the random initial conditions. Usually enough evolution is used to obtain reasonably
fixed ('asymptotic') results, but without being influenced by the periodic boundary conditions of the finite lattice (an alternate approach is outlined below). In rare cases we can obtain theoretical predictions of some entropies. Usually these cases are rather trivial, but they can illustrate the limitations on the importance of these entropies. Consider the CA with either rule R = 170 or R = 240 (k = 2, r = 1). These CA shift any configuration one cell to the left or right (respectively) in the next time step (Fig. 10.41). Thus, in any spatial block, X, or temporal interval, T, the random initial conditions are faithfully reproduced by the dynamics, and hence N(T) = kT
N(X) = kX, Fig. 10.41 1
0
1
011
0
1
1
(10.10.9)
Coupled maps and cellular automata
476 and
PIx) = k-X
Therefore all of the entropies are maximum (!) S, (X) = s, (X) = s, (T) = s, (T) = 1.
(10.10.10)
The entropy in the case is obviously a reflection of the initial randomness rather than
randomness `generated' by the CA dynamics. This example illustrates that these entropies are not always useful; alternative measures must be obtained. Exercise 10.24 Another simple case (of no practical interest!) is when R = 204 (k = 2, r = 1). Determine the four specific entropies s, (X), su(T), s, (X), st(T), and show that si sµ in this case. As another example, consider rules R = 2, or R = 16. In these cases the only nonzero condition is 001/1 or 100/1 respectively. These rules propagate 1 states to the left or right only if they are separated from other 1 states by at least two 0 states. All other 1 states are eliminated (become 0). Configuration of this type can be represented by ((001)*(0)*)*, where the * signifies an arbitrary number of copies of the configuration in the parenthesis.
We can use this information to determine the temporal topological entropy. Let N1(t) and No(t) be the number of sequences which end in 1 and 0 respectively at time t (in a given cell). At t = 0 (initial condition), the state is either 1 or 0, so N1(0) = 1 = No(0).
(10.10.11)
If s = 1 the next two values must be zero, whereas, if s = 0, in the next time step it can be 0 or 1, but in the following time step the s = 1 must be followed by a 0, whereas 0 can become 0 or 1. 1
0 0
0
t
01
t+1
0 1 0 t+2
We conclude that N1(t + 2) = No(t)
and No(t + 2) = 2No(t) + N1(t).
(10.10.12)
To obtain s,(T) we need to obtain N(T), (10.10.7). This equals, for large t = T, N(t) = N1(t) + No(t). We find from (10.10.12)
N(t+4)=N1(t+4)+No(t+4)=No(t+2)+2No(t+2)+N1(t+2) =2N(t+2)+N0(t+2)-N1(t+2) = 2N(t + 2) + 2No(t) + N1(t) - No(t)
(10.10.13)
Entropies of one-dimensional CA
477
so that we obtain the recurrence relation for N(t) N(t + 4) = 2N(t + 2) + N(t).
(10.10.14)
The initial condition (10.10.11) yields N(O)=2, and (10.10.12), (10.10.13) yields N(2) = 4. From (10.10.14) we can obtain N(t) for later times. Rather than work out the exact number, we simply note from (10.10.14) that N(t + 4) >, 2N(t + 2),
so N(t) > 2`12N(0),
and
N(t + 2) < 3N(t),
so N(t) < 3`12N(0).
Thus
log2N(T)> Z,
11og23>s,(T)-
(10.10.15)
T
where the large t = T limit is used. (10.10.15) gives bounds on sz(T) for R = 2 or R = 16. This example is, of course, only of pedagogical interest, but the value of s, is now at least partially a reflection of the influence of the CA dynamics. However, neither of these rules generate randomness, they only suppress some of the initial randomness. Again we see that alternative measures are needed (see below). A more interesting, but somewhat hypothetical example involves the rule R = 18 (with nonzero conditions 001/1 and 100/1) with k = 2, r = 1. It appears that in this case all sequences occur which do not contain 11 subsequences Q. Milnor). Assuming that this is true, then s = 1 can only follow s = 0, whereas s = 0 can follow either s = 0 or s = 1. Using the same N1(t) and No(t) defined above, this implies that
N,(t + 1) = No(t);
No(t + 1) = N1(t) + No(t).
Adding these to obtain the total number of sequences N(t + 1) = N1(t + 1) + No(t + 1) = N(t) + No(t) or
N(t + 1) = N(t) + N(t - 1).
(10.10.16)
This recurrence relation, when R = 18, replaces (10.10.14), when R = 2, or R = 16. The interesting result obtained from (10.10.16), and the fact that N(0) = 2,
N(1) = 3,
(10.10.17)
is that
N(t) = F,
(10.10.18)
where F, is the tth Fibonacci number, obtained from
Ff=Fr-1 +Fr-2
(Fo= 1 =F1)
For large t we then obtain
N(t) - 4'
= (51/2 + 1)/2.
(10.10.19)
478
Coupled maps and cellular automata
Thus st(T) =1og2 4
0.69.
(10.10.20)
Exercise 10.25 Obtain this asymptotic result, (10.10.19), for N(t) from (10.10.16). Other spatial-temporal entropies can be introduced (Wolfram, 1984), but the above examples illustrate both the basic ideas and some of the shortcomings of these measures
of randomness. In particular, these entropies are not useful for identifying turbulent systems. A careful study of useful measures of turbulent dynamics, in the limit of
infinite systems, has been made by Oono and Yeung (1987). Other measures of randomness for finite sequences can be based on Popper's concept of n-free sequences (Popper, 1968). Such approaches may prove to be important in the future for empirical (FINITE) applications.
10.11 Particle-like dynamics from partial CA rules Rather than study the broad dynamic categories of all possible CA rules, we might wish to select CA rules which give some specific elementary form of dynamics. This process is not well understood at present, and in this section we will only examine some simple aspects of this type of construction. It should be emphasized that the present approach for obtaining `physically significant' CA rules is very different from what was done in Section 10.3. In that section some `reasonable' coupled lattice maps, with several control variables, were studied. The issue of `CA' rules only entered when we considered possible dynamic bifurcations, produced by changing the control variables (see Exercises 10.9 and 10.10). Here we are considering the process of selecting
only a partial set of rules (corresponding to a family of CA) which will produce certain desired dynamics. As the specified dynamical situations are extended, the partial set of rules needs to be correspondingly extended, as will become clear. To illustrate the present point of view, let us first try to describe some collision of one-dimensional localized `particles'. By a `particle' we simply mean a specific moving collection of (nonquiescent) cells. A reasonable starting point might appear to be to take the tessellated space of CA dynamics to be cells in physical space (configuration space), and to use the simplest CA (two states per cell, k = 2; interaction range, r = 1). If this is done, however, then it is not possible to model all of the dynamics of even two interacting particles. The
reason is that the dynamics of configuration variables, q;(t), are governed by second-order ODE, whereas CA dynamics in the tessellated space involves only a (first-order) map. Roughly speaking, the present CA dynamics does not contain `general velocity information' - where the particle was in the last time step. This information can be introduced by using more general forms of CA dynamics (e.g., Fredkin, 1982; Margolus, 1984), but we will not pursue this topic here. Instead we note
Particle-like dynamics from partial CA rules
479
that, in order to have any velocity-like information, it is necessary to either `spread the particle out', by using a larger value of the interaction range r, or else to introduce more states in the cells (k > 2). To illustrate this with some elementary examples, consider the CA with r = 1, k = 3,
so the possible states are 0,
1,
2. If we require the CA to be legal, there are
317 - 1.3 x 108 possible CA rules. Even if we could survey the dynamics of these rules at the rate of one a second, it would take about 49 months (without sleep!) to finish the survey. Instead of considering all rules, we will consider sets of CA rules, defined by specifying only a limited number of the 17 conditions. We will select the conditions based on the physical requirements we wish to study, and specify additional conditions only as they are required by the dynamics. To obtain a simple moving particle, we begin with those legal CAs which have the conditions 001/1, 002/0,012/2,02 1/0
(C -1)
and abc/d implies cba/d, and 000/0. In trinary notation, (C - 1) can be designated by cl = 1, c2 = 0, c5 = 2, c7 = 0. This dynamic selection yields a `particle' which can move with one speed in the positive or negative direction, as given by 02100 00210
or
00120
(t)
01200
(t + 1)
or
Of course `complementary' conditions (C - 1)*, obtained by interchanging 1 and 2 in (C - 1), could be used, producing complementary particles. We note that the speed is determined by r, whereas the additional state, s = 2, is needed to build an `asymmetric
particle' for positive or negative velocities. The only way that we can introduce a selection of speeds is to increase the range r, and to increase the size of the particle accordingly. It is interesting to note (Toffoli, 1977) that this relationship between the spatial extent of the particle, Ax, and the possibility of limiting the value of its velocity, Av, is reciprocal (Av Ax > a), as in the quantum uncertainty principle.
Exercise 10.26 By a simple change of two conditions of (C - 1) we can make the particle (2, 1) an oscillator. Obtain these new conditions, and determine whether additional conditions can produce a `radiating' oscillator. The conditions (C - 1) are adequate to determine the dynamics of these isolated particles, but other conditions (more of the CA rule) must be specified when they `interact' (come within a distance r). Two types of collisions can occur 0021001200 0002112000
or
021000120 002101200
(t) (t + 1).
These two types are due to different amounts of `vacuum' between the particles! To
480
Coupled maps and cellular automata
proceed to the next time step, we need to specify the additional conditions 112/c14
and
101/clo,
so there are 32 = 9 possible next steps, and we begin to see again the richness in the CA dynamics. Again, as in the selection (C - 1), we can be motivated by physical interests rather than mathematical completeness. We might require that (a) the collisions behave the same, regardless of the amount of `vacuum' between them, and (b) the particles are time-delayed by the collision (similar to a soliton). One possibility is to take (C - 2S)
112/2, 022/1, 011/2,122/0
which produces the `delay' in the `soliton' motion for even-vacuum collisions whereas
(C - 3S)
101/1, 212/1, 010/2,121/0, 202/0
produces a similar `delay' for odd-vacuum collisions.
ll t
02100120
t+1
00211200
t+ 2 t+ 3 t+ 4 t+ 5
00022000 00011000 00122100 01200210
t
0210120
t+ 1
0021200
t+ 2 t+ 3
0001000
t+4
0120210
0012100
The conditions (C - 1), (C - 2S) and (C - 3S) are all compatible, and hence define a set of 34 CA `soliton' rules. By a slight modification of the conditions (C - 2S) and (C - 3S), the particles (C - 1) can be made `hard spheres', which exhibit no delay upon collision. The conditions now are 112/1,011/2,122/0
(C - 2H)
Particle-like dynamics from partial CA rules
481
and, for odd-vacuum collisions 101/2, 022/1, 222/2, 121/0,202/0
(C - 3H).
Thus there is a larger set (35) of 'hard-sphere' CA rules. The two particles cannot co-exist because the first condition in (C - 2H) and (C - 2S) as well as (C - 3H) and (C - 3S) differ. Rather interestingly, you can also have a mixed `hard-soliton' system by selecting (C - 2X) and (C - 3 Y), with either (X, Y) = (S, H) or (H, S). This is because
there is only the one common neighborhood of (C - 3H) and (C - 2S), 022, and it has a common condition. There is apparently no known systematic method of determining whether there is more physically interesting dynamics within these sets. We can nonetheless explore possibilities in a simple-minded fashion. How about producing `walls' so that we can contain these particles? The simplest fixed configuration ('wall?') compatible with (C - 1, 2S, 3S) is 020, provided we require 020/2. Can this configuration reflect a particle? Consider a particle approaching it, as illustrated. 02100200 00210200
t
t+ 1
To proceed to the next time, the condition 102/c11 needs to be selected. Whatever the choice, the next step yields 02c120. Since this is symmetric no asymmetric reflection can occur, so 020 cannot act as a wall (it can, for example, act as a reactant, producing a new reflected particle, if c11 = 1). How about a more substantial `wall', say 02020? Since r = 1, the second 2 does not influence the initial fate of the first 2 state, and the 02020 configuration is apparently no more effective as a wall. It would be nice to have an elementary proof of such dynamical properties. While this CA is quite simple, there are other CA rules with more interesting dynamical `particles'. For example, we can obtain travelling oscillators ('molecules') which move as illustrated. 02121000 00221100 00021210
t t+1 t+2
In contrast to the four conditions (C - 1), for the above simple particle, this molecule requires ten conditions (i.e., this set of CAs has only 3' members). Now it is much less obvious whether these molecules can survive a collision. This period-two molecule
may be compared with the period-two molecule in Section 10.3 that `naturally' occurred when k = 4, r = 1, and the cells are bistable. As noted above, the approach
in Section 10.3 was not from the rules, but rather the lattice maps. Whether the present approach warrants further exploration remains to be seen. Other studies involving `soliton-like' dynamics can be found in Aizawa and Nishikawa (1986) and Park, Steiglitz, and Thurston (1986).
482
Coupled maps and cellular automata
10.12 Two-dimensional CA While one-dimensional CA are interesting and important introductions to this form of dynamics, the original interest, stemming from von Neumann's problem, was in two-dimensional systems. There are also a number of other possible applications of two- and three-dimensional CA which are of interest. Examples of these are the formation of crystal structures, or specified patterns, the recurrences of patterns (as in fluid turbulence), and a variety of self-reproduction and evolutionary situations.
In what follows, we will only consider some very elementary aspects of twodimensional CAs, and refer the reader to the references for more advanced discussions in this area (e.g., Packard and Wolfram, 1985; Toffoli and Margolus, 1987). We begin by noting that the 'self-reproduction' of `machines', that is configurations in the cellular space (see `A general association for legal CA', Section 10.6), is very easy to accomplish, with the help of a suitable transition function, (10.4.3). Consider the
five cells of a two-dimensional CA, consisting of a center cell (C), and the T(op), B(ottom), L(eft), and R(ight) cells (Fig. 10.42). The dynamics (transition function) is Fig. 10.42
given by specifying C at the next time step, given the values of C, T, B, L, and R at the present time. Thus in this CA the range is r = 1, and the diagonal cells do not influence C at the next time step (this being part of the definition of this transition function). Two-dimensional neighborhoods of this type (without diagonal interactions) are known as von Neumann neighborhoods. The allowed states of the cells are taken to be s; = 0, 1, and the `reproducing' transition function is C(t + 1) = T(t) + B(t) + L(t) + R(t)
(mod 2)
(10.12.1)
In other words, if T + B + L + R = 0, 2, or 4, then C(t + 1) = 0, otherwise C(t + 1) = 1. Since this rule satisfies the quiescent condition, (10.5.1), and is rotationally invariant, it is a two-dimensional legal CA. It was discovered in the 1960s by E. Fredkin of MIT,
that this CA will reproduce any pattern ('machine') at the end of 2" steps, where n depends on the pattern. In fact it will produce four copies of the initial pattern. The four copies will be displaced 2" cells from the original pattern, which will vanish. This
is illustrated in Fig. 10.43 for several initial patterns. In the figure only a block of cells containing the excited states is shown, and the time steps are to the right. In the
last section, one more time step will produce the four copies of the original configuration.
Two-dimensional CA Fig. 10.43
0110 1101 0011 0010
11
01
10 1
0100 1100
483
001100 000100 110011 010001 001100 000100
00010000 001000 00110000 001100 01100000 100110 11000000 1 0 0 1 1 -01- 1 0 0 0 0 0 0 1 011001 00000011 001100 00000110 000100 00001100 00001000
1 -- 1 0 0 1 -- 1
01
0011
0010
Exercise 10.27 It should be noted that the number of copies of any self-reproducing configuration does not grow exponentially with time, because the population is limited by the available space (the Malthusian effect). Show that, if N(t) is the number of copies of a configuration at the time step t, then N(t) < Ltd, for some k states, where d is the dimension of cells. This result depends on the range, r in (10.4.3), being finite. Exercise 10.28 Show that there are 2" legal two-dimensional CA, which have k = 2, and von Neumann neighborhoods. It is clear that calling the above reproducing process 'self-reproduction' is inappropriate, because the latter implies that the `self' has something to do with the reproduction. However, in the above CA, all configurations are reproduced, so that
the reproduction process is entirely due to the transition function, (10.12.1). In this
sense, this type of 'self-reproduction' is trivial. To the contrary, the type of self-reproduction which von Neumann established is perhaps the most nontrivial type. He considered a particular CA, with a transition function which involved only the interaction of a cell with its four adjacent orthogonal cells (as in the last example), but in which the cells have 29 possible states. He showed that this CA has the property that, for any given Turing machine, there is an initial configuration of his CA which will perform the same sequence of steps as that Turing machine. Since, given any logical sequence of steps (an algorithm), there is a Turing machine which will perform these steps, there is `embedded' in von Neumann's CA this extremely general capability for performing all algorithms (for a recent popular account of Turing machines, see `Turing Machines' by J. E. Hopcroft, Sci. Amer. (May, 1984), p. 70). Von Neumann referred to his CA as being `logically universal'. Building on this result, it can then
484
Coupled maps and cellular automata
be shown that embedded in this logically universal CA is a `universal constructor', U. U has the property that, given any quiescent machine, M, and a tape, T (M), containing
a description of M, U can be activated by a tape, D(U), and will produce both M and T(M), and then will activate this machine. If M and T(M) are taken to be essentially U and D(U), then von Neumann's CA has embedded in it a universal constructor which is self-reproducing. This is obviously a very nontrivial form of self-reproduction - indeed, one can argue that this type of self-reproduction is much more nontrivial than what occurs in nature (see below). In any case, it is frequently required (by definition) that `nontrivial' self-reproducing automata must be Turing machines.
Since von Neumann's work on self-reproducing automata, there have been other studies to reduce the complexity of his CA. Codd reduced von Neumann's 29 state cells to 8 state cells by making use of a computer to search for transition functions with prescribed properties (a brief account of Codd's approach can be found in Burks' (1970) essays). Codd's CA, like von Neumann's, has the great generality of having embedded in it all Turing machines. There certainly can be doubts raised about whether the process of self-reproduction in nature occurs within some dynamical scheme which is as general as this universal form of dynamics. It seems likely (to some people at least) that it occurs within some much simpler dynamical context. The difficulty is to decide how we can (or how nature does) reduce the conditions which permit such complexity, without introducing the trivial examples, such as discussed above, where all configurations 'self'-reproduce. An exploration of this problem has been made by Langton (1984), who requires that the `construction of the copy should be actively directed by the configuration itself'. In other words, the `responsibility' for the reproduction should reside `primarily' with
the parent structure, but `not entirely'. While these conditions are rather vague, Langton modified Codd's CA to give a nice example of self-reproducing 'loop-withhandle', which is not a universal constructing CA, but yet whose reproduction is due `mainly' to the parent configuration (as one can see just by watching the process!). This very pretty dynamics is unfortunately too long to be described here, and one is referred to Langton's clear discussion (also see Langton, 1986). The problem of understanding reproduction in nature is, of course, not only the logical question addressed by most investigations using CA dynamics, such as von Neumann's, but also the physical question of how this process arises within the context of existing particles, radiation, force laws, etc. These issues are briefly discussed in Section 8.14.
10.13 Garden-of-Eden configurations The above reference to the activation of a quiescent machine avoids a feature discovered by E. F. Moore, which is true for many CAs, including von Neumann's.
Garden-of-Eden configurations
485
The feature is that there are configurations, called Garden-of-Eden configurations, which cannot arise from any other configuration, and hence can only occur as an initial state. Such configurations cannot therefore be self-reproducing. If, however, we restrict our considerations to configurations which are moreover activated, the reproduction of this combination does not suffer from the Garden-of-Eden initial value difficulty, and such combinations can always be reproduced. The necessary and sufficient condition for the existence of Garden-of-Eden configurations is that two configurations in some array of cells, C and C' (see Fig. 10.44), which are identical Fig. 10.44
C=
F
f
G
8
H
C' =
F*
--t
G
f 8
H
in regions G and H but differ in some interior region (F and F*), become identical over the indicated regions (f and g) in the next time step. Note that the configuration in the outer region (H) is not specified at the later time, because no information is specified about the configuration outside of this array of cells. Moore called two configurations which have this property `mutually erasable'.
To prove that mutually erasable configurations imply that the system has Garden-of-Eden configurations is not very difficult to establish, given Moore's clever idea, which we outline here (also see Moore's paper, and its converse, by J. Myhill, both in Burks' (1970) collection of essays). Consider an (n x n) array of cells (n:integer) and two configurations in this array, C. and C' (the use of square arrays is only for convenience). We define an equivalence class, E, of these configurations as follows
(Fig. 10.45): C. and C are in the same class E if C,, -
or if they are mutually
erasable. In the (n x n) array there are k"2 possible configurations. By assumption, we
are interested in the case where some C,, is mutually erasable, so there are at most (k"2 - 1) E-equivalence classes in this array.
Coupled maps and cellular automata
486 Fig. 10.45
n
C.
Fig. 10.46
E
N
C;,
n
n
mn
mn
n
E*
mn
mn
Next consider an array of (mn x mn) cells (mn:integers), and two configurations C and C' on this array (Fig. 10.46). We define another equivalence class E*: C and C' are E*-equivalent if each of the m2(n x n) subconfigurations in C is E-equivalent to the subconfigurations of C' in the correspondingly located (n x n) arrays. Since there are
at most (k"2 - 1) E-equivalence classes in an (n x n) array, there are at most (k"2
- 1)"'2 E*-equivalence classes in the (mn x mn) array. The purpose of introducing
the (mn x mn) array is that we now know that outside of the (n x n) arrays the configurations are identical in C and C' (except for those (n x n) arrays on the border - in which case, only two boundaries are known to be bordered by identical configuration). (See Fig. 10.47.) Fig. 10.47
Imo- mn - 2 --.]
mn
CorC'
mn
Thus, in the next time step, all of the H regions of the (n x n) arrays, which are `interior' (shaded region) must be identical. Hence, any E*-equivalent configurations, (C, C'), map to identical (mn - 2) x (mn - 2) configurations (if r = 1). Since there are 1)"'2 E*-equivalence classes in the original (mn x mn) array, there are at most (k"2 1)"'2 different `evolutionary' configurations in the (mn - 2) x (mn - 2) array. On (k"2 the other hand, in a (mn - 2) x (mn - 2) array there are k(m" - 2)2 possible configurations. Therefore, if (10.13.1) < k(mn-2)2, (k"2
-
-1)m2
H. Conway's `Life'
487
then there are fewer `evolutionary' configurations than there are possible configurations in the (mn - 2) x (mn - 2) array. In other words, if (10.13.1) is true, then there are configurations in the (mn - 2) x (mn - 2) array which could not have evolved from any (mn x mn) configuration of the previous time step. Such 'non-evolved' configurations are pecisely the Garden-of-Eden configurations. Finally we note that (10.13.1) can always be satisfied by selecting m to satisfy m > 4n/logk(k"2/(k"2 - 1)).
(10.13.2)
Thus, if C" and C are mutually erasable, then a Garden-of-Eden configuration can always be found, using a large enough value of m. This, of course, does not imply that Garden-of-Eden configurations must be that large. Exercise 10.29 Using the smallest possible n which can have an erasable configuration,
determine (for k = 2) the smallest possible (A x A) array which is known to have a Garden-of-Eden configuration. Also determine A for the case k = 3. Generalize (10.13.2), for arbitrary values of r. Garden-of-Eden (GoE) configurations are, of course, not limited to two-dimensional arrays. Thus, for the one-dimensional CA (k = 2, r = 1) with rules R = 204 (identity), R = 2 (shift-left), R = 16 (shift-right) there are no GoE configurations. However, for R = 18 (001/1, 100/1), any configuration with a (111) subconfiguration is GoE (as well as many others).
10.14 J. H. Conway's `Life' One of the most famous two-dimensional CA is the one invented by J. H. Conway, which he called `Life' (see Gardner, 1970; 1971; 1983; Berlekamp, Conway and Guy
1982, Chap. 25). The possible states are again simply s; = 0, 1, and the range of interaction is r = 1. However there is now an interaction between the center cell and all eight contiguous cells. This type of neighborhood is frequently called a Moore neighborhood (Fig. 10.48). The `life' transition function can be written in the form
f (Y-'siPsoo) Fig. 10.48 Moore neighborhood.
Coupled maps and cellular automata
488
where soo is the state of the center cell, and the sum is over the eight neighboring cells (not including the center cell). The function is then defined as follows
x=012345678 f(x,y)=00y 1 0 0 0 0 0
(10.14.1)
soo(t + 1) = f ( 'si,(t), soo(t).
(10.14.2)
and
To put this more picturesquely, the center cell `dies' (of loneliness) if it has one or no `live' neighbor, or (from overcrowding) if it has four or more neighbors. The cell is unchanged if there are two neighbors, and it definitely comes to life if there are exactly three live neighbors (rationalize that, if you can). This transition function was carefully selected by Conway by experimentation, to give interesting and nontrivial dynamics from simple initial configurations. He in fact accomplished much more than that, for
it has since been proved that in his CA are embedded all Turing machines - as in the case of von Neumann's CA. Thus Conway's CA, defined by the `simple' rules (10.14.1) and (10.14.2), has embedded in it this very general capacity to perform any algorithm, by starting it in a suitable initial state. While this is very impressive in the abstract, the interest in `Life' has come from discovering the interesting and varied behavior of many simple initial configurations
(the Turing machine algorithms generally require large initial configurations). Of course, some initial configurations just `die out', because there are not enough cells with two or three live neighbors. For example:
010
010
000
1 0 1
0 1 0
000.
When there are enough live cells, the system may tend to a stationary configuration, 11
_
10
11 1
(stationary).
1
Other stationary configurations are:
0110
1011011 1
1001
101 101
1001
and
01 10
(snakes of various lenghts)
(the pond)
Some initial configurations lead to oscillatory states,
000 1
1
010
000
1-010-1
000
010
(the blinker).
1
1
000
H. Conway's `Life'
489
Adding one more `live' cell to the last initial configuration, generates a stationary structure called a `beehive',
0000 11 1
0110
0110
0110
0110
1 -01 10-+ 1001
0000
(stationary beehive),
(10.14.3)
which is also produced from the initial states
100
10
11 01
111 000
and
(10. 14 . 4) '
Other examples of stationary states are
0100 1010 0101 0010 (the barge)
and all of its longer, and one shorter, generalizations (changed by adding or subtracting two live cells along the diagonal). The ship shapes, in which a bow and/or a stern is added to any barge (one live cell at higher end), are also stable configurations. However,
if a `live shell' lands next to, or inside the ships, the result is usually fatal (try it). Another interesting dynamical configuration is obtained by taking a stationary ship configuration (with both a bow and stern), and removing alternate `life' columns which do not contain the bow or stern, as illustrated in Fig. 10.49. The resulting configuration is periodic, with period two, and is called a barber pole. (The `stripes' in the figure have, of course, nothing to do with the dynamics.) Fig. 10.49
+ + t
i snip
aroer p
F
Coupled maps and cellular automata
490 Fig. 10.50 blinker
0
1 \1
0
0 o o
-10
l ei
I
I
ea ter
o
0 0
o 0 0
0
0
4 Th e e ate r a te!
BAR
Some stationary configurations have an `appetite' when placed near other configurations. An example is `the eater' which ingests a variety of other configurations. This is illustrated in Fig. 10.50 for the case of a `blinker'. In this figure, a 0 is put only where a live cell died in the last step, in order to assist in following the dynamics. Other examples can be found in Gardner's articles (1983). The `eater' however is not stable to many simple perturbations. Thus, placing a single live cell at (0, 0) (marked
by X in figure), or at (0, - 2), etc., will destroy the `eater'. If the above `blinker' is initially displaced down one cell (so that it is at (0, 0), (1, 0), and (2, 0)), then there is a `standoff'', and the stationary `beehive' configuration develops by step 11. Thus, while the `eater' may be fearsome, it does not represent some totally dominant configuration. Fig. 10.51
Excitable medium
491
Example 10.30 An interesting standoff occurs between two `eaters' which begin nearly nose-to-nose (see Fig. 10.51). This develops into a rare period-three oscillatory state, in which only three cells oscillate. You might like to determine this state, since it is established very fast. What happens if the two `eaters' are nose-to-nose? One of the most interesting simple configurations is the `glider', which in every four steps repeats the configuration, except that it is displaced one cell to the side and one cell vertically. This is illustrated in the figure
0100 0010 1
1
10
0000
_+
0000 0010
0000 1010
0000 0010
0000 0100
01 10
1010
001
1
0001
0100
0110
0110
0111
_+
(10.14.5)
The first indication that Conway's game of `Life' might contain a universal calculator
came with the discovery that this CA has a `glider gun', meaning a configuration which glides and periodically fires out the above glider. An example of such a configuration is illustrated in Fig. 10.52. This configuration fires its first glider at Fig.10.52
0000000000000000000000000000000000
000000000000000000000000000000000000 0.00000000.0000000000000000000000000 9000000000.0000000000000000900000000 00000000000.000000000000000000000000 000000000000000000000000.00000000000
step 40, and the gun is an oscillator which fires a new glider every 30 steps. The importance of such a glider gun, from the point of view of computations, is that it is capable of transmitting information to distant locations. These topics, however, are outside of the scope of this study. Many other examples and variations of Conway's `Life', as well as other CA games have been collected by Poundstone (1985) and Gardner (1983).
10.15 Excitable medium In Section 10.3 we considered a CLM in which the cells have three states (say 0, 1, 2), where 0 is a fixed point of the lattice map. Moreover, the CLM yielded the sequence 1-* 2 -> 0, so the cell becomes quiescent ('dead') in at least two iterations. We saw
that if these cells are diffusively coupled, they can produce interesting dynamics, because a cell can be excited out of its quiescent state by its neighbors. There are many neurological, biological, and chemical situations which can be modeled (more
Coupled maps and cellular automata
492
or less realistically) by a two-dimensional array of such `excitable' cells. Many of these are nicely discussed and further referenced in Winfree (1987).
In the neurological context the three states of the cells are frequently referred to as quiescent (q), excited (e), and refractory (r). The uncoupled tendency of a cell is e -> r q (fixed state), which is the same as above. The coupling with neighboring cells is somewhat different from the diffusive coupling of Section 10.3. The rules are e
r -q;
q->e
if any neighbor is e
(10.15.1)
where the arrows in (10.15.1) represent the next time step. To show this dynamics graphically, it is useful to use the notation in each cell q = blank,
r = 1.
e = 0,
The use of a blank in the quiescent cell saves considerable writing. It is also nice if you have hexagon cells, so that waves can move outward in a near-circular pattern, as in nature (Fig. 10.53). We will, however, simply use square cells, but use Moore Fig. 10.53
Fig. 10.54 0 0 0 0 0
000 0
00
1
0
0 0 0 0 0
0
1
111
00
1
0
1
1
0
1
0
1
1
1
1
1
1
0 0
1
0
1
0 0
1 01 01 01 01 01
1
neighborhoods (so cells which touch only at a point are nonetheless affected). To illustrate, consider only two excited cells (Fig. 10.54). It can be readily seen that this will yield outward propagating waves (square wave fronts!), in a rhythmic pattern, until some boundary effects disrupt the simplicity of the pattern. To make such models more realistic to such systems as electrical waves in heart muscles and other neurological nets, anisotropic or inhomogeneous effects in the duration of the refractory state have been considered. As a trivial example, if the
Invertible CA and physical dynamics
493
refractory state held for two iterations, the above spatial pattern would be altered modestly. But if different refractory periods held in different regions, or excitations occurred preferentially in one direction, much more profound and interesting effects will result (see references and discussions in Winfree (1987)).
10.16 Invertible CA and physical dynamics The CA `Life' belongs to a general class of CA which can map two or more configurations into the same configuration. In other words, the parallel map, (10.4.6), which describes the behavior of the entire system, maps two different configurations C = {s; } and C' _ s' j to the same configuration C' = {s'. }, C'
C -> C",
C".
(10.16.1)
A simple example of this is obtained from the initial states (10.14.4) which both ultimately lead to the same `beehive' configuration in (10.14.3). In this case, the middle
configuration of (10.14.3) corresponds to
{s'.
}
in (10.16.1), which has at least two
preceding configurations (namely the first configuration in both (10.14.3) and (10.14.4)).
Because of this behavior, it is clear that `Life' has at least two `mutually erasable' configurations, and therefore must have a Garden-of-Eden configuration, as discussed above. However, the smallest such configuration discovered to date contains around 300 live cells (s, = 1) (see Berlekamp, Conway, and Guy, 1982, Chap. 25), and therefore does not appear to be a concern in the limited configurations normally considered. If a CA has a one-to-one (bijective) parallel map, (10.4.6), we will call it an invertible CA. Unfortunately these CA have also frequently been referred to as `reversible' CA.
This terminology is unfortunate, because it may be confused with the historical concept of reversibility in particle dynamics. An invertible CA simply means that the mapping of configurations, C - C', is always one-to-one (a unique correspondence).
Such a uniqueness is typically taken for granted in dynamics, but nonetheless the system may not be dynamically reversible. Thus, for example, all solutions of dx/dt = v;
dv/dt = - v
are unique, but none are dynamically reversible. By dynamic reversibility we mean that, if we take a movie of the configurations, x(t; x0, vo), of a system of particles, there exists another dynamic solution (new initial conditions (x' , v' )) whose configurations behave the same as seen in the movie when it is run in reverse. Put more formally, the equations of motion are invariant under the interchange (v, dt) to (- v, - dt) (outside currents, producing magnetic fields, must also be reversed). Thus invertibility of a CA configuration map does not mean the same (nor imply as much) as does reversibility in the case of particle dynamics. In the case of a CA, invertibility simply means uniqueness of the dynamics.
494
Coupled maps and cellular automata
An important property of invertible CA is that all of the information about the initial configuration is retained for all time, so there are as many `constants of the motion', or conserved quantities, as there are cells in the system (recall the analogous feature with systems of ordinary differential equations, discussed in Chapter 2). Many of these constants are uninteresting, since only the features of the `live' cells are usually of interest. In any case, an invertible CA clearly has the largest possible number of conserved quantities, and in this sense is most analogous to fundamental dynamic theories. As mentioned before, an important attribute of all CAs is that their dynamics is exact - not being subject to the roundoff errors which occur in numerical solutions
of differential equations. Thus, if a CA is invertible, all conserved properties are exactly conserved. This means that we can examine the exact significance of various
conserved quantities on the resulting dynamics, provided that we can obtain an invertible CA with the desired conserved quantity. The first problem, therefore, is to obtain invertible CA. If we write down an arbitrary transition function, (10.4.3), satisfying (10.5.1) and (10.5.2), for a given number of k
states in each cell, then in most cases the CA will be noninvertible (many-to-one configuration map). However, if for this same transition function F({sj}), we consider another mapping (apparently due to E. Fredkin; Margolus (1984)) defined by
s.(t+1)=F({s;(t)})-s,(t-1)
(mod k)
(10.16.2)
then the configuration {s;(t + 1)} is determined by the two configurations {s;(t)} and {s;(t - 1)}. On the other hand, the configuration {s3(t - 1)} can be determined from the two configurations (s#)) and { s,(t + 1) } because (10.16.2) can be written in the form
s.(t-1)=F({s3(t)})-s,(t+1)
(mod k).
Therefore the mapping (10.16.2) is invertible, but it is not of the form (10.4.3) because it involves two time steps.
This difficulty can be overcome if we next consider a CA which has k2 states in each cell, rather than the original k states. The k2 states are taken to be all of the possible pairs of state values (s;(t), s;(t - 1)), numbered in any convenient manner. The cells can now be independently assigned any of these k2 states initially, since s.(t) is no longer determined by the set {s;(t)}, because of (10.16.2). If we denote the new states in the cell i by q;(t), so that
qi=0,1,2,...,k2, then, since the pair (s;(t + 1), s.(t)) (or, in other words, q.(t + 1)) can be determined from the set of values q;(t) = (s;(t), sj(t - 1)), it must be possible to define a transition function, G( {q;(t) } ), such that
q;(t + 1) = G({q;(t)}).
(10.16.3)
495
Invertible CA and physical dynamics
This is, of course, in the standard form (10.4.3). Therefore, at the expense of squaring
the number of states in a cell, it is possible to associate with every CA another invertible CA, (10.16.3).
Exercise 10.31 We want to use the above method to obtain the invertible CA associated with the game of `Life', (10.14.1) and (10.14.2). Let the four states of a cell, qj, be labeled according to the following scheme: q,(t) = if (s,(t), s.(t - 1)) =
0
1
2
3
(0,0)
(0,1)
(1,0)
(1,1)
Using the definitions (10.14.1), (10.14.2), and (10.16.2), determine the mapping properties of the transition function G in (10.16.3). In particular, determine the dynamics of the initial configuration 3 z
It should be noted that, while there is a type of time reversibility to the states {s;} (using two configurations), there is no such time reversibility in the invertible CA dynamics, (10.16.3). More specifically, consider interchanging two s configurations, so that the new configurations, {s*}, have the `initial conditions' {s*(0)} = {s1(N)}
and {s*(- 1)} = {s(N+ 1)}, where N is an arbitrary time step. Note that two C* configurations are being given. Then, using (10.12.2), we find that {s*(t)} = {s(N - t)},
so that the s configurations simply retrace the past history of the states {s;(t)}. But the s configurations are not the invertible CA states, for these are the q configurations. These go through the sequence q*(t) = (s;(N - t), s;(N - t + 1)), and this does not equal q,(N - t). Therefore invertible CA do not have this time reversal propety. What does exist, however, is a CA rule which will take any configuration {q,(N)} to its (unique) predecessors {q;(N - t)}. There are, of course, no Garden-of-Eden configurations for invertible CA. The fact that a mapping of the form (10.12.2) is invertible is true even if the function F({sj}) is many-to-one. Fredkin (see, Vichniac, 1984a) pointed out that this means that numerical roundoff in computers does not necessarily mean that there is a loss of information. In particular, if the computation is of the form (10.16.2), where the function F({s;}) produces a roundoff, the dynamics will still be invertible, so that no initial information is lost. This rather surprising fact is a consequence of the two time steps in (10.16.2).
For example, consider the second-order Newton's equations M
d 2x
= F(x)
which we approximate by xn + 1 = { (h2/m)F(xn) + 2xn} - xn _ 1
(10.16.4)
Coupled maps and cellular automata
496
where the bracket indicates any numerical roundoff (finite precision arithmetic). Then
(10.16.4) is of the form (10.16.2), and the computation is reversible. Note that the subtraction, on the right side of (10.16.4), is exact in fixed-point arithmetic. This single exact use of the `oldest information', in generating xn+, is what retains the information.
Exercise 10.32 To make this quite clear take, for example, zl
Xn+1=1XnI-Xn-1 J)
where the roundoff is to the nearest integer. Start with (x0 = 1, x, = 2.5) and show that x3 = 22.5, x4 = 501, from which xo and x, can be recovered. In addition to the interest in the dynamics of CA per se, these systems are receiving additional attention as a possible source of a fundamental, rather than an approximate,
description of nature. Thus, instead of the approximate connections (indicated by wavey arrows) discrete space-time: theory: nature --+ differential ---+ finite difference equations equations
project variables onto a finite number of states (e.g., the roundoff of a computer)
cellular automata models
perhaps the best description of nature would be
nature -
theory: cellular automata.
(10.16.5)
That is, perhaps nature is best described (exactly) by CA, rather than by differential equations. Differential equations, after all, imply an infinite amount of information in any arbitrarily small volume of space-time. As Feynman expressed it (1967): It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed,
and the laws will turn out to be simple, like the chequer board with all its apparent complexities. But this speculation is of the same nature as those other people make -'I like it', `I don't like it', - and it is not good to be too prejudiced about these things.
Invertible CA and physical dynamics
497
It has been recognized for a long time that `atomistic' (discrete) space-time descriptions of nature avoid such classic continuity `difficulties' as Zeno's Achilles and the Tortoise paradox in a very simple fashion (there are no limit problems, since limits simply do not exist!). It is interesting that the discrete character of time, which
is perhaps the most `daring' concept according to our present prejudices, was propounded by philosophers in the Middle Ages, notably by the Jewish philosopher Maimonides. He wrote (in Arabic) that `Time is composed of time-atoms, i.e. of many parts, which on account of their short duration cannot be divided...' (duration?), and he went on to conclude that there were at least 6010 such time-atoms in one hour (for a nice discussion of various characteristics of time, see G. J. Whitrow, The Natural Philosophy of Time, Oxford Univ. Press, 1980). The possibility of the connection (10.16.5) has been considered more recently by several people (Fredkin, 1982; Feynman, 1982; Toffoli, 1982, 1984; Vichniac, 1984). Of course if space-time is discrete, then the scale must be very small to agree with existing experimental observations. (The present empirical division of a second, which is better than one part in 3 x 1012, is coming close to Maimonides' division of one part in 2 x 1014.) As Feynman noted, for example, a discrete structure in space-time would imply an anisotropic speed of light, which can be accepted in principle, provided it is not too large. It is also interesting to note that, while CA uses different (namely discrete) primitives than differential equations, there is no mathematical reason why they cannot have the same expressive power (Toffoli, 1984). By this is meant that the collection (set) of all configurations {si } in an infinite CA, has the same cardinality as the continuum, which is of course the cardinality of the functional values in a differential equation. In other words, the set of all configurations (the CA `phase space') is as capable of as much refined description as are the functions in differential equations. This cardinality fact is easily established by using a standard Cantor method. We can first count (order) the cells in space. For example, if the space is two-dimensional, we could count out from some cell in successively larger rings and order the cells clockwise in each ring, thereby obtaining a cell number, i. The ordering is, of course, even simpler in one dimension. The configuration of the system is the set {si}, which
can then be written in an ordered form (s1,s2,...). To show that all of these configuration sets cannot be ordered (counted), we simply assume the contrary, and write down a hypothetical ordering. {Si}1 =(0,0,0,...);
{Si}2 =(1,1,0,...);
{si}3 =(0,1,0,...)
and so on (somehow). Then we construct a new configuration, {si}*, which consists
of an entry in the jth location (cell) which differs from that found in the jth configuration, {si};. Thus, in the above case, the new configuration begins
{si}* =(1,0,1,...).
498
Coupled maps and cellular automata
The configuration {s;}* does not belong to any of the above ordered configurations, by construction, hence all configurations are not in the above hypothetical ordering of configurations. Therefore the set of all configurations cannot be counted and, in fact, has the cardinality of the continuum. This result means that we can introduce a concept of configurational continuity into the CA dynamics, as exists in differential equations, but without requiring an infinite amount of information in any finite volume of space-time (see Toffoli, 1984). What has been accomplished, of course, is to introduce
the same (uncountable) amount of information in an infinite CA tessellated space as differential equations squeeze into an infinitesimal configurational space. If we reject the empirical importance of infinite amounts of information (a la Feynman), then this accomplishment of infinite CA is not really of empirical importance. Put another way, any empirical importance of CA dynamics should be found in finite CA (tessellated space). This reverts back to the discussion in Chapter 4, Section 11, which dealt with some other issues concerning the finite nature of empirical science uis-a-uis the infinite nature of mathematics. As interesting as these considerations are, there appears to be serious problems in the use of CA for some quantum mechanical observations. Feynman (1982) gives
an analysis of the two-photon correlation experiment, from which he concludes that quantum mechanics apparently cannot always be imitated by the usual locally connected classical computer. (He also considered a probabilistic computer, with other negative results - in several senses.) Certainly the resolution of this difficulty is required before we can seriously consider that CA may be the most fundamental way to describe natural phenomena. On a more specific, and less grandiose level, we close this discussion of cellular automata with the following questions of Toffoli (1977), which are concerned with the future clarification of the relationship between cellular automata and the physical world:
(1) What physical-like constraints (in addition to those already implicit in their definition) must cellular automata obey to be consistent with their intended role as models of the computational aspects of nature? (2) What correspondence should be used for associating objects and phenomena in cellular automata with objects and phenomena in nature? For example, what counterparts, if any, are to be found in cellular automata for energy, momentum, gravitation, etc.? (3) To what extent can the general mathematical methods of physics (e.g., harmonic
analysis, calculus of variations, etc.), which were originally conceived for continuous problems, be applied to cellular automata? More recently, Wolfram (1985), has proposed a list of twenty problems concerning cellular automata.
Appendix
499
Appendix The specific questions raised by von Neumann in his work `The Theory of Automata:
Construction, Reproduction, Homogeneity' (begun in 1952, and left in manuscript form at his death) are contained in the following excerpt:
We will investigate automata under two important, and connected, aspects: those of logics and of construction. We can organize our considerations under the headings of five main questions: (A) Logical universality. When is a class of automata logically universal, i.e., able to perform all those logical operations that are at all performable with finite (but arbitrarily extensive) means? Also, with what additional - variable, but in
the essential respects standard - attachments is a single automaton logically universal?
(B) Constructibility. Can an automaton be constructed, i.e., assembled and built from appropriately defined `raw materials', by another automaton? Or, starting from the other end and extending the question, what class of automata
can be constructed by one, suitable given, automaton? The variable, but essentially standard, attachments to the latter, in the sense of the second question of (A), may here be permitted.
(C) Construction-universality. Making the second question of (B) more specific, can any one, suitable given, automaton be construction-universal, i.e., be able to construct in the sense of question (B) (with suitable, but essentially standard, attachments) every other automaton? (D) Self-reproduction. Narrowing question (C), can any automation construct other automata that are exactly like it? Can it be made, in addition, to perform further tasks, e.g., also construct certain other, prescribed automata?
(E) Evolution. Combining questions (C) and (D), can the construction of automata by automata progress from simpler types to increasingly complicated types? Also, assuming some suitable definition of `efficiency', can this evolution go from less efficient to more efficient automata?
Question (E) is one of the basic remaining problems in this area of research. The commentaries about von Neumann's work, by Burks, are highly recommended.
Comments on the exercises (10.1) `Patterns' are associated with spatial (configurational) correlations. We typically do not study the correlations between the configurational components
of the general Lagrangian variable x(t)ER'. For example, in a classical Lagrangian system, x(t) - {q(t), q(t) } ERZ". Patterns in this context means
Coupled maps and cellular automata
500
some type of correlation between various qk(t) (k = 1, ... , n). This is rarely considered. We, of course, do consider flow patterns in phase space, but that is quite different from the Eulerian spatial patterns. (10.2) Let g(x; x', x") - cx(1 - x) + D(x' + x" - 2x). A sufficient condition for all solutions to be bounded is that 0 < g(x; x', x") < 1 for all x, x', and x" in the interval [0, 1]. Now g(x; x', x") < g(x; 1, 1), and g(x; 1, 1) is maximum at xo = 1- D/c, if D < c/2. Hence, if c > 2D (and since D < 2), g(x; x', x") g(xo;1,1) = c[(D/c)2 + (D/c) + (4)]. Thus g(x; x', x") < 1 if D < c112 - (c/2) (when c > 1). If c < 2D the maximum of g(x; 1, 1) is at x = 0, so g(x; x', x") <, 1 for all D < Z. The other condition is that g(x; x', x") >, 0. We have g(x; x', x") g(x; 0, 0) = (c - 2D)x - cx2. The minimum value of g(x; 0, 0) occurs at x = 1,
and g(1; 0, 0) = - 2D. Hence, this analysis shows there is no lower positive bound for g(x; x', x"). A more profound analysis might examine two or more iterates, with the corresponding larger set of variables. Explore. 2D) + 2D. If f(x) ,<1 and D,< z, then x1(i) ,<1. (10.3) x.+ 1(i) ,, f 2D) which is not negative under the same conditions. Hence, if all 1 > xo(i) 3 0, then 1 > 0 for all i and n. (10.4) There are an infinite number of isochronal regions, I j, which become dense as x -+0 or x 1. The points bi map to the unstable fixed point of f 2(x) (i.e., the fixed point of f (x)). (10.5) The pattern scales for D = 0, roughly scales for D = 0.1, but does not scale for D = 3. You will have to determine the D = 1 pattern. (10.6) Such regularity requires investigation. Why are the regions the size that they are? Does this size depend on initial conditions? In this periodicity and region size retained if N is varied? Try N = 47 (a prime number), and either the same
initial conditions, or random initial conditions. You will find that some pattern features are retained, but more dynamic features occur (and are more `generic').
(10.7) Clearly, if D is decreased, the periodic motion must be replaced by more chaotic motion. As D is decreased, does one or another of these regions go chaotic first? Does the size of these regions change? As D is decreased, it clearly will take
more iterations to reach any `asymptotic' behavior. If we use only a personal computer, some conscious decision must be made (and recognized) when we say `Enough!'. This returns to the finite aspects of empirical sciences (Section 4.11). Also see Section 10.16 below. (10.8) Explore! The case D1 = 0 means there is no action of cell 1 onto cell 1. This is equivalent to a nonautonomous system. You can find interesting bifurcations. What generalization of these bifurcations does this suggest? Clearly, other generalizations to larger cellular systems would also be interesting.
Appendix
(10.9) Curve
D= 2- (3/M)
D=3/(M+1) D = 3/2M
D= 1- (3/2M)
D=1/M
Si
M 0 0 M 0
Si 1 1
M 0 0
Sk
501
b
0 M M 0
1.5
M
0.5
1.5 1.5 1.5
(10.10) Bifurcations can only occur if x', given by (10.3.1) equals one of the boundaries, Thus possible bifurcation values of D are when b,, when b; = (1 - D)s, + (D/2)(s, + or
subject to the condition 1 > D > 0. There are (k - 1) values b 9 0; k values of We must discount the case where all s3 = 0, s; and k(k - 1)/2 values of (s, + so there are not more than (k - 1)[k2(k - 1)/2 - 1] values of D. For those who like that sort of thing, obtain the actual value of L. (10.11) There are k" possible ways k states can occur in N cells, so this is the largest possible period. 350 7.17 x 1023 (large!). Of course, it rarely (ever?) happens that these systems are ergodic, and periods are generally much shorter than k" (e.g., see the lattice CAT map, Chapter 6). Very long periods (how long?) must be permitted in the definition of `turbulence' for finite CA. It should be
noted moreover that not all initial states necessarily recur. There is not a Poincare recurrence theorem for CA dynamics. See the discussion of Garden of Eden configurations. (10.12) Period-one: (0.2, 0.9) or (0.2, 0.9, 0.9) Period-two: (0.2, 0.9, 0.9, 0.9, 0.9) (s3, s2) Period-three: (s3, s2)
Does this result raise questions (correlations?) or suggest other possibilities? Try them with your program. (10.13) D - 0.814; at b2 = 0.66, sj = 0.9, s, = 0.3, s = 0.9 in Exercise 10.10 (10.14) Only the industrious reader will learn the answer! However you might try initial states (0.6, 0.9, 0.1) and/or (0.1, 0.9, 0.6) in a `sea' of x0(i) = 0.2 cells. (10.15) You should find the dynamics indicated in the exercise. D = 0.35 has localized period-ten and period-fourteen configurations. (10.16) (a) There are (2r + 1) cells in a neighborhood, hence k12r+') neighborhoods.
Coupled maps and cellular automata
502
For general CA, each neighborhood can result in k possible values for the central state in the next time step, yielding kkj2' " possible rules. However, there are k''+ 1) symmetric neighborhoods and hence [k12i+ 1) - k"'+')] asymmetric neighborhoods. Because of (10.5.2), the asymmetric neighborhoods are required
to have the same rule, so the dynamically distinct neighborhoods number V+1)+z[k(2r+1)-k'r+1)]=zk(r+1)(k'+1).
In addition, (10.5.1), fixes the dynamic condition for the quiescent neighborhood, leaving one less neighborhood. Thus there are kk'r+ "(k'+ 1)12 -1 legal CA rules.
(b) In general, for k = 2, r = 1, the rules are permutationally equivalent if ci = c*_ i
(i = 0,..., 7), where c* = (c + 1) mod 2 is the permutation operation.
However legal CA must satisfy co = 0, c1 = c4, and c3 = c6. Hence we are free to
pick (c1,c2), for example, and the rest are determined. This yields four equivalent pairs, one of which is (0, 0, 1, 0, 0, 1, 0, 1) - (0, 1, 0, 1, 1, 0, 1, 1). The
equivalent rule numbers, R (see Section 10.8), are: 164 - 218; 132 - 222, 160 - 250, and 128 - 254. (10.17) For illegal CA, 000/4, 001/4, and 004/0. For legal CA; 0.2, 0.2, 0.2/0.3, 0.2, 0.2, 0.3/0.3, and 0.2, 0.2, 0.9/0.9. (10.18) The first 20 iterations are illustrated in Fig. 10.55 (rule R = 150) Fig. 10.55
(10.19) The transition (10.7.2) can be written
s,(t+ 1)=(s1_1(t)-si+1(t))2 whereas the one in Exercise 10.18 can be written
si(t+1)={[s1_,(t)-s1+1(t)]2-si(t)}2, or also as a third-order polynomial. (10.20) Fig. 10.56 illustrates the first 28 iterations.
(10.21) The value of R for the dynamics in Exercise 10.18 is obtained from CO=C2=C4= 1, and c, =C3=01 so R=2 4 +2 2 + 20 = 21. For all totalistic cases with k = 2, r = 1, c3 = c, and c2 = c4, so there are eight possible values of R, corresponding to the two values of c4, c3 = c1, and c2 = c4. These are
Appendix
503
Fig. 10.56
16c4 + 10c3 + 5c2 = 31, 26, 21, 16, 15, 10, 5, and 0. If k = 2, r = 2, the sum is either 5, 4, 3, 2, or 1, so there are 25 totalistic CA. (10.22) Code 20 corresponds to f(2) =f(4) = 1, and the rest zero. The first initial state has period-two, while the second has period one (stationary). The third, fourth, and fifth initial states lead respectively to the patterns in Fig. 10.57. Fig. 10.57
(10.23) There are two counterdrifting cases, depending on whether they are separated by an even or odd number of 0 cells. Thus, if at t = 0 10011110110001101111001
this leads to a quiescent state at t = 33. If there are initially four 0 cells between these patterns the dynamics becomes periodic (period 22) at t = 18 (see the left figure in Fig. 10.57). (10.24) R = 204 corresponds to the conditions 111/1, 110/1, 101/0, 100/0, 011/1, 010/1, 001/1 and 000/0. This rule reproduces any configuration (the identity rule). Since all initial configurations are reproduced, N(X) = kX and p;" = k - X, so st(X) = 1. On the other hand, in any cell the number of sequences in an interval T is simply N(T) = 1 (either s = 0 or s = 1), and therefore st(T) = 0. Since these are two possible such sequences, p;`° = (i = 1, 2) and therefore i sµ(T) = 1/T (10.25) (10.10.16) yields N(t + 1)/N(t) = 1 + [N(t - 1)/N(t)]. Let R(t) = N(t + 1)/N(t). For large t we have R(t) 1 + R-'(t), so R(t) - (1 + 51/2)/2, which yields the asymptotic result N(t) - [(1 + 51/2)/2]`.
Coupled maps and cellular automata
504
(10.26) Oscillator conditions interchange (21) and (12), such as 001/2, 012/1, 021/0, 002/0, or the complementary conditions 1+-+2. This CA (k = 3, r = 1) is too limited to have both an oscillator and autonomous particles ('photon') which can move directly away from it. However, it can `radiate' like an oscillating paddle in water, continually sending out a disturbance. (10.27) Let the volume of the smallest d-cube to contain all of the initial configuration (s; : 0) be De, and let n be the number of cells for which s; : 0. After a time t all s; o 0 are contained in a cube of volume (D + 2rt)', because the edges can only expand by 2r each time step. Thus the number of copies cannot exceed (D + 2rt)dm". Hence there is a k (e.g., (2r + D)dl") such that N(t) < dtd for all t. (10.28) No comment. (10.29) The smallest n is n = 5 (r = 1). Evaluate m from (10.13.2) and use A = mn. (10.13.1) becomes (k"'- 1)'"2 < k''°" - 202, which is satisfied if logk [(kn2 - 1)/k"2] < 4r2/m2 - 4rn/m. This is certainly satisfied if m > 4rn/
- 1)]. (10.30) The oscillatory state involves stationary live cells at (- 4, - 4), (- 4, - 3), (- 3, - 3), (- 2, - 3)(- 2, - 1), (- 1, - 2) and its `mirror group' (3, 4), 1ogk[k"2/(k"2
(3, 3)(2, 3), (1, 3), (1,1), (0, 2), and three oscillating cells (- 2,0), (0, 2) and (- 1, 0)
which repeat with period 3. If they are nose-to-nose, nothing happens. (10.31) The function G of (10.16.3) has the following effects on the central cell, depending on its adjacent cells.
No. adjacent
change of central cell
cells
with states
all
0 and/or 1
1
2or3
0--> 0 1-->2
2and/or3
0->0
2
2
same as first case
The dynamics is 12
03
03
21
12
32
13
23
31
32
(10.32) No comment.
3-->3
3
3
2 and/or 3
2--1
(periodic)
Epilogue: `Understanding' complex systems
Life is complicated, so it is not surprising that most dynamic systems are also `complicated'. One of the challenges of the future is to obtain some level of 'understanding' of these dynamic complications. This understanding hopefully will involve, for limited groups of systems, the identification of categories of behavior of observable properties, which can be qualitatively classified and/or quantitatively measured. This process has hardly begun and what follows should certainly be approached with a spirit of adventure; value judgements should be developed only after a considerable
period of personal reflection. Above all, the reader should be very cautious about accepting pronouncements made in the existing literature, even when the authors are world-renowned for other research. The world is full of inflated egos which result in pontifications that are not supported by solid knowledge. We all might benefit, before we look at details, by reflecting with Newton:
I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary whilst the great ocean of truth lay all undiscovered before me.
The following analysis suggests that any `understanding' first requires the explicit recognition that there are various `modes of complexity,' which depend not only on the physical systems, but also on our methods of representing, observing and interacting with these systems. In this spirit, we need first to distinguish clearly (Fig. El) between the physical systems (PS) which we study, and the models (M) which we develop in
order to `capture' some observable dynamic aspects of the PS (see Footnote at the end for more details concerning `models'). Frequently `models' are referred to as `theories', particularly when people believe that they are getting a more `fundamental' description of nature, but we will retain the term `model' (as in the figure). It is an appropriately humble word, retaining some of Newton's spirit. Also, it should be
emphasized that a PS may have many models, such as MI and M2, which are
Epilogue
506 Fig. El
atkFK(xt,...,x4)
(K=1,...,4)
x1
N(t + l) = CN(t)(l - N(t)) au at
3
+ ax3 =Q
S1(t+ 1)=F(S_I(t),S;(t),S1+1(t))
Dynamics in the real world
This process contains the physical insight ('artistry') of the theorist in attempting to describe real phenomena
Nonlinear dynamics in the phase space of the physical variables
associated with different dynamic facets (e.g., phase transitions, electrical conductivity, sound waves, etc.) Moreover, a model M 1 might be more fundamental than M3, in
the sense that M 1 can account for M3 and other additional dynamic features (e.g., the kinetic theory of gas accounting for sound waves, but also for their dispersion, etc.).
Models may also be viewed as more fundamental when they describe dynamic properties of a variety of systems (e.g., M4); thus Newton's `model' of gravitation applies to planets, comets, a cannonball, or a falling apple, etc. This is a big topic, and we'll return to it later. An important point to note, however, is that complicated systems are sometimes felt to be `understood' when the data obtained by experimental investigations (e.g.,
`Understanding' complex systems
507
the empirical sets {E1;}, {E2;}), can be used to obtain some characterization (e.g., parameters, polynomial exponents, etc.) which is believed to be relatively clear, even when no explicit model is known. Historically, such data were considered to yield some `understanding' of the PS only when they were coupled with (embedded in) a
dynamic mathematical model. Models, however, are analytically oriented (see Footnote), whereas at least some `understanding' may fall within qualitative (e.g., topological, scaling, etc.) frameworks. Indeed, with the studies of increasingly complicated dynamics, this direct empirical `understanding' has found increased usage. We shall discuss these various aspects of `understanding' in what follows. There are other classes of models (e.g., many-particle Hamiltonian equations, or the many-particle Schrodinger equation; where `many' is >, 3); these are metaphysical in character ('metamodels', MM) in the sense that they cannot be precisely related, either mathematically or empirically, to observed properties of most PS (Fig. E2). In Fig. E2
order to obtain some predictable properties of a PS, the MMs must first be `approximated' to obtain a conjectured model, M, which can then be used to predict
observable phenomenon. For most models, M;, of complex systems, no such approximation of any MM is known (e.g., M3, M4, and M6). It is a matter of faith that any MM could account for the phenomena of most physical systems, much less even account for their simpler models. Nonetheless, many scientists believe that such MMs are the most fundamental description of nature, despite their total nonempirical character. However, a growing number of even `hard' scientists are beginning to appreciate that there are `emergent behaviors' of complex systems which will never be derivable or predictable from a `unified fundamental theory' and that one's efforts are better spent on uncovering and defining limited aspects of complexity. Indeed, a very convincing analysis has been made by Popper (1982), supporting the idea that scientific determinism (which he carefully defines) is baseless. That is, certainly at the level of so-called unified fundamental theories of many-particle systems (more than two!), there not only exists no empirical evidence for such a philosophy, but he points out both logical and metaphysical arguments against scientific determinism. Those interested in the foundations of scientific descriptions of complex systems should certainly become familiar with Popper's incisive analyses.
508
Epilogue
Over a century ago, Boltzmann recognized that large numbers of particles can yield a form of complication which can be put to statistical advantage, despite the ensuing controversies. Poincare, however, was the first to obtain a precise insight into how even `simple' three-body systems can be incredibly sensitive to initial conditions. This complexity was abstracted and expanded upon by Birkhoff, and then `brought back' to physical systems (models!) by Cartwright and Littlewood and by Levinson. Since then (- 1950), many scientists and mathematicians have clarified the character of these forms of complexity in the temporal domain. At present, there is `widespread' agreement that the term `chaos' should be used to refer to dynamics which exhibit sensitivity-to-initial-conditions, at least for selected values of the control
parameters and initial conditions. This chaos can be quantified with the help of Lyapunov exponents. Thus, `chaos' is associated with the exponential temporal divergence of nearby solutions of models, and the associated temporal unpredictability
of most solutions. This unpredictability of the model solutions has profound implications on our `understanding' of the associated PS because a PS is never closed to the environment, as are many of these models (Section 10.1). Thus, the ever-present perturbations of the environment continually `scramble' the solutions of such chaotic models (e.g., making such concepts as the Poincare recurrence time physically quite meaningless). There are, of course, many known variations of this concept of chaos (e.g., intermittency, transient chaos, attractive and nonattractive varieties, bifurcation features, etc.), so that at present this is one of the better understood areas of complicated dynamics.
In the case of space-time dynamics, the nearest analogy to `chaos' is frequently referred to as `turbulence', particularly in fluids (including chemically reactive fluids). However, there are other spatially extended systems, such as biological systems or CA models, where the terminology `turbulent' is either not used, or ill-defined. Thus, autocorrelation functions, , of some variable v(x, t), taken over some ensemble of initial states, or averaged over space and time, have long been used to characterize these complications. In the case of CA dynamics, various 'entropies' have been proposed to describe the character of these complications (section 10.10),
with limited success. (Anytime one comes across the word `entropy' in this area of research, proceed with extreme caution! The `entropy' of `entropy' is large!). Some careful studies of space-time complexities have been made, particularly in the area of CA models (Oono and Yeung, 1987), but much remains to be done for other models of physical systems. However, we will proceed to consider the broader aspects of dynamics 'complications', and how the above examples fit within a more general picture. Many authors (see references under Related Topics) have turned to these considerations with varying degrees of skill and care, and some blending and generalizations of their ideas will now be attempted. One broad distinction which is to be kept in mind is between
509
Epilogue Fig. E3
Open adaptive model
closed, open passive, and open adaptive models, as has been outlined in section 10.1 and also in the Footnote of this section (Fig. E3). In considering the following variety of ideas concerning complexity, it is useful to reflect occasionally about how they
may (or may not) apply to these various types of models; for brevity, this will be done explicitly on only rare occasions. First of all, it is useful to distinguish between two forms of complexity, which we will refer to as Order and Organization. The importance of such a distinction has been particularly emphasized by Denbigh (1975), among others (e.g., Wicken, (1987), May (1988)), but is certainly not universally used (e.g., Nicolis, (1986); Kampis and Csanyi, (1987)). Here we will loosely describe these concepts as follows. Order Concerns the spatial-temporal, or space-time `structure' of a model's (or PS's) dynamics, including stationary states; a space/time sequential characterization of dynamics; possibly arranged in `degrees.'
Organization Concerns the responsive, environmentally operative qualities of a system's dynamics; the `value', `flexibility', `purpose', `function', or `adaptability' of various dynamic parts; possibly organized in hierarchical levels; possibly in commensurate components.
The above complications of chaos and turbulence fall within the `Order' category. Indeed, the complications of most models of inanimate systems have been judged under `Order' tests. Some exceptions to this involve aspects of bifurcations, or dynamic stability under external perturbations, particularly in the area of control systems (e.g., under stochastic actions), but `Order' tests appear to dominate the `complexity studies' of inanimate models. It should also be emphasized that many systems which physicists
and chemists call 'self-organizing', such as phase transitions, superconductivity, superfluidity, lasers, chemical oscillations, and Rayleigh-Benard convention (e.g., Haken, (1983); Davies, (1989)) would fall in the above category of 'self-Ordering'. This is, of course, only a semantic problem, but it can cause confusion. Thus, the concepts under `Organization', until very recently, have been largely the topics of interest and development within the life-sciences, and more recently, various cognitive
and memory-model dynamics. These scientists have been grappling with the most complicated systems, and have developed the greatest understanding, if not yet many quantitative measures of the terms in quotation marks under `Organization' above.
510
Epilogue
The point of view of most life-scientists concerning questions of Organization is
typically totally different from the relatively precise concepts which have been suggested in Ordered complexities. The functional aspects of a system are considered central, particularly in living systems (but this can certainly be extended, for example, to such areas as mechanical control systems, which need to function under various
environmental conditions). The evolving philosophy of how to investigate very complex biological systems has been authoritatively discussed by Mayr (1988). These
highly complex issues are well beyond the scope of this study (and ability of the author!). We will instead consider some of the issues involved in the term `function'. It would seem to be an unavoidable association with the description `the functioning of a system', or `a system's function', that the system's dynamics must be evaluated in terms of its interaction with its `environment'; be that an experimenter, or statistically defined stochastic influences, or well-defined orderly influences. As noted in Section 10.15, if one part of a system simply acts on another part of the system, and everything is deterministic, then we simply have a larger closed system. However, let us delay these considerations, and first take up the `Order' side of the complexity issues.
Order Before considering some rather precise concepts of complexity, we will consider some less precisely formulated, but imaginative ideas concerning `degrees' of order. Bohm
and Peat (1987) have suggested that it is useful to recognize a spectrum of Orders. They proceed along several lines, the first of which is empirical in character, and appears to be the following (Fig. E4). A line may be composed of segments which have only one `similar difference' (namely their similar displacements). Fig. E4
a
b
.
c
.
d
The same is true of any regular polygon (or circle in the limit); that is, a is to b as b is to c, etc. Even if the length decreases in a regular fashion, say length (a)/length (b) = length (b)/length (c), and the angles of rotation are constant, then there is only one similar difference (in other words, a `difference' is a set of comparisons). These curves may also be taken out of the plane, but the complexity of the curve is clearly quite limited if the segments have only one similar difference. Bohm and Peat define
Order
511
this class of segmented curves as having second degree Order. To define the curve, they attach a significance to the starting point as well as the one similar difference,
to obtain the designation `second degree'. This appears to be a rather curious admixture of a qualitative and a quantitative character. If the curve is broken down into segments as illustrated in Fig. E5, then there are Fig. E5
two different similarities: the similar differences of a:b:c or d:e:f, but these similar differences are different from one another (in their orientation). Moreover, this difference between the similarities a:b:c and d:e:f is the same as between d:e:f and g:h:i. Adding the starting point, they assign this a third degree order. They do not appear to indicate what determines the segments, but let us assume that the end points of each segment are obtained empirically by making observations at regular time intervals. Then we have a reasonably well-defined way to characterize the degree of order of an empirical dynamic set of data, based on similar difference, and different similarities. The above examples show that, by changing the observation rate, the degree of Order is changed, but reaches some limit if the data is not too complicated. Clearly, this approach to `understanding' falls within the empirical methods, {E;}, characterized above. It is only one of many such methods e.g., Lyapunov exponents, embedding dimensions of attractors, fractal dimensions, and multifractal parameters e.g., Paladin and Vulpiani, (1978), none of which require dynamic models for their determination. The second and quite distinct viewpoint of Bohm and Peat is to consider models of PS which are represented by a system of ordinary differential equations (ODE). They assign a `degree' to the `Order' of that model which simply equals the order (!) of their system of ODE. (One recognizes here an unfortunate overlap of the `order' of differential equations, and our dynamics `Order' of complexity; note the capitalization). Thus, for example, the one-dimensional particle motion acted upon by a force has an Order of degree two, according to Newton's model.
512
Epilogue
An important distinction needs to be made between the above empirical degree of Order of a physical system (PS) and the degree of Order of the models of the PS. To illustrate, Bohm and Peat refer to the concept of 'context-dependent' degree of the Order, and give the example of a ball rolling down an inclined plane which has a number of irregular bumps on its surface. If all of the irregularities of the plane are taken into account, and the ball remains on the plane then Newton's model for this system has an order of degree four (the x and y motions). Hence, within this `context', the system (defined by Newton's model) has a low (model) degree of Order. Their observation is that, if we do not consider the ball's dynamics within this context, but describe the observed trajectory through some experiment, E, and determine its degree of complexity (using Bohm's method of `similar differences'), then we would obtain a very large degree of Order for this same PS. This point of view leads to their conclusion `that order is neither subjective nor objective, for when a new context is revealed, then a different notion of order will appear'. From the present viewpoint, their observation can be expressed as representing two modes of complexity, or two modes of Order, one which is empirical, the other based on a model, M. Thus, their `context' is what we might call the `E-M context' of the PS.
The last example might also appear to be rather singular, in the sense that the more macroscopic experiment, E, has a higher degree of Order than the more microscopic (detailed) Newtonian model of the PS. The opposite is more commonly the case, and is indeed the whole raison d'etre for the `model industry'; namely, we are interested in obtaining models of selected observables with low degrees of Order, for PS which presumably have a high degree of Order (in terms of the microscopic Newtonian or quantum models). While our appreciation of the degree of Order of a physical system may be initially revealed in terms of our models of some of its dynamic properties, there is no basis for the belief that this `context' can be ever-refined to models of greater degree (and hence more `fundamental' in the above sense). This issue is important even on the Order side of complexity (Bohm, (1984); Leggett, (1987)), but is even more important for Organizational complexity (e.g., Maynard Smith, (1986); Wicken, (1987); Rosen, (1987)). The recognition of our limitations in `understanding' complex physical systems
is indeed part of the education of understanding complex systems. That is, in understanding the nature of these limitations, we will better understand the meanings ('modes') of complexity, and certainly will be less frustrated through not pursuing nonexistent goals ('And our goal is nothing less than the complete description of the universe we live in'; Hawking, (1988)). It is much more meaningful to keep Newton's perspective of science and to note, with Leggett (1987) that For a physicist to claim that he `knows' with certainty exactly how a particular physical system will behave under conditions very far from those under which
Order
513
it has been tested - for example, the conditions under which certain types of space-based defence systems might have to perform if they were every used seems to be arrogant, and indeed in some contexts dangerous.
One hardly need observe the relative degree of arrogance involved in claiming knowledge about the `inner workings' of living systems. As Bohm pointed out some time ago (Bohm, (1984)),
no experiment could possibly prove that a given set of quantitative laws never depend on the qualitative state of matter, since evidently, under new conditions not yet investigated, or in studies carried out to a higher level of approximation,
such a dependence might eventually appear. Thus the assumption that all qualitative changes are, at bottom, just passive `shadows' of quantitative changes
of some basic set of entities, like the one that higher level laws are reducible completely to those of some fundamental level, cannot be founded on any conceivable kinds of experimental fact.
These issues relate, of course, to the `metamodels' described above, and to the fact that they (at best) may give some sense of `security' for obtaining usable models.
The Bohm-Peat classification of model-degrees of Order does not recognize the dynamic chaotic-bifurcation which occurs as a function of degree (now restricted to ODE models). Thus Poincare recognized that `chaos' defined in a precise sense can exist in closed models with degree 18 (i.e., the three-body problem). Birkhoff, considering area-preserving maps, reduced the degree of order to at least four since the mapping can be embedded in a Hamiltonian flow with two degrees of freedom. Later, Lorenz (1963) showed that chaos can occur in third-order ODE models, and this has been expanded to many other models. Thus, in the Bohm-Peat description, models which are third degree in Order can exhibit chaos. This is not the case for autonomous models which are either first or second degree in Order, so there is an important bifurcation in complexity at degree three. For open, and nonautonomous models, the bifurcation occurs at degree two; illustrating a striking difference between
open and closed passive models. Indeed, these bifurcations are among the most important dynamic discoveries of the last century. This naturally raises the question whether other bifurcations exist at higher degrees, and the answer is obviously yes. Boltzmann certainly realized that sound waves, which can be modeled by low order partial differential equations (PDE), can be the outcome of very complex molecular dynamics. Here a simplified space-time order of certain physical variables (e.g., pressure or density), is presumably the consequence of a very high degree of Order, that is, in the limit of large numbers. Many other examples of this
`emerging' simplicity are well-known, such as collective variable in ionized gases,
514
Epilogue
normal modes in lattices, and of more recent interest, the Belousov-Zhabotinski chemical oscillations and patterns, the Rayleigh-Benard convection, or Taylor vortices. Some of these have been the focus of the Prigogine and Stenger (1984) examples of `order out of chaos', as a small part of a broad, and frequently metaphysical
discourse on the nature and `meaning' of chaos. It is not at all obvious that chaos, as defined above, has any basic relationship to the above list of simple 'self-organizing' dynamics observed in systems with large numbers of particles. It indeed is an illusion to think that we will ever have a detailed understanding of some `complexity-bifurcations' which are responsible for the `birth' of simple models, based on the above `fundamental' metamodels, in the limit of large numbers (or large degree of Order, to use Bohm and Peat's description). Let us now turn to several rather precise descriptions of complexity, which have their
origins in computer science. One idea, suggested independently by Kolmogorov, Chaitin, and Solomonov and discussed by Brudno (1983) and by Ford (1988), gives a measure of the complexity of a dynamic sequence which can be experimentally obtained, or taken from the computer output of some model associated with some physical system. Consider some finite dynamic `run', which yields a 'word', x, of length, L(x), as measured in its number of bits (when expressed in binary notation). Let L(x*) be the length of the smallest binary input, x*, to a universal computer which can generate
x. Thus, x* contains both the initial conditions and the necessary program (which is usually a minor contribution to L(x*); see below). L(x*) is variously referred to as the algorithmic complexity of x, K(x) (e.g., Zvonkin and Levin, (1970), or the algorithmic information content of x, H(x) (e.g., Bennett, (1985), Chaitin, (1987, 1988). It should be
noted that this approach does not refer to any specific model of a PS, but rather to a dynamic `word', which may have been obtained by some empirical method directly from the PS (the PS may be a computer). Now the dynamics is considered to be random if K(x) is proportional to L(x) when L(x) becomes sufficiently large (i.e., a mathematical condition is that L(x*)/L(x) > 0, as L(x) - oo; one again should note the nonempirical character of this definition; see Section 4.11). Alternatively, x is called algorithmically random if L(x*) is approximately equal to L(x); in that case the `word' x is said to be incompressible. Thus, a dynamic sequence such as
1,2,1,2,1,2,1,2,1,2 has a length L(x) = 10 bits and can be generated by a simple four-line program; (for i = 1 to 5; print 1; print 2; next). As the length of this repetitive x, L(x), is increased, the length of the program, L(x*), is only increased by replacing 5 by loge (L(x)/2). For large L(x), the algorithmic complexity, K(x), grows only as loge (L(x)), and hence this sequence is neither algorithmically random nor algorithmically complex. Notice that
we have not discussed a model M, nor any general property of the PS, simply the complexity of x, related to some observable.
Order
515
A classic example of a dynamic model which produces random dynamics is the special logistic map
which was discussed in Chapter 4, and particularly in Section 4.11. Notice that here we have switched from the point of view of observing a word, x, to considering a given model which generates random words. This distinction is significant, as will soon be discussed. As pointed out in section 4.11, this logistic model puts out words, x, which are essentially the same length as the input, x*, because of its shift-dynamics of binary strings. There is no informational compression of x associated with the input x*; we are getting out only a revised form of the information we put in. As Ford emphasized, (see Ford, 1988)), the differential models which yield chaos, in the sense of exponentially separating orbits, are precisely those models (when programmed) that yield accurate outputs, x, of length approximately equal to input information L(x*). Moreover, since our physically available input is limited by experimental errors (also see Section 4.11),
the output information, L(x), is also limited in such systems. Thus, if a dynamic measurement yields an algorithmically random word, x, there exists no model which will yield us effectively new information from our (limited) initial data. Thus, when data is algorithmically random, it is essentially a negative statement about our ability to `compress' this information. On the other hand, if some data is not algorithmically random, so that L(x) is large compared to L(x*), we have not learned
from that fact anything about a possibly appropriate model which relates these dynamics to the physical system; that is, the program which generates x from x* on the
universal computer bears no apparent relationship to an appropriate model of the physical system. Hence, we could save considerable effort attempting to make 'longtime' predictions of certain observed physical phenomenon, if we can show that x is incompressible (see below for difficulties). An interesting case in point was von Neumann's early attempts at weather prediction; the problem, of course, was for the computer to make a one-day weather prediction in less time than one day! Ultimately von Neumann was able to obtain some fairly reliable predictions for a period of up to two days before new inputs were required. Norbert Wiener was highly critical of von Neumann's approach, feeling that a statistical analysis was the only appropriate way to make such predictions. Presently, both methods are used for long-range predictions. See Heims (1984) for an interesting discussion of the `Chaos or Logic' viewpoints of Wiener and von Neumann. It is worth emphasizing that the `interesting' forms of order-complexity occur when K(x) is neither very small nor very large (Fig. E6). If the importance of a form of order (e.g., in indicating the system's ability to respond stably to changing inputs) is directly related to K(x) - and there is presently no assurance that this is the case - it might take some dependence such as Kp exp ( - AK"), as illustrated. It seems likely that it is the
Epilogue
516 Fig. E6
Interesting
orders
K(x)
moderately order-complex forms of dynamics which play many of the dominant roles in organizationally complex systems. However, highly complex dynamics may be of singular importance in certain cases (see, for example, Nicolis, (1987)).
Order/Organization Whereas algorithmic information, or complexity, is essentially a measure of the randomness of dynamics, Bennett (1985) has made the interesting suggestion that a better measure of organization should involve the `value' of a message, rather than its `information content'. He noted
A typical sequence of coin tosses has higher informational content but little message value; an ephemeris, giving the positions of the moon and planets every day for a hundred years, has no more information than the equations of motion and initial conditions from which it was calculated, but saves its owner the effort of recalculating these positions. The value of a message thus appears to
reside not in its information (its absolutely unpredictable parts), nor in its obvious redundancy (verbatim repetitions, unequal digit frequencies), but rather
in what might be called its buried redundancy - parts predictable only with difficulty, things the receiver could in principle have figured out without being
told, but only at considerable cost in money, time, or computation. In other words, the value of a message is the amount of mathematical or other work plausibly done by its originator, which its receiver is saved from having to repeat. Bennett suggests using the term logical depth to measure this value of a message. He states
A message's most plausible cause is identified with its minimal algorithmic description and its `logical depth,' or plausible content of mathematical work, is (roughly speaking) identified with time required to compute the message from this minimal description. There are at least two important aspects of the concept of logical depth. The first is that it involves the `importance' or `value' of a message to `something'. If this message is
Order/Organization
517
to have `value' the `something' cannot be part of the system's environment, as defined
above, but rather must make use of this 'value-content' (which stochastic and predetermined environments cannot do). Thus, Bennett's logical depth might be important in hierarchical systems, where the dynamic output of one `level' can be used
as an input of another level, or in compound systems, formed from commensurate
parts. Thus, the concept of logical depth is an important attempt to address Organizational concerns, in addition to Order measures. Other possibilities will be discussed shortly. A second point to note, however, is that, to obtain a measure of `value', the logical depth is based on the idea that a `message's most plausible cause is identified with its minimal algorithmic description...'. Here we see an essentially nonempirical, and nonphysical element enter the picture. The minimal algorithmic description has no apparent connection with possible physical dynamics. Indeed, a minimal description looks most completely random (for if x* had a significant regularity, it could be used to encode it more concisely). Occam's razor was never intended to cut this deep! Thus, while Bennett approves of the fact that `the use of a universal computer frees the notion of depth from excessive dependence on particular physical processes (e.g., prebiotic
chemistry) and allows an object to be called deep only if there is no shortcut path, physical or non-physical, to reconstruct it from a concise description...', there would appear to be good reasons to attempt to find another `value' which is not as `excessively independent' of physical processes. Neither of the above algorithmic concepts, as it might appear, generally evaluate the complexity of a system directly from its output, x. As Bennett pointed out, neither the
algorithmic information nor the logical depth are effectively computable properties. The difficulty is associated with the unsolvability of the halting problem, which makes
it impossible to prove that an output is random (although it may be proved to be nonrandom). In other words, it is generally possible to obtain either the algorithmic complexity (information) or logical depth only if we have a model and the initial conditions which yield the output to be analyzed. Even in this case, the logical depth is
referred to as a minimal algorithmic description by making use of additional assumptions (Bennett, (1985)) rather than the given known model. Part of the difficulty with the computational insights into complexity arises from their dependence on limiting concepts (outputs of infinite length). Nothing in the empirical sciences depends on this concept. Thus, algorithmic complexity, logical depth, and such concepts as computational universality involve aspects which are unnecessarily `refined' for an understanding of complex physical systems. While there exist suggestions on how to define a `complexity' (essentially a randomness) from finite sequences (e.g., Popper, (1968); Lempel and Ziv, (1976)), there appear to be no insights yet developed into the `value' of such finite sequences. It should be emphasized that the `value' or other organizational qualities of a system need not be intrinsic to that system
Epilogue
518
in its environment, but may well depend on the duration that the system is required to exist (e.g., before it dies, or is thrown away, etc.). This is, of course, a well-known
philosophy of manufacturing, and empirical scientists might take note of it in understanding how nature evaluates organizational qualities. A second factor which is not emphasized in many computational discussions is the interplay between the `length' of an output, L(x), and the actual physical time that is predicted. This relates to a question of accuracy, which is of great importance. For a given L(x), the less the accuracy required of the observables, the longer the time-length
of the prediction. Given some maximum empirical accuracy, this can be used to maximize the accuracy of the initial conditions, but the accuracy must subsequently be
reduced to obtain predictions over substantial times. A point to note here is that a system's dynamics in an environment, to be of `value' for organizing, must be able to accommodate these ranges in dynamics ('accuracies'). Thus, value judgements should probably also consider such organizational `stability' features; for, as Wiener noted long ago, noise is part of the `warp and woof' of empirical sciences. Indeed, the whole subject of stochastic processes has been sadly omitted from most of the present study, and much of the existing literature in this area. That will have to be corrected some day.
Organization The above emphasis of a theoretical minimal algorithmic description, rather than making any association with the actual dynamic mechanism of a system, appears (to the author at least) to make the above concepts excessively removed from physical constraints. Indeed, one might take quite the opposite viewpoint from those which search for a minimal L(x*) that can generate x. (See Fig. E7.) It might well be that the Fig. E7 Environment (x )
value of some dynamic `component' (e.g., to another `component') is to take a complicated input, zER", and to generate a much `smaller' output, yERm (m << n), so L(x) >> L(y)
Organization
519
generally in a many-to-one fashion
so that different inputs in a set k yields the same output, yk. Thus, one dynamic component acts as a dynamic `filter' for data passed onto another component of the system. This may be accomplished in several ways, one of which is for a component to be an attractor, taking initial states in R" and compressing them all into R' (m << n) (see,
for example, Nicolis, (1987)). In simple cases the many-to-one feature is related to isochrons (see index). If a second component is only responsive to the variables in the reduced attractor space, R'", then clearly the first component is `valuable' to the second component in making this isochronal reduction of the input x.
A second possibility is that one component is only `resonant' to some of the information in the input z, and then communicates its subsequent evaluation to the second component. In any case, this 'filtering-value' of one component, which surely is a useful organizational operation, is the antithesis of the above algorithmic viewpoints (e.g., Bennett's ephemeris example).
Moreover, a new element has clearly entered the discussion. Once `value' is considered, it only has meaning in some collective context; either in terms of a system's
response to environmental changes, or what one `component' of a system does for or to another `component'. This has long been appreciated by life-scientists and we are interested here in attempting to abstract this insight to the general realm of
dynamic systems. Denbigh (1975) suggested that we might assign a degree of organization, which he called integrality, based on the number of distinctive component parts, n, and the total number of connections, c, including feed-back channels and perhaps also connections to the environment, provided that they all `facilitate the function or functions of the whole assembly'. He suggested as a rough measure of integrality simply the product cn =1, for brevity. Denbigh thus obtained, by way of simple examples, 1= 24 for a pair of shears, and 1= 81 for a rotary can opener. As he noted, `How immensely large it must be for a clock and how immensely larger still for an aircraft.' In general, if a system has N components, then N2 > c > N, and n is the number of distinctive components of N. Thus, 1= cn grows rapidly with respect to N. There are, however, fundamental difficulties with such large values of 1, to which we will return shortly. Denbigh also noted that the integrality of a closed system is not generally conserved; for example, in the case of the growth of an embryo in a bird's eggshell, which he views as effectively a closed system. Moreover, he argues that integrality does not need to bear
any quantitative relationship with changes in the entropy, or entropy production. He argued that `there is no reason why the making or breaking of connections within an organized system should not be carried out reversibly, at least at the limit'; and, of
course, reversible processes in closed systems involve no change in the entropy.
520
Epilogue
Denbigh argued that the fundamentally unrelated character of entropy and integrality is another reflection of the distinction between Order and Organization. To those who are comfortable with the use of entropy concepts within the present dynamic contexts, this observation is undoubtedly important.
As noted by Denbigh, a more precise measure of integrality would require the definition of `components' of a system, and of `connections'; particularly if one tries to distinguish those which `facilitate the function or functions of the whole assembly'.
It is interesting to pursue, if only briefly, what we might mean by a dynamic `component' of a complex system. Ecological systems, or the major organs of our bodies come immediately to mind as examples of systems with such `components'. However, inanimate machines and computers certainly fall into this class of systems. From a physical point of view, a `component' of a complex system apparently should be
spatially disjoint, but open. From a mathematical point of view, a component (e.g., (x,, x2) or (x3, x4)) is a set of coupled dynamic variables, whose time-dependent dynamics is sustained (asymptotically) by connections with exogenous variables. In other words, a `component' is an dynamic system, open possibly, to the environment, but certainly to other variables which are endogenous to the complex system (Fig. E8). Fig. E8
c1IEDIIEDD
EIEIIEDD The component's asymptotic dynamics may tend to a fixed point (it `dies') if the connections to the outside dynamics are severed. On the other hand, it has some `nontrivial' dynamic behavior for at least some limited time, even if the connections are severed. There are clearly many possible variations of these ideas; this is only a point of departure. There is a fundamental problem concerning the definition of complex systems which
have large numbers of connections; one is that it returns to a reductionistic mode of thinking, discussed above, with all of the illusions that we are somehow `understanding'
a complex system when we break it down into many `components'. Unless these components indeed have simple dynamics, and the nature of the couplings is likewise simple, then in all likelihood we are simply deluding ourselves into thinking that we
have obtained any more profound insight into this system's `complexity'. Thus, if Denbigh's integrality is very large, it may be an interesting measure, but it may not give much insight into what such `complexity' means.
Organization
521
There are also other known difficulties when I becomes large, which are related to the
question of stability. It had long been felt, particularly in the field of ecology, that systems which have many components (e.g., species), and with many connections between the components, form a more stable whole (e.g., to fluctuations in the environment, or among the components). The studies of Gardner and Ashby (1970) and
May (1972) challenged this perception by establishing that, at least for random connections, most of the systems in such a random ensemble have fixed points which are unstable, provided that 1= cn is sufficiently large. More specifically (May, 1972), let n be the order of the linear system
x=
(xeR")
(*)
so A is a random n x n matrix (with aKK = 0) representing the coupled perturbation of a system from its fixed point (x = 0). Let c represent the probability that aid = 0, so that
(1 - c) is the probability that aid is a non-zero random number, with a symmetric distribution about zero, and a root mean square value of s. Thus, c is the probability that there is some connection between any xi and any x, (i : j). May then drew upon a theorem due to Wigner, to conclude that the probability, P(n, c, s), that (*) is stable satisfies lim P(n, c, s) = 1
if s(nc)1i2 < 1
lim P(n, c, s) = 0
if s(nc)"2 > 1.
n- cc
n- 00
Moreover, the width of the transition region between P = 0 and P = 1 apparently scales as n-213 for large n (Gardner and Ashby's results indicate that n = 10 is quite sufficient for a near-step function behavior of P vs c). This analysis, based on fixed points and random connectiveness suggests that the `complexity', as measured by s2nc, cannot be large if the system is to have a stable fixed point equilibrium. It is not clear that either the focus on fixed point stability, which yields the above linear equations (e.g., vs quasiperiodic stability; see Maurer, (1987)), nor on random connections (e.g., vs evolutionary selected connections, see May, (1974)) is particularly
compelling. However, within this fixed point/random constraint, May made the additional observation that compartmentations of large numbers of components may lead to a more stable system. Thus, for example, a 12-species community with 15% connectance might have a probability of essentially zero of being stable (for a given s), whereas if the interactions occurred only in 4-species groups, with 45% connectance, the stability probability rises to 35%. The possible importance that such `blocks' of species may have on the actions of real systems has been experimentally strengthened (Moore and Hunt, (1988); McNaughton, (1988)). Thus, the complexity of organization may be quite different than a simple increase in nc (even if s (nc) 112 < 1); it may involve
Epilogue
522
`inhomogeneous' groups of connections, or else very select channels of connections which insure some form of stability (static or dynamic). For example, the elements of a B-matrix, where z = Bx, might have only weak couplings in the shaded regions in Fig. E9 (e.g., Nicolis, (1986)). Such a matrix is a linear version of the (x1, x2) and (x3, x4)
component dynamics in the previous figure. Fig. E9 b11
b12
h13
h14
b21
b22
b23
b24
h31
h32
b33
b34
b41
h42
b43
b44
Rosen (1985,1988) has proposed perhaps the most complex classification of complex
systems - one which excludes the possibility of any model based on global dynamic equations, dx/dt = F(x) (xER). Any system which has a set of observables that can be modeled by such dynamic equations, he proposed to call a `simple system'. For such a system the functions U, (x) = aFi(x)/ax;
obviously satisfy the conditions Uij(x) = Uji(x). Some time ago Higgins (1967) noted that the signs of these functions are often sufficient information to predict the existence of periodic solutions. The classic example of this fact is Bendixson's criterion, discussed
in Chapter 5. Higgins generalized this result and explored this `informational' approach, noting that if Uij > 0, x3 is an activator of xi (i.e., a(dxi/dt)/axe > 0) and if Uij < 0, xj acts as an inhibitor of xi. He studied a number of examples, pointing out that one must generally deal with cases where the U13(x) changes sign in various regions of phase space. Rosen has proposed to generalize Higgins' observation, approaching the study of systems from the `informational' viewpoint by generally defining the `observable quantities'
U.j = a(dxi/dt)/ax;
(i, j, = 1, ..., n).
However, the operational meaning of this `partial derivative' has not been precisely described at present. Assuming that this may generally be accomplished, if Ui; = U;i for all xER", a function F(x) can be obtained such that dx/dt = F(x), so the system would
Endnote
523
be `simple' by Rosen's definition. Like Higgins, Rosen viewed the operational use of the
quantities U1 as an `informational' approach to the study of systems, and that the systems are to be regarded as complex only if U;, 0 U fi. In addition to the activatorinhibitor character of the U;j, he proposed that one obtain further information by determining (somehow) such quantities as Ui jk = a U;;/axk
and referred to Ui jk > 0 (< 0) as `xk is an agonist (antagonist) of xj with respect to x;'. Then `complex systems' will generally exhibit the property U;Jk U;kj. Rosen observed that our usual dynamic modeling can only be valid for limited times and regions of the observables in the case of such complex systems. However, since this
is clearly true for all physical systems (see Endnote), a further elucidation of the limitations of `complex systems' is needed to shed any new insight into this matter. Moreover, the `observables' Uj; and U;;k need not have definite signs as a function of time even for simple systems, and for `complex systems' they are not generally even functions of xeR" (the set of observables being considered). Thus, even for given x, they need not have a definite sign, in which case their `informational' content is questionable. Rosen feels that `causality is precisely the name we give to the regularity of the succession of events in the formal world' (i.e., the mathematical theory). However, some forms of `regularity' among events are often referred to as `determinism' (see Endnote) without any cause-effect implication. While he suggests that a 'causal structure' exists in such `complex systems', much remains to be clarified as to how his `complex systems' fall within any causal framework of empirical science. One is referred to Rosen (1988) for a further discussion of this viewpoint and other references. One of the unsatisfactory aspects of this extreme definition of `complex systems' is the possible misinterpretation that `simple systems' (U;j = Ui,) are in fact simple, which of
course is not the case. It would seem that we still need a classification within this dynamic group of systems which admits some form of complex distinction. Thus, we close on this rather inconclusive note, and return to Newton's thoughts at the beginning of this section; which, of course, capture some of the spirit which keeps all of us endlessly stimulated.
Endnote: models, causality, irreversibility We consider here a reasonably general and precise definition of `models' for a physical system (PS). In the empirical sciences, we are concerned about the dynamics of some observables, x;(t), of a particular PS. By a deterministic model of the PS we mean a subset of its observables, 0(t) - { x,(t) I i = 1, ... , n }
524
Epilogue
together with dynamic rules (mathematical equations), that yield a unique relation
D(O(t1),...,O(tn))=0 for some minimal set of times (t 1 < t2 < . . < which can be ordered without loss of generality. However, note that, since all O(ti) have the same status, there is no usual sense of causality implied by this relation (e.g., 1) can be viewed as the `result' of the remaining O(ti)); it is simply the mathematical consequence of the dynamic rules. We will refer to this relationship between the {O(t1)} as determinism.
The dynamic relation will be called reversible tl < t2 <
if,
given any set of times,
< t,,, and a sequence of observables satisfying D(O(t 1), ... , O(tn)) = 0,
the sequence 0*(tk) = O(tn+ 1- k)
(k = 1, ... , n).
also satisfies D(O*(t1),... , O*(tn)) = 0.
In other words, the sequence of O*-sets is simply the time-reversal of the sequence of 0-sets. If the dynamic relation is not reversible, it will be called irreversible. These are strictly mathematical definitions. The next question is whether a dynamic relation is empirically `valid'. To discuss this, we must first define the set of accuracies, A = {ai(t) I i = 1, ... , n} with some care.
This set is controlled by two factors, one empirical, the other mathematical. The experimental instruments yield only finite reproducible numbers. The interval between these numbers, for the observable xi, we will denote by ai, which will be assumed to be time-independent (over the lifetime of the experiments). The numbers in this range only have mathematical significance, since they cannot be experimentally observed.
Put another way, experimental observations do not map onto the real number, as is often stated, but instead map onto real number intervals, the {ai}.
The second factor controlling the accuracies, ai(t), involves the reproducible requirement involving repeated experiments. Starting from a fixed experimental initial state, O(t1), repetitions of the experiment will yield values (measured by the equipment) in the range xi(tk) - ri(tk)
to
xi(tk) + ri(tk)
(k = 2, ... , n)
where zi(tk) is simply the observed mean value. We will now define the uncertainty ai(tk) to be ai(tk) = r,(tk) + ai
(k = 1, 2, ... , n)
Endnote
525
where ri(tt) = 0 above (but this may be trivially generalized, if we wish to study other ensembles of experiments). Because of the inclusion of the mathematical numbers, ai, these accuracies are more precise than the equipment can determine, and are defined for mathematical reasons, to be discussed shortly. Since we observe the system for only a finite time (t - t1) < T, we may not be able to detect any variation in the A(tk), if the set A = {ai I i = 1, ... , n} has sufficiently large elements (i.e., the equipment may always register the same set of numbers, so ri(sk) = 0). More commonly the a(tk) >1 A, and cannot be controlled; their relative magnitudes
are controlled by the dynamics of the physical system (the PS may, in numerical experiments, involve a specific dynamic system that is more complicated than its model, such as in fluid dynamic or molecular dynamic calculations). Using these ideas, we can say that reproducible experiments generate a sequence of 'range-sets' of the observables, R(ti), so that all experiments yield observables in these ranges, O(ti) c R(ti),
where the range intervals are, schematically, R(ti) = [0(ti) - A(ti), 0(ti) + A(ti)]
n)
and
0(t) ± A(t) = {xi(t) ± ai(t) I i = 1, ... , n}.
Here, the set of times { ti I i = 1, ... , n} are the values given by the experimental clock,
whose (mathematical) accuracy is ± T.
Now we will say that the model is an empirical dynamic relation (the model is T, T)-valid') if, for every experimental set {ti } which is studied, experimentally within a range (t - ti) < T, there exists some time-set {si (ti - t < Si < Ti + T; i = 1, ... , n}
such that the model solutions D(O(si),... , (sn)) = 0
satisfy
0(si) c R(ti).
This condition is illustrated in Fig. E10, for some member of 0(t). Having set up this rather oppressive collection of detailed definitions, we now come to the really interesting topic of reversibility and irreversibility, which will only be discussed briefly. What is a reasonable empirical definition of these concepts? Some people (e.g., Prigogine and Stenger (1984), supported by quotations from Planck and Popper) argue that these concepts are not empirical, but somehow `fundamental'. To quote Prigogine and Stenger, `irreversibility is either true on all levels or on none. It cannot emerge as if by a miracle, by going from one level to another'. By `levels' they mean `macroscopic' vs `microscopic'. The latter falls within the metamodels discussed
Epilogue
526 Fig. E10
x
D(0(ti),0(t2),0(t3))=0 X(t2) X(t3
x(tl) -2A (t, T
S,
t1
S2T
t2
t 3 't .S3
t
above, which are without empirical foundations. Likewise, Planck took the point of view concerning the second law of thermodynamics that `the gist of the second law
has nothing to do with experiment'. We choose to deal here only with empirical sciences and their observable phenomena, and will leave the metaphysical discussions to others. In any case, many such discussions are all too frequently garbled beyond any hope of logical comprehension. One point of view concerning reversibility might be that, if the model is an empirical dynamic relation (an empirical statement), and the dynamic relation is reversible (a mathematical statement), then the empirical dynamic relation is reversible. However, this type of definition is not related to any observations of `reversed' dynamics. To illustrate this perspective, assume that the PS has two empirical range-sets, {R(f,) }
and {R*(f,)}, (i= 1,...,n), such that the intervals
Ik(tk)=R(tk)nR*(tn+1-k) 0
(k= 1,...,n).
Then, if the empirical dynamic model has a solution D(O(s,), ... , O(sn)) = 0
satisfying O(Sk) C Ik(tk),
we might view this as a reversible empirical dynamic model. These are clearly not the only possibilities; a more detailed investigation remains to be done. We end with a few simple examples of the above dynamic relation, since its noncausal
character is quite distinct from the more common `initial condition' solutions we are familiar with. Thus, Newton's equation for a particular body of unit mass acted on by a `force' F(t), d2x/dt2 = F(t)
References
527
yields the dynamic relation
D(x(t1 ), *2), *A _ Y_ [x(ti) - F(ti)](tj - tk) = 0
('k)
CS
where
F(t) = f dt' f F(t") dt" 0
o
and the sum is cyclic (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2). It should be noted that this
avoids the concept of an instantaneous velocity, which is not an observable, but a derived concept (from several measurements and a nonempirical process). However, as is characteristic with Newton's equation, the dynamic relation (*) is in fact a statement concerning the existence of some function F(t) satisfying (*) for any (t1, t2, t3),
because F(t) cannot be independently measured. Finally, a damped harmonic oscillator
d2x/dt2 = - x - p dx/dt has the dynamic relation Y x(ti) exp (ati) sin [w(tj - tk)] = 0 Cs
where 2a = p, w = (1 - (X2)1/2 and the sum is again cyclic. Note that while this is deterministic, it is not a reversible dynamic relation. It is left as an interesting exercise to determine whether such an irreversible dynamic relation necessarily (or under what
conditions) implies an irreversible empirical dynamic relation, using either of the above suggested definitions (or others of your own making).
References Bennett, C.H. (1985). Dissipation, information, computational complexity and the definition of organization. In Emerging Syntheses in Science, ed. D. Pines, pp. 297-313. The Santa Fe Institute. Bennett, C.H. (1986). On the nature of complexity in discrete, homogeneous, locally-interacting systems. Found. Phys. 16, 585-92. Bohm, D. (1984). Causality and Chance in Modern Physics (University of Pennsylvania Press). Bohm, D. and Peat, F.D. (1987). Science, Order, and Creativity (Bantam). Brudno, A.A. (1983). Entropy and the complexity of the trajectories of a dynamical system. Trans. Moscow Math. Soc. 4, 127-51. Chaitin, G.J. (1987). Algorithmic Information Theory (Cambridge University Press). Chaitin, G.J. (1988). Randomness in arithmetic. Sci. Amer. 259, (July) 80-5. Davies, P. (1988). The Cosmic Blueprint (Simon and Schuster). Denbigh, K.G. (1975). An Inventive Universe (London: Hutchinson). Denbigh, K.G. (1980). How subjective is entropy? Chem. in Brit. 17, 168-85. Ford, J. (1988). What is chaos, that we should be mindful of? In The New Physics, ed. Davies, P.C.W. (Cambridge University Press).
528
Epilogue
Gardner, M.R. and Ashby, W.R. (1970). Connectance of large dynamic (cybernetic) systems: critical values for stability. Nature 228, 784. Haken, H. (1983) Advanced Synergetics (Springer-Verlag). Hawking, S.W. (1988). A Brief History of Time (Bantam). Helms, S.J. (1984). John von Neumann and Norbert Wiener (MIT Press). Higgins, J. (1967). The theory of oscillating reactions. J. Ind. Eng. Chem. 59(5), 18-62. Kampis, G. and Csanyi, V. (1987). Notes on order and complexity. J. Theor. Biol. 124, 111-21. Leggett, A.J. (1987). The Problems of Physics (Oxford University Press). Lempel, A. and Ziv, J. (1976). On the complexity of finite sequences. IEEE Trans. Inform. Theory 22, 75-81. Lorenz, E.N. (1963), Deterministic nonperiodic flow. J. Atoms. Sci. 20, 130-41. Maurer, B.A. (1987) Scaling of biological community structure: a systems approach to community complexity. J. Theor. Biol. 127, 97-110. May, R.M. (1974). Stability and Complexity in Model Ecosystems (Princeton University Press). May, R.M. (1972). Will a large complex system be stable? Nature 238, 413-4. Maynard Smith, J. (1986). The Problems of Biology (Oxford University Press). Mayr, E. (1988). Toward a New Philosophy of Biology (Belknap Press of Harvard University). McNaughton, J.J. (1988). Diversity and stability. Nature 333, 204-5. Moore, J.C., and Hunt, H.W. (1988). Resource compartmentation and the stability of real ecosystems. Nature 333, 261-3. Nicolis, J.S. (1986). Dynamics of Hierarchical Systems (Springer-Verlag). Nicolis, J.S. (1987). Chaotic dynamics in biological information processing: a heuristic outline. In Chaos in Biological Systems, ed. H. Degn, A.V. Holden, and L.F. Olsen, pp. 221-32 (Plenum Press). Oono, Y. and Yeung, C. (1987). A cell dynamical system model of chemical turbulence. J. Stat. Phys. 48, 593-644. Paladin, G. and Vulpiani, A. (1987). Anomalous scaling laws in multifractal objects. Phys. Reports 156, 147-225. Popper, K. (1968) The Logic of Scientific Discovery (Harper and Row). Popper, K., (1982) The Open Universe (Rowman & Littlefield). Prigogine, I. and Stenger, I. (1984) Order of Chaos (Bantam). Rosen, R. (1985). Organisms as causal systems which are not mechanisms: an essay into the nature of complexity. In Theoretical Biology and Complexity, Ed. R. Rosen. (Academic Press). Rosen, R. (1988). The epistemology of complexity. In Dynamic Patterns in Complex Systems. pp. 7-30. J.A.S. Kelso, A.J. Mandell and M.F. Shelsinger (World Scientific). Wicken, J.S. (1987). Evolution, Thermodynamics, and Information (Oxford University Press). Zvonkin, A.K., and Levin, L.A. (1970). The complexity of finite objects and the algorithm-theoretic foundations of the notions of information and randomness, Russian Math. Surveys 25, 85-127.
Appendix J: On the Cartwright-Littlewood and Levinson studies of the forced relaxation oscillator
The contents of this appendix has been outlined in both Chapters 5 and 6. However, these studies are so important, both historically and because they remain unique in the dynamic perspectives which they provide, it seems worthwhile to review, and to give a few more details about these studies. In 1927 van der Pol and van der Mark published a short note, entitled `Frequency Demultiplication', in which they reported on an experimental study of the generation of
the subharmonics of a harmonic oscillator (period, T) by connecting it with a relaxation oscillator. As they changed the capacitance, and hence the natural frequency of the (uncoupled) relaxation oscillator (To ir/2RC), they observed a sequence of
jumps in the period of the current in the coupled system to the values nT (n = 1, 2, 3, ... ,40,. ..). The system they used, containing a neon discharge tube, and their experimental results are indicated in Fig. J 1. The broken curve is the natural period of the uncoupled relaxation oscillator. While they commented on the occurrence of noisy
regions (indicated by the hatched regions in the figure), which they viewed as `a subsidiary phenomenon', they did not discuss the hysteresis effect which they clearly indicated in the figure. Whether or not these results were the impetus for the theoretical Fig. J1
E
Period of system E0 = 7.5 V
1111111 111
15T
i
5T
T
77
}
10T
1 r t i
1
T
Bands of `noise'
'
I
Hysteresis
i
2
3 X 10-3 µF
- No C
Appendix J
530
research of Cartwright and Littlewood, they remarked on the fact that they found substantiation for one of their important results in the two stable subharmonics shown in these hysteresis regions of van der Pol and van der Mark. This fact is rather curious, and indeed has a touch of irony, since the above experimental results do not in fact correlate with the theoretical results of Cartwright and Littlewood (which is not to say that the experiment contradicts their results, but neither does it clearly substantiate their findings). This will be clarified below. Conversely they remarked about some of their theoretical results related `to non-stable motions (which the experiments naturally did not reveal)' despite the noisy regions in the experiments. This will not be discussed below in any detail. At present there does not appear to be any experiment which has been analysed and correlated with the theoretical findings of Cartwright, Littlewood, and Levinson. The studies of Cartwright and Littlewood and that of Levinson established for the first time the existence of a type of dynamics in physically related systems which had previously only been described (by Birkhoff) as a possible behavior of general (abstract)
'dynamical systems'. As such, these works represent a milestone in the theory of nonlinear dynamics, and deserve much more recognition than they have received to date. This is particularly true because these investigations rigorously establish the existence of these extraordinary types of dynamics, without obtaining any solutions nor reverting to computational methods. Unfortunately the price which is paid is that their
papers contain a series of detailed bounds on the solutions of their equations, for various periods of time, which are frequently tedious and sometimes obscure. What will be described below is a bare outline of their work, together with some further results establish recently by Levi (1981). Cartwright and Littlewood first reported in 1945 a brief summary of their research
on the forced van der Pol equation z + k(x2 -1)z + x = b2k cos (At)
(J.1)
in the case of large k (leading to relaxation oscillations for the uncoupled (b = 0) system). The details of this research was published in a series of papers (1947-51). The research by Levinson (1949) dealt with the equation sR + O (x), + Ex = b sin (t)
(J.2)
where 4(x) is, generally, a polynomial having sign features similar to those in (J.1) (for the most general presentation, see Levi), and s is small (similar to large kin (J.1)). For all of his detailed estimates, Levinson made use of a function which is discontinuous and constant in intervals 4(x) = 1,
if 1XI > 1;
O(x) = - 1, if I x I < 1
(J.3)
This has the great advantage that (J.2) can then be solved analytically in each interval
Appendix J
531
(where (J.2) is linear equation) and then solutions can be joined at the discontinuities.
However one does not do that for individual solutions, because the interest is in arbitrarily long-time solutions, and the process would be prohibitively difficult. Nonetheless, Levinson uses this solvability feature of (J.2) and (J.3), to establish the long-time character of certain families of solutions, as did Cartwright and Littlewood. The studies of both Cartwright and Littlewood and of Levinson divide into two parts: (1) They prove that (J. 1) and (J.2), for sufficiently large k or small 8, and for particular
ranges of the parameter b (the strength of the forcing term), possess periodic solutions - with known stability properties - and, most importantly, a `chaotic family' of solutions, F. What is meant by a `chaotic family' of solutions will be made clear shortly, but it should be emphasized that we are not speaking of a family of chaotic solutions, for each solution of the family F is well-behaved and determined by its initial conditions. The chaotic behavior of this family refers to the fact that, for nearly identical initial conditions, the members of F behave in certain arbitrarily assigned different ways from one another. The arbitrariness of the solutions in the family F can be characterized by the fact that there is some member of F which can be put into correspondence with any infinite sequence of coin tosses (i.e., heads or tails), despite the fact that their initial conditions are all very close. The way this correspondence is made will be described shortly. (2) The second part of their studies is to make use of Poincare's first return map in
the extended phase space, to establish a relationship between the asymptotic behavior of all solutions of Q. 1) or (J.2) and this chaotic family of solutions, F. This very pretty analysis shows that these equations possess (for certain ranges of b) what may certainly be called a curious attractor. That is all solutions of these equations (except an unstable limit cycle) tend asymptotically in time to a region of zero area in the Poincare surface of section, KO. This region, KO, contains the map-points of the solutions of the chaotic family F, two stable limit cycles with different periods, and a number of other interesting properties (see below, and Levinson, 1949).
The terminology `curious attractor' was not used by Cartwright, Littlewood, or Levinson, and indeed is not generally used now. For all `practical purposes' this attractor may essentially be an example of a `strange attractor', introduced much later by Ruelle and Takens (1971). This terminology is presently used in a variety of ways, but it generally requires that the attracting set consists of solutions with positive Lyapunov exponents (see e.g., Ruelle's nice brief review, 1980). At present this does not appear to be established for this curious attractor, which contains two stable limit cycles. Indeed
numerical and analog studies have been much more successful in detecting strange attractors by using Shaw's (1981) variant of the forced van der Pol oscillator (see Chapter 5). In any case, the following results of Cartwright, Littlewood, and Levinson
Appendix J
532
(with added refinements by Osipov (1976), and Levi (1981)) apparently are the only mathematically rigorous and precise descriptions of such a very curious attractor. As outlined above, Cartwright and Littlewood first established the character of some solutions of (J.1) for various values of b (provided that k is larger than a ko(b)). For b > 2/3 all solutions tend to a stable limit cycle, which has the period (T = 27E/),) of the forcing term. The more interesting region, 0 < b < 2/3, divides into open intervals, Ai
and Bj (j = 1, 2,...), which are separated by relatively small gaps, gj, illustrated schematically in Fig. J2 (following Levi). If Fig. J2
Al
BO
A2
I.
0
B1
I
82
81
80
r
b
2 3
there is a pair of periodic solutions, one stable and one unstable, which have periods (2nj ± 1)T for some integer nj (constant in the interval A), where again T is the period of the driving force. More interestingly, if beBB
(1) There is a pair of stable periodic solutions of least periods (2n; ± 1)T, for some integer nj. This pair of stable periodic solutions, call them P1 and P2, prove to be very important in establishing the character of the curious attractor of (J.1). Note
that these two stable periodic solutions differ in their periods by 2T, and therefore do not correspond to the two periodic solutions in the hysteresis regions of van der Pol and van der Mark's experiment, since they only differ by T.
(2) There is an unstable limit cycle of period T. (3) There is a chaotic family of solutions, F. Perhaps the easiest way to define what is meant by this `chaotic family' of solutions is
to follow the approach used by Levinson to establish the existence of F. Levinson essentially considered a family of solutions characterized by the fact that for some time in the interval (mod 2n) ?r < T < it + T
(T < 0.1)
(J.4)
they each have the value x = - 1 and a small positive velocity which lies in a prescribed range VO + Av (t, b, E) as illustrated schematically in Fig. J.3. Note that the values of time Fig. J3
x
Ov(t, b, e) I
I
V0
1f
?r+T
t
Appendix J
533
and positions are relative to (J.2) and (J.3). In particular the period of the forcing term is
now 2n (corresponding to 2n/J in (J.1)). Levinson referred to (J.4) as an `odd base' interval. By a judicious choice of the range of initial velocities, Av(t, b, E) in the odd base
interval, he was able to show, with a series of careful and tedious estimates that, for suitable values of b, there are members of this family of solutions which cross the point x = + 1 (again note (J.3)) with small negative velocities at any of the times in the intervals (At = T) which begin at (2n - 1)n
or
(J.5)
(2n + 1)n.
He referred to these intervals as `even base' intervals. In (J.5), n is some positive integer depending only on b and E. The essential point here is that the initial family of solutions now cover two base intervals, which are shifted by an odd number of half periods (of the
forcing term) from the original even base interval. To make this clearer, consider Fig. J.4 illustrating schematically the base intervals and velocity ranges. As this figure Fig. J4
x=-1
x
(2n - 1)7r (2n - 1)7r+r
V0
-V0
x=+1
x=+1
(2n+1)7r (2n+1)7r+r t
7r
7r + r
illustrates, Levinson in fact proved considerably more. He showed that, by a careful selection of the range vo + Av(t, b, e), not only did the odd base family cover these two even base intervals, but moreover all of the solutions had velocities inside the range - vo - Av(t - (2n - 1)n, b, E) and - vo - Av(t - (2n + 1)n, b, e) respectively. Thus the first band in the figure maps into the interior regions in the second and third intervals. The next step is to note that this result can easily be extended to all future times. The forcing term in (J.2) is negative in the odd base intervals (J.4) and positive in the even base intervals (J.5), since they differ by an odd number of half periods. Moreover the solutions are at x = - 1 and x = + I in these respective intervals. Because of the symmetry properties of (J.2), it follows that solutions can be found which begin in these even and odd base intervals with opposite velocities, which subsequently retain their antisymmetric relationship. In other words there are solutions x+(t), such that x_(t) = - x+(t + (2n ± 1)n) and such that some solution satisfies x_(t) _ - 1 for any it < t < it + T. This of course is a general feature of solutions of (J.2), if ¢(- x) = - (P(x), and not restricted to the above family. However this shows that the same analysis which proved that the above solutions map into the inverted regions in the last figure can be used again to conclude that these solutions will in turn map back inside the original velocity
534
Appendix J
ranges, but not in the odd base intervals which begin at (2n ± 1)271 or (2n)2n, and where
x = - 1. This argument obviously can be applied for all values of future time, which allows us to draw important conclusions without obtaining any particular solution! The first conclusion is that there exists a chaotic family of solutions, F. It is called a chaotic family because it contains solutions which can be put into correspondence (i.e., can be uniquely associated) with any infinite sequence of coin tosses (a Bernoulli sequence). That such a family exists can be easily seen from above discussion. In any base interval we can find a solution which goes to an interval that is shifted in time by either (2n - 1)71 or (2n + 1)71 (corresponding to heads or tails). Therefore, given any finite sequence of (-, +), or (h, t), we can find a solution which passes through a series of base intervals that are time-shifted by a corresponding amount, (2n ± 1)71 relative to the last base interval. In fact one can obtain an open set S(N) of such solutions (because the
base intervals are open) for any sequence {dk}N_N, where dk = ± 1 (for each k = 0, 1, 2, ...). Clearly the set S(N + 1) is a subset of S(N), so that, as N oo, one has a nested sequence of open sets, so S(oo) is not empty. Thus F exists. A second important point to note is that most of the other solutions `converge', in some sense, toward the family F as t - + oo. This will be discussed further below. (4) Within the family F there are unstable periodic solutions with any of the periods m(2n ± 1)271 (m = 1, 2, ...). The fact that there are periodic Bernoulli sequences of
these base intervals is not sufficient to prove that any solution is (exactly) periodic. In order to establish the periodicity, it is necessary to show that, as the
initial time is varied continuously across the even base interval, there are corresponding solutions which continuously cross over either odd base interval. When this continuity feature is employed successively over the periodicity of the base intervals, it is not difficult to see that there must be at least one initial state
which has the same periodicity as that of the base intervals. This continuity feature was proved by Levinson, thereby proving the existence of these periodic solutions. The instability of these periodic solutions follows from the fact that they are contained in the chaotic family F (with the power of the continuum), so that some perturbations are bound to take the solutions away from its periodic orbit. This completes part (1) of this analysis! It establishes the existence of certain stable periodic solutions and a chaotic family of solutions F. Part (2) of the analysis addresses the question of the relationship of all solutions to these particular solutions. Clearly, if most solutions `moved away' in the phase space
from the solutions in F, then this chaotic family would be a curiosity, but of little practical (or observational) importance. What Part (2) of the analysis establishes is that
this is not the case, but rather that most solutions `converge' in some sense towards these solutions. To establish these results, use is made of Poincare's first return map which is defined
Appendix J
535
as follows. Let PO = (xo, vo) be a point in the phase space, and x(t; xo, vo) be a solution of
(J.1) and (J.2) with the initial conditions x(0; xo, vo) = xo;
x(0; xo, vo) = vo.
(J.6)
If T is the period of the driving force, then consider the point in the phase space P, = (x(T; xo, vo),
x(T;xo, vo))
(J.7)
This can be used to define a map from the point PO to P,, written simply as
P, = TPo
(J.8)
which is known as Poincare's first return map. The remaining analysis centers on the use of this map and its iterates, Pk = T'P0. We begin with the observation that Levinson's equations (J.2) and (J.3) have a very simple periodic solution, x = b cos (t), provided that b < 1. To show that this periodic solution is unstable, we note that dt
[(x - b cos t)2 + (x + b sin t)2] = 2(x - b cos t)(z + b sin t)
+2(x+bsint)(-x+(z/E)+(b/E)+ sint+bcost) _(2/r)()i+bsint)2>0 provided that 4(x) = -1 (i.e., x < 1). Hence any perturbation of the periodic solution moves away from the periodic trajectory, so it is unstable. From this we can conclude that, if Co is a small circle about the point (x,.z) = (b, 0) in the phase plane, then it will map under T into a somewhat larger enclosing curve, C, (see Fig. J5). Note that (b, 0) is Fig. J5
therefore an unstable fixed point of the return map T If C, = T'"C0 is the curve obtained by the mth iterate of Co, then the interior points of C. are contained in the interior of C.+, (Interior C. c Interior C.+,). Levinson then defines the set of points CO
Interior Cm
Appendix J
536
which is simply connected and open. At this point there is really no indication of how large this set of points may be. The second step is to show that there is another large enclosing curve, CO (note now the superscripts rather than subscripts), which, after a sufficient number of iterates (say N), lies entirely inside of Co. In other words, we now have a contracting region
Interior T"C° c Interior CO; (n > N). This is illustrated schematically in Fig. J.6. Levinson next considers the set of points Fig. J6
which are the union of the curves and their interiors (so that the set is closed) and then considers the intersection of all of these sets to define a new set, 00
K=
n=N A
(Interior C" U C")
(J.10)
which is simply connected and closed. From their definitions it is clear that both sets H and K are invariant under the map T (TH = H, and TK = K). It is also clear that the set K contains the set H (see the figure), so now it is clear that H is a bounded set in the phase space. Levinson finally defines the very important set of points
Ko=K-H
(J.11)
which is a closed set (because K is closed and H is open), and `ring-shaped' (an annular region). K° is that set which is the limit of the shaded region of the last figure as both n and m tend to infinity, `squeezing' the annular region. How much is this annular region squeezed? To answer this, consider any region, D (t),
whose boundary is governed by the equations
at x =
x; at x
- k4(x)z - x + bk sin (t)
(J. 12)
Appendix J
537
where 4(x) is defined by (J.3), and k = 1//c. The area of this region
dxdz
A=
(J. 13)
n«)
which can be written in the form
A = fno W dxo dvo,
W
,Jc
ax
ax
axo ax
avo ax
axo
avo
(J.14)
where (xo, vo) are the initial conditions (see (J.6)). In the same way that the Liouville theorem is established, one readily finds that the Jacobian W satisfies dW
at
ko(x)W
when the equations of motion are (J.12). Therefore W = Woexp{-k f O(x(t'))dt'I.
(J.15)
Now let CIO be the shaded region in the last figure for large n and 1, which includes all of
the points of K0. Levinson showed that the solutions in this region rapidly cross the region -1 < x < + 1, and spend most of their time in the regions I x I > 1, where they reverse their velocity many times (see Chapter 5, Section 14, for illustrations of these dynamics). Therefore, most of the time 4 (x(t)) = + 1 for these solutions. It follows then, from (J. 15), that W(t) --+0 as t -* + oo, so that the area of f 0, given by (J. 14), tends to zero as t - + oo. Since the set KO is interior to 00, it follows that its area also tends to zero. This set, KO, is the curious attracting set for the equations (J.1) and (J.2), where O(x) need not be limited to (J.3), but can be any function satisfying 0(- x) = O(x), ¢(x) < 0 for x I < 1, and 4(x) > 0 for I x I > 1. It has the following remarkable list of properties: (1) KO is a closed set with zero area (hence it contains no open sets - it has an `empty interior'). (2) The set KO is the boundary between the interior set H, (J.9), and an exterior set
(the complement of H - KO, equal to R2 - K). Therefore any neighborhood of any point in KO contains both interior and exterior points. (3) KO contains (2n - 1) points corresponding to one stable limit cycle (if beBj, see under (1)), and (2n + 1) points corresponding to the other stable limit cycle. Because of this, KO cannot be a simple closed curve, C. For, if it were, then one can define a rotation number p(T) for this invariant curve (TC = C). In order for it to have the above two periodicities, this rotation number must satisfy p(T) = N1/(2n - 1) and p(T) = N2/(2n + 1) for two positive integers (N1, N2) which
Appendix J
538
are smaller than their respective denominators. This requires that (2n + 1)/(2n - 1) = Ni/N2 or 2n = (N1 - N2)/(N1 + N2), which is impossible for any integer n.
(4) The set K° is invariant under map T (because it comes from invariant sets, (J.1 1)).
(5) K° is an attracting set. With the exception of the single unstable fixed point of the map T (e.g., the point (b, 0) in Levinson's case), all points in the phase plane tend towards K° under the map T (i.e., T"P has its limit points in K°), so the w-limit
points of all P are contained in K0, except one fixed point. (6) K° contains the continuum of points which correspond to the chaotic family of solutions, F. It is this fact which insures that the attractor K° is indeed strange. We know that the points corresponding to the family F are in K0, because K° contains the w-limit points of all Pin the phase space (except the fixed point). The
family F however is the set of a particular family of solutions, and hence the points corresponding to F under the map T must be in the set K0. (7) The points of K° have yet another more abstract property associated with Birkhoff `discontinuous recurrent' point groups. One expression of this is that the points of K° are neither exterior accessible nor interior accessible - which is to say that they are not the limit point of a C° curve which lies entirely in the exterior (R2 - K) or entirely in the interior (H) regions. This is not Birkhoffs definition of a discontinuous recurrent point group, which is related to the map T, rather then the above continuity description, but that will not be discussed further here. (8) (Levi) K° contains a Cantor set, and within this set are solutions with rotation numbers which exactly cover the interval 1/(2n + 1) 5 p(T) < 1/(2n - 1), where the rotation number is about the unstable fixed point. This elementary discussion does not do justice to the many refinements and new
aspects which have been discovered by Osipov (1976) and Levi (1981). These discussions are recommended for those desiring more details concerning this fascinating `simple' system.
Appendix K: Smale's horseshoe map
Levinson's analysis of the forced van der Pol oscillator was described in Appendix J. In this analysis he established that there exists a set of solutions of this system (which he called KO) that can be put into one-to-one correspondence with any sequence of zeros and ones, (... 0, 1, 0, 0, 1, 0,...). In other words, the set KO consists of solutions
which, although they are deterministic, appear to be as random as the Bernoulli process of flipping a coin. These solutions were established with the help of a Poincare-type map, but involved a rather long and tedious analysis. Levinson's results inspired Smale (1965) to invent a simple abstract map (i.e., not obviously related to
any known physical system) that also has a set of solutions that can be put into one-to-one correspondence with any Bernoulli sequence, (...0, 1, 1,0, 1,...). Smale's map-dynamics also have some other similarities, and differences, with Levinson's Ko-set of solutions of the forced van der Pol oscillator. The following is a short account of Smale's so-called horseshoe map (for reasons which will be shortly obvious).
More details can be found in Nitecki (1971) and Guckenheimer (1979). We will use a version which illustrates some other similarities with Levinson's analyses. Consider a bounded region, R in the (x, y) plane, which is the union of three parts, R = A u B u C (Fig. K 1). The map, f, consists of squeezing R in the x direction while stretching it in the y direction, then bending it into a horseshoe shape, and finally Fig. KI
PC)
Appendix K
540
placing it back inside R, as illustrated. Thus, f (A) and f (C) are both in C, whereas the bend of f (B) is in A. If this is repeated, the map f 2(R) looks something like the dark region in Fig. K1 (f 2(R) c f(R)). We note that the points in A are continually swept into C by f, f (A) c C, whereas those in C remain in C, f (C) c C. Moreover, many points of B are either mapped into A or C, and hence are permanently lost from B, whereas C attracts all of these points (in the future). However, all of the points are not lost from B under the action of f"(R), even as n--> oo. We see from above that B n f 2(B) consists of four vertical strips, whereas B n f 3(B) = n = o f k(B) consists of 23 vertical strips, and so on (Fig. K2). It is not difficult to convincek oneself Fig. K2
that the number of strips obtained in the limit, limn n k = of k(B), is infinite, but also uncountable (nonenumerable); for the construction of the strips is analogous to the construction of Cantor sets (Section 2.6).
The region which maps onto R, when the upper half is rotated clockwise by it, compressed in the x direction, and stretched in the y direction, is illustrated in Fig. K3. Fig. K3
f-l (C)
f-1(A)
f-' (B)
It is a horizontal horseshoe-looking region. Now B n f -1(B) consists of two horizontal strips, and Br )f -k(B) has 2k horizontal strips. =f-2 (B) n f -1(B) n B n f 1(B) n f 2 (B) which is Next, consider the set of points S2 the collection of dark regions in Fig. K4. The set of points n
Sn = n f k(B) k= -n
similarly consists of 4 " regions, interior to the above set (i.e., Sn c S" -1). Therefore,
Appendix K
541
Fig. K4
the set of points 0 =n - c f k(B) is an uncountable invariant set (f (Q) = £2). The f-dynamics of points in the set 0 have the same feature which Levinson established for the set of solutions, K0. Namely, we can find points P in S2 whose
map dynamics can be put into one-to-one correspondence with any Bernoulli sequence, ( . . . 0,1,0,0 ....). One way to do this is to identify the `left' and `right' regions
of B n f (B), and obtain a sequence { ak } k the prescription
o
associated with a point peQ, by using
(0 if f k(p) is in the left B n f (B) Ilk =
1 if f k(p) is in the right B n f (B).
In Fig. K5, {...a_2,a_1,a0,a1,a2,...} = {..., 1, 0, 1, 1,0,0,...}. It is not difficult to Fig. K5
f3p
f-1P f2p
ak = 0
ak = 1
see that the correspondence of any sequence {ak}k o is unique to one p. Briefly, start with {a0,. .. , a } which identifies one vertical strip uniquely, whereas la -w ... , a-,) likewise identifies a unique horizontal strip; then increase n. This converges onto one point, the p in question.
The association of a sequence {ak} with p is a `symbolic' representation of the dynamics of p. Let £' represent the set of all sequences {ak}. A `shift dynamics', u:,F -+ 1, of the sequences in E is defined to be a({ak }) _ {bk } where bk _ 1 = ak. In other words, all elements of {ak} are shifted one place to the left under a. We note that if {ak} is associated with p, then {bk} is associated with f(p). Using this dynamic
542
Appendix K
concept, it is not difficult to see that there are 2" different sequences in I which have period n, 6"({ak}) _ {ak},
simply by considering the groups of n arbitrary elements which must reproduce under the n-shift, a". Hence, there are exactly 2" distinct periodic orbits with period n. Of course, the same is true for Levinson's set of solutions, Ko and all of these periodic
solutions are unstable, both in Ko and here in I. However, Ko also contains two stable periodic orbits.
With the aid of the shift dynamics, we can show that 0 contains an orbit of f (call it F'), which is dense in 0; meaning that every point of 0 is the limit point of some sequence of points selected from F. To see that this is true, first note that two points of S2 are very close if their sequences in I, {ak}, are the same for all ski N, where N is large; this is ak = ak for all ski N. These finite sequences {ak}N, contain 2N + 1 elements, so there are 22N+ 1 such distinct sequences. We can order these finite sequences in some fashion, for example according to their binary value. For N = 0, the possible sequences are 0 and 1' for N = 1, there are eight sequences (0, 0, 0), (0, 0, 1), (0, 1, 0).... ; for N = 2, thirty-two sequences. We now form a countable infinite sequence, consisting of these groups of sequences collected in the order N = 0, 1, 2, ... . { bk I= (0, 1,0,0,0,0,0, 1, 0,1,0'...)
This can, of course, be made infinite in both directions using a symmetric collection, if one wants. The main point however is that, by applying the shift o to {bk} a suitable we can produce a sequence which is arbitrarily number of times, o'({bk}) = close to any other sequence {ak } ; that is a * = ak for all I k I < N, and for any N. Thus {bk}, or its symmetrized form, is one orbit, F, which is dense in D. This result clearly applies also to Levinson's Ko-set. A significant difference between Smale's set S2 and Levinson's set Ko is that the latter is localized to annular region with zero area. Moreover, while both Smale's maps, f"(R) and Levinson's T"-map have sinks, as n-> oo, the sink of f"(R) is not a point in 0, whereas T" has two stable periodic sinks in K0.
Appendix L: notes on the KolmogorovArnold-Moser theorem
In this appendix an elementary outline of the KAM theorem will be given. Its purpose
is simply to give some of the underlying ideas involved in this theorem, with no pretense of detailed mathematical rigor. For these details, see Arnold (1963). First we need a little background information. If F(O) is a real analytic function, periodic in all Oi with period 2n, and I F I < M when I Im Oi I < p, then the function can be written F(O) _
ak exp (ik-0)
(a - k = ak ),
k= -w
and the coefficients satisfy
IakI<Mexp(-Iklp) where I k I = I k 1 I + I k3 l +
+ I k" 1. This inequality follows from
f...fF(O)exp(-ik-O)dOj Za d0".
ak = (27r) -" 0
Deform the contour to lie along 0S = Xs ± ip(01< Xk < 2n), plus canceling end segments,
where the sign ± is selected so that + ks < 0 for each s. Using the bound I F I < M then yields the above inequality. F(O) is called a quasi-periodic function of time. This is a subclass of H. Bohr's almost-periodic functions, which allow any denumerable set of frequencies, and require
only bounded and continuous functions. However quasi-periodic functions are particularly important in dynamics, as illustrated below.
Standard perturbation of near-integrable systems We now consider a Hamiltonian which is a power series in a parameter µ,
H(p,q)=Ho(p)+,uH1(p,q)+
u2H2(p,q)+...,
Appendix L
544
where H 1 is 27r-periodic in the q,, and satisfies I pH
1
I
M for I Im q; I
p. Then H 1(p, q)
is a quasi-periodic function of q, so H 1(p, q) _
hk(p) exp (ik-q),
and the coefficients have the bounded property I
hk I<
M exp (- I k l p).
If pER", then the solutions of the equations of motion, Pk = - aH/aqk,
qk = aH/apk
are confined to N-tori, T", if p = 0 (i.e., Pk = constant, Ok = (J)k t + Ok ). For p 0 0 we can attempt to determine the motions by various perturbation methods. The purpose of the present outline is to show that, if p is sufficiently small, most solutions remain on N-tori, which of course differ from the p = 0 tori.
The perturbative method we will use begins by looking for a canonical transformation (p', q1),
(p, q)
which eliminates the term which is linear in p from the Hamiltonian, so that H(p, q) = Ho (p') + p2 H; (p', q') + O(p3).
This means that a solution in the (p',q') phase space lies on an N-torus, `except' for a factor of O(p2), which we will need to show is smaller than the original term of O(p).
To accomplish this transformation we use the generating function S(q,p1)= E Sk(p')eXp(ik-q), k#0
involving the old q and new p, yielding the canonical transformation
as
p=p + p aq ,
q1 = q +p
-. as
We now write H1(p, q) = ho(p) + lY1(p, q), where
H1 =
hk(p) eXp (ik-q) k*O
Substituting the transforms into H(p, q), we obtain H(p, q) = H0(p') + pho(p') + p H 1(p', q') +
aH OS ap
°
aq 11
+ O(p2).
Appendix L
545
We want the last bracket to vanish, leaving the q dependence only in the terms of O(µ2). For this to vanish for all q, we need 0
where
w(P) = aH0/ap Therefore we have a formal solution:
S(q,P')= I
ihk(P1)
exp(ik-q)
k#0(k w(P'))
If all hk 0 0, then this series has arbitrarily small denominators, may not converge.
0, and hence
First question Are there some w's for which the above sum converges, even though the denominators get arbitrarily small? If the (w's are not rationally related 0 0, the denominators, become very small only if I k I is large. If (k (o) does not decrease too fast as I k I increases, the series
can converge because the numerator decreases as exp (-1 k I p) (from the introductory remarks). If I k co I >, K I k I - " for some (K, v) and all k, then the series converges. For in this case, I Sk I= I hk/(k w) I,<M/K exp (- I k I p) I k I ". Moreover 1 k I" ,<(v/e)"exp I k I b/b" for
any k, v > 0, and q > 0. To show this note that f (x) = x - v In (x) has a minimum at x = v. Hence exp (x)/x" > exp (v)/v". If we set x = 1 k 16, the previous inequality follows. So
M v "exp(-Ikip+IkIb) K
ISI k
8"
e
exp(IlmgkIlk))
Therefore, if
{ note the slightly reduced strip range, from p to p - 261
I Im qk I < p - 26
then ISI
(v)"exp( IkIb)
Y
K
so
ISI1
KI e
fa"L1+2 Y exp(-kb)]n k>O
M vK(v)'"1 1+exp(-b) e
b"
[l_exp(_l
"
Appendix L
546
where n is the number of degrees of freedom. Therefore the series converges for co's which satisfy (k (o) ,>K I k I -"
(all k).
Second question Are there many such w's?
Answer Yes, if K is small and v > n - 1. Proof Consider those co's such that (k (o) < K I k I - '. They are nearly orthogonal to k, and lie in a region illustrated schematically in Fig. L 1. Let I w I <, Sl, then the volume
Fig. LI
of the region is K I k I -'- 'D(12, n), where D depends on S2 and the dimension, but not on k. Next, summing over all k with fixed I k I gives A"k" -' such volumes (because
k"-' is proportional to the area of an n-sphere). Finally summing over k gives the volume of the (o-space satisfying (k co) < K I k I -
w-volume for all k=KD(Q,n)A"
x
k"-'k-"-'.
k=0
This series converges if v > n - 1. We have therefore shown that, if v > n - 1, the number of co's satisfying (k co) < K I k I
goes to zero as K - 0. Therefore, for small K, most co's satisfy
(k.(o)>KIkI
(v>n-1)
for all k. For these w's the series for the generating function S(q,p') converges.
547
Appendix L
So what's the problem? (A) Even if this series converges for some w(p) at p = p*, if the w(p)s are functionally independent, namely' 2
Det
a(O`
= Det a H
0,
aPiaP;
aP;
then in any neighborhood of p* there is a p such that
0. At this p
the perturbation series generally fails (exception: if the corresponding hk(p) = 0, e.g., Whittaker's adelphic integrals). So, generally the perturbation series fails in any region of p-space. If it does converge the resulting function is everywhere discontinuous!
(B) Moreover, this is just the start of the perturbation series to simply H(p, q)! It reduces H0(P) + pH1(P, q) + ... to
Ho(P')+ u2H1(p',q')+ ... The standard perturbation would next reduce this to Ho(P2) + p3H (p2, q2) + ...
i The convergence of this series of reductions of H is doubtful, at least generally (it did converge for Whittaker). Siegel (1954) indeed proved that generally this series diverges.
In 1954 Kolmogorov made two simple but very important, suggestions on how to overcome the difficulties (A) and (B): (A*) Instead of considering p-regions and the associated w(p), which vary, fix the frequencies from the start to be w* satisfying
KIkI-"
(all k).
This is possible since we assume the w(p)s are independent (see (A)). This insures that the generating function's series indeed converges. That is S(q,
p') = E
lhk(P')
k 10 (k w*)
exp (ik-q)
t See the discussion at the end concerning an independent condition for functional independence of the co(p)s. In other words, the present condition is sufficient, but not necessary for the functional independence on an energy surface.
Appendix L
548
has the bound ISI < M/K(v/e)"(1/5v)(3/8)", when the qk are limited to the range I Im qk I < p - 26. Note that S > 0 is arbitrary, but if it is very small (so that the region IIm qI is not reduced much below p) then the bound on S is very large; we want to keep this bound small in order to prove convergence of the further series needed to reduce H(p, q) to a function of p alone (problem (B)).
(B*) The second suggestion Kolmogorov made is to use a rapidly converging perturbation method (Newton's method). Instead of the series of functions H i (p", q") whose coefficients are successively µ2,µ3,µ4,µs, much more rapidl y conveygent series µ ,µ ,µ , ...,µ "
. , one obtains the
The way to obtain rapid convergence is to build the next approximation upon the results obtained up to that point, rather than the initial series. Thus, after the first canonical transformation we have
Ho(P1)+µ2H,(p',q')+ .. If we now introduce a canonical transformation (p 1, q') - (p 2, q 2) we can transform the last Hamiltonian into H2(P2) + µ4H
by exactly the same
H' (p') + µ2H '(p', q') +
i(p2, p2) + ...
analysis used in going from Ho(p) + pH 1(p, q) into .
Note that, just as
µ2H i - (µH1)2 (which must yet be shown), (µ2H')2, which gives the rapid convergence, going as (µ 2") we also obtain µ4H2 We need to first show that if
Ho(p)+H1(p,q);
IH1I <M1,IImgI
is transformed to
H'(p')+H'(p',q'), then, in fact Hi is related quadratically to H1, by a relationship IH'I<M2=Mi/a2v,
provided that
IImq'I
IP1-P#11<M. In the last expression p*1 satisfies (8Ho/8p')P*i = co*, where w* are the fixed frequencies
in (A*) above. Here 6 is arbitrary, but the sequence of S's for subsequent transforms needs to be selected with care to ensure convergence.
Appendix L
549
A bare sketch of the proof that H' - (H 1)2 will now be given. In the initial Hamiltonian Ho(P) +,uho(P) + uH1(P, q) = Ho(P' + zp) + µho(P' + zp) + PH1(P' + Ap, q) we set 1
aS(q, P')
µ
AP=P-p
Oq
Expanding in Ap, the last expression becomes
Ho(P')+pho(P')+pH1(P',q)+ 1
(02 HO
+2 a12
<( aHa+(P'-P*') l \\ ap )P*
aH1
ho
)(zip)2
8Ho
02 H
ap) z) Pert 1
AP
a2s
3
+uapl4P+p apt dp+ ap1 µagap1Ap+O(µ )
Because we want to deal with fixed co* = (aHo/0p')P.,, a secondary Taylor expansion about p*' has to be included above. If we require
IP'-p*'I '< M, then this term is included in H'1. Next, to obtain bounds on derivatives of S we use the following reasoning. Since
f(W ) =
27Ci
f -xd
if If(x)I'< M
and is analytic for xED, then for xeD - 6 a2f
2M
axe
8
2
Using this result, replacing f by S, and the above bound M with the bound
M 3n " ISI
K
e
1
_ ML
62"
K62"
for I Im q I < p - 6 and I p' - p *' I < M, we can get an upper bound on each term of H i in the above expression, and then take the largest of these. This largest term is essentially p(ML/K62"+ 1)2 Therefore
IHiI
<(iM2/K/3)a-a"-6
(M/Sv)z
The exact power of 6 is not important, only its n dependence (v = 4n - 6).
Appendix L
550
Now consider the sequence of canonical transforms yielding the Hamiltonians Ho(ps) + Hs (p", qs)
in the domains Ips - p*SI < Ms;
aps )P*S CaH'O
=
w*.
Ilmgsl
Then, as we just proved,
IH l,<MS+1=(MS/bs)2
Now what is needed is a sequence of bS such that MS -+ 0, but at the same time 0. Start with ps -+ M1 = b; (T large and fixed, say T = 4v + 1) 61: `sufficiently small'; say 61 < p/2, so b1 < 1.
Now pick subsequent bS according to the relation 5s+
1 = 8s /2
(so that bS -> 0, as s - oo).
Next we show that the bounds on all Hi satisfy MS < bs . then MS+ 1 = (M/bs)2 = bs T 2" If T = 4v + 1,2T-2v=6v+2 > i T, so MS+ 1 < (6311)T = bs Therefore we have
Proof It holds for s = 1. If it holds for
s,
S
S -'OD
1
S-00
Finally we need to show that the decreasing strips I Im qs I < ps - QS _ 1 - SS remain positive as s -+ oo. ps = pl so we need a bounded sum as s - oo. We have bS+ 1 = bs i2 = bSbs/2 bSb 1/2 therefore bs+ 1 < (6112)S b 61/(1- 61/2), proving that p,,,, > 0 if bl is sufficiently small. This ends this rough outline of the proof of the KAM theorem. Finally we return to the independence of the function co(p). We want to show that there is another condition, independent of
so yn
a2H
*0
Iapiap;
which insures functionally independent frequencies. Consider a function of the Hamiltonian, F(H), and a new 'Hamiltonian'
H = F(H)/F'(h)
Appendix L
551
where h is an allowed value of H, and F' is the derivative of F(H). On the energy surface H = h, H gives the same equations of motion
p = - aH/aq,
4 = aH/ap.
The Hessian of H(p) is aZH
apiapj
=
(F'(h))-n
02H
F'
OH OH
+ F" - apiapi api apj
For simplicity, consider the case (i, j = 1, 2) and write aH/api - Hi. The determinant is then
F'H11+F"H
F'H12+F"H1H2 _ (F,)2 I Hi
F'H21 + F"H1H2
F'H22 + F"H2
I j.
-
(F'F")
H11
H12
H1
H21
H22 H2
H2
H1
0
which can be confirmed with a little effort. Therefore, if the Hessian I Hij I = 0, but (using a shorthand notation) H,
Hi
Hi
0
: 0,
then the frequencies cu - aH/ap are functionally independent. A system which satisfies the last condition is called isoenergetically nondegenerate. Thus this condition can be used in place of the condition I Hid 10 0, to ensure independence of the w(p)s on the energy surface. The following examples illustrate that the zeros of these two determinants represent independent conditions on H(p,q). (1) If H = p1 g(q) + f (p2, ... , pn, q), then I Hij I = 0, because H1 j = 0. However, generally ...
H22 H.
Hi
H;
0
H2n
= H2
:0.
1
...
Hn2
Hnn
(2) If H=log(P1/P2),IHi;I=-(P1P2)-2:0,but Hij Hi Hi 0
_Pi
0
p1'
0
P22 p21
pi 1
p21 =0. 0
Bibliography
Ablowitz, M.J. (1981). Remarks on nonlinear evolution equations and inverse scattering transform. In Nonlinear Phenomena in Physics and Biology, ed. R. Enns et al. Ablowitz, M.J. & Fokas, A.S. (1983). Comments on the inverse scattering transform and related nonlinear evolution equations. In Nonlinear Phenomena. Lecture Notes in Physics 189, ed. Wolf, K.B. pp. 4-23. Springer-Verlag. Ablowitz, M.J. & Nachman, A.I. (1986). Multidimensional nonlinear evolution equations and inverse scattering, Phvsica 18D, 223-41. Ablowitz, M.J., Kaup, D.J., Newell, A.C. & Segur, H. (1973). Method for solving the Sine-Gordon Equations, Phys. Lett. 30, 1262-4. Ablowitz, M.J., Kaup, D.J. & Newell, A.C. (1974). Coherent pulse propagation, a dispersive irreversible phenomena. J. Math. Phys. 15, 1852-8. Ablowitz, M.J., Kaup, D.K. Newell, A.C. & Segur, H. (1974). The inverse scattering transformfourier analysis for nonlinear problems. Studies Appl. Math. 53, 249-315. Ablowitz, M.J. & Satsuma, J. (1978). Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180-6. Ablowitz, M.J., Ramani, A. & Segur, H. (1978). Nonlinear evolution equations and ordinary differential equations. Lett. Nuovo. Cimento 23, 333-8. Ablowitz, M.J., Ramani, A. & Segur, H. (1980). A connection between nonlinear evolution equations and ordinary differential equations of p-type. I. J. Math. Phys. 21, 715-21. Ablowitz, M.J. & Satsuma, J. (1978). Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180-6. Ablowitz, M.J. & Segur, H. (1977a). Exact linearization of a Painleve transcendent. Phys. Rev. Letters 38, 1103-6. Ablowitz, M.J. & Segur, H. (1977b). Asymptotic solutions of the Korteweg-deVries equation. Studies Appl. Math. 57, 13-44. Ablowitz, M.J. & Segur, H. (1981). Solitons and the Inverse Scattering Transform. SIAM. Abraham, E. & Smith, S.D. (1982). Optical bistability and related devices. Rep. Prog. Phys. 45, 815-85. Abraham, N.B. (Ed.) (1989). Quantitative Measures of Complex Dynamical Systems. Plenum. Abraham, R. & Marsden, J.E. (1978). Foundations of Mechanics, 2nd edn. Benjamin/Cummings. (Contains: A.N. Kolmogorov's 1954 Address to the International Congress of Mathematicians: `The General Theory of Dynamical Systems and Classical Mechanics'.) Abraham, R. & Shaw, C. Dynamics: The Geometry 0/ Behavior; Part one: Periodic Behavior (1982); Part two: Chaotic Behavior (1983); Part three: Global Behavior (1983); Part four: Bifurcation behavior (1984). Aerial Press: Santa Cruz, CA.
554
Bibliography
Abraham, R. & Simo, C. (1985). Bifurcations and chaos in forced van der Pol systems. In Singularities and Dynamical Systems, ed. S.N. Pnevmatikos, pp. 313-44. Elsevier Sci. Pub. Abraham, N.B., Gollub, J.P. & Swinney, H.L. (1984). Testing nonlinear dynamics, Physica 11D, 25264.
Adey, W.R. (1975). Evidence for cooperative mechanisms in the susceptibility of cerebral tissue to environmental and intrinsic electric fields. In Functional Linkage in Biomolecular Systems, ed. F.O. Schmit, D.M. Schneider & D.M. Crothers, Raven Press: New York. Adler, R.L., Konheim, A.G. & McAndrew, M.H. (1965). Topological Entropy, Trans. Amer. Math. Soc. 114, 309-19. Ahlers, G. & Behringer, R. (1978a). Evolution of turbulence from the Rayleigh-Benard Instability. Phys. Rev. Lett. 40, 712-16. Ahlers, G. & Behringer, R.P. (1978b). The Rayleigh-Benard, instability and the evolution of turbulence. Prog. Theor. Phys. Suppl. 64, 186-201. Aikawa, T. (1987). The Pomeau-Manneville intermittent transition to chaos in hydrodynamic pulsation models, Astrophys. Space Sci. 139, 281-93. Aizawa, Y. & Nishikawa, 1. (1986). Toward the classification of the patterns generated by onedimensional cell automata. In Dynamical Systems and Nonlinear Oscillations, ed. G. Ikegami, pp. 210-22, World Scientific. Albrecht, F., Gatzke, H., Haddad, A. & Wax, N. (1974). The dynamics of two interacting populations, J. Math. Anal. and Appl. 46, 658-70. Alekseev, V.M. & Yakobson, M.V. (1981). Symbolic dynamics and hyperbolic dynamic systems. Phys. Rep. 75, 287-325. Alfaro, V. de & Rasetti, M. (1978). Structural stability theory and phase transition models. Fortschritte der Physik 26, 143-73. Alfsen, K.H. & Fr¢yland, J. (1984). Systematics of the Lorenz Model at a = 10. Univ. of Oslo, Institute of Physics: Norway. Amann, H., Bazley, N. & Kirchgassner, K. (Eds.) (1981). Applications of Nonlinear Analysis in the Physical Sciences. Pitman: Bath. Amari, S. & Arbib, M.A. (Eds.) (1982). Competition and Cooperation in Neutral Nets, Lecture Notes in Biomathematics 45. Springer-Verlag. Ames, W.F. (1965, 1972). Nonlinear Partial Differential Equations in Engineering, Vol. I, Vol. 11, Academic Press. Ames, W.F. (1978). Nonlinear superposition for operator equations. In Nonlinear Equations in Abstract Spaces ed. V. Lakshmikantham, pp. 43-66. Academic Press. Anderson, D.L.T. (1971). Stability of time-dependent particle-like solutions in nonlinear field theories II. J. Math. Phys. 12, 945-52. Anderson, P.W., Arrow, K.J. & Pines, D. (Eds.) (1988). The Economy as an Evolving Complex System, Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley. Anderson, R.L., Harnad, J. & Winternitz, P. (1982). Systems of ordinary differential equations with nonlinear superposition principles. Physica 4D, 164-82. Anderson, R.L. & Ibragimov, N.H. (1979). Lie-Backlund Transformations in Applications, SIAM. (Contains historical references.) Andrews, J.G. & McLone, R.R. (Eds.) (1976). Mathematical Modeling. Butterworths. Andronov, A.A., Leontovich, E.A., Gordon, I.I. & Maier, A.G. (1971). Theory of Bifurcations of Dynamic Systems on a Plane. US Dept. Commerce; Israel Program for Scientific Translations. Andronov, A.A.,Khaikin, S.E., Vitt, A.A. (1966). Theory of Oscillators, Pergamon.
Bibliography
555
Anosov, D.V. (1985). Smooth dynamical systems. Introductory article, Amer. Math. Soc. Transl. (2) 125, 1-20. Appel, K. & Haken, W. (1977). Every planar map is four colorable. 1. Discharging. 11. With J. Koch. Reducibility Ill. J. Math. 21, 429-90; 491-567. Appel, K. & Haken, W. (1986). The four color proof suffices. Math. Intelligencer 8, 10-20. Aris, R. (1978). Mathematical Modelling Techniques. Pitman: Bath. Arnold, V.I. (1961). The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case. Sov. Math. Dokl. 2, 247-9. Arnold, V.I. (1963). Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys 18, No.6, 85-189. Arnold, V.I. (1964). Dokl. Akad. Nauk. SSSR 156, 9. Arnold, V.I. (1965). Small denominators. I. Mappings of the circumference onto itself. Trans. Amer. Math. Soc. 2nd Series 46, 213-84. Arnold, V.I. (1972). Lectures on bifurcations in versal families. Russian Math. Surveys 27, 54-123. Arnold, V.I. (1978a). Mathematical Methods of Classical Mechanics. Springer-Verlag. Arnold, V.I. (1978b). Ordinary Differential Equations. M.I.T. Press. Arnold, V.I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag. Arnold, V.I. (1984). Catastrophic Theory. Springer-Verlag. Arnold, V.I. & Avez, A. (1968). Ergodic Problems of Classical Mechanics. W.A. Benjamin. Arnold, V.I., Kozlov, V.V. & Neishtadt, A.I. (1988). Mathematical aspects of classical and celestial mechanics. In Dynamical Systems Ill, ed. V.I. Arnold, Springer-Verlag. Aronson, D.G., Chory, M.A., Hall, G.R. & McGehee, R.P. (1982). Bifurcations from an invariant circle for two-parameter families of maps of the plane. Commun. Math. Phys. 83, 303-54. Aubert, K.E. (1984). Spurious mathematical modelling. Math. Intelligencer 6, 54-60. Aubin, J.P., Saari, D. & Sigmund, K. (Eds.) (1985). Dynamics of Macrosystems. Lecture Notes in Economic and Mathematical Systems 257. Springer-Verlag. Aubry, S. (1983). The twist map, the extended Frenkel-Kontrova model and the devil's staircase. Physica 7D, 240-58. Babloyantz, A. (1986). Molecules, Dynamics and Life, Wiley. Bai-line, H. (1984). Chaos. World Scientific: Singapore. Balazs, N.L. & Voros, A. (1986). Chaos on the pseudosphere, Ph}s. Reports 143, No. 3, 109-240. Balian, R. & Peube, J.L. (Eds.) (1977). Fluid Dynamics. Gordon and Breach. Barber, M.N. & Ninham, B.W. (1970). Random and Restricted Walks. Gordon and Breach. Bard, J.B.L. (1981). A model for generating aspects of zebra and other mammalian coat patterns. J. Theor. Biol. 93, 363-85. Bardos, C. & Bessis, D. (Eds.) (1980). Bifurcation Phenomena in Mathematical Physics and Related Topics. D. Reidel. Barenblatt, G.I., Iooss, G. & Joseph, D.D. (Eds.) (1983). Nonlinear Dynamics and Turbulence. Pitman. Barnsley, M. (1988). Fractals Everywhere, Academic Press. Barnsley, M.F. (1986). Making chaotic dynamical systems to order. In Chaotic Dynamics, ed. M.F. Barnsley & S.G. Demko, pp. 53-68. Academic Press. Barnsley, M.F. & Demko, S. (1985). Iterated function system and the global construction of fractals. Proc. R. Soc. Lond. A399, 243-75. Barnsley, M.F. & Demko, S.G. (Eds.) (1986). Chaotic Dynamics and Fractals. Academic Pres. Barone, A., Exposito, F., Magee, C.J. & Scott, A.C. (1971). Theory and applications of the Sine-
556
Bibliography
Gordon equation. Nuovo Cimento Series 2, 1, 227-67. (Contains extensive references to all applications.) Barrow, J.D. (1982). Chaotic behavior in general relativity. Phys. Rep. 85, 1. Barrow, J.D. (1988). The World within the World. Clarendon Press. Barrow, J.D. and Tipler, F.J. (1988). The Anthropic Cosmological Principle. Oxford Univ. Press. Bartlett, J.H. (1975). Classical and Modern Mechanics. Univ. of Alabama Press. Basar, E., Durusan, R., Gorder, A.P. Ungan, P. (1979). Combined dynamics of E.E.G. and evoked potential. Biol. Cybernetics 34, 21. Beau, W., Metzler, W. & Ueberla, A. (1987). The Route to Chaos of Two Coupled Logistic Maps. Univ. of Kassel. Bedford, T. & Swift, J. (Eds.) (1988). New Directions in Dynamical Systems. Cambridge University Press.
Bednar, J.B. et al. (Eds.) (1983). Conference on Inverse Scattering: Theory and Applications. SIAM. Behringer, R.P. & Ahlers, G. (1987). Heat transport and critical slowing down near the RayleighBenard instability in cylindrical containers. Phys. Letters 62A, 329-31. Bblair, J. & Glass, L. (1985). Universality and self-similarity in the bifurcations of the circle maps. Physica 16D, 2, 143-54. Belousov, B.P. (1959). Sb. Ref. Radiats. Med., 1958. Medgiz: Moscow, 145. Benjamin, T.B. (1978). Bifurcation phenomena in steady flows of a viscous fluid 1. Theory, II. Experiment. Proc. Roy. Soc. Lond. A359, 1-26, 27-43. Bennett, C.H. (1986). On the nature and origin of complexity in discrete, homogeneous, locallyinteracting systems. Foundation Physics 16, 585-92. Bennettin, G.A., Casartelli, G.M., Galgani, L., Giorgilli, A. & Strelcyn, J.M. (1978). On the reliability of numerical studies of stochasticity. Nuovo Cimento 44B, 183-5. Bennettin, G., Galgani, L. & Strelcyn, J.M. (1976). Kolmogorov entropy and numerical experiments. Phys. Rev. A14, 2338-45. Bennettin, G., Galgani, L. & Giorgilla, A. (1985). A proof of Nekhoroshev's theorem for the stability times in nearly-integrable Hamiltonian systems, Celest. Mech. 37, 1-25. Bennettin, G., Galgani, L. & Giorgilli, A. (1987). Realization of holonomic constraints and freezing of high frequency degrees of freedom in light of classical perturbation theory, Commun. Math. Phys. 113, 87-103. Bennetin, G., Giorgilli, A. & Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them. Meccanica 15, 9.
Benzinger, H.E., Burns, S.A. & Palmore, J.I. (1987). Chaotic complex dynamics and Newton's method. Phys. Letters 119A, 441-46. Berge, P. (1979). Experiments on Hydrodynamic Instabilities and the Transition to Turbulence. Lecture Notes in Physics 104, 289-308. Springer-Verlag. Berge, P. & Dubois, M. (1984). Rayleigh-Benard convection. Contemp. Phys. 25, 535-82 Berge, P., Pomeau, Y. & Vidal, C. (1984). Order Within Chaos: Towards a Deterministic Approach to Turbulence. Wiley. Berlekamp, E.R., Conway, J.H. & Guy, R.K. (1982). Winning Ways, for your Mathematical Plays, Vol. 2, Chap. 25. Academic Press. Berry, M.V. (1976). Waves and Thom's theorem. Adv. in Phys. 25, 1-26. Berry, M.V. (1977). Semi-classical mechanics in phase space: a study of Winger's function. Phil. Trans. R. Soc. Lond., 287, 237-71.
Bibliography
557
Berry, M.V. (1978). Regular and irregular motion, pp. 16-120. In Topics in Nonlinear Dynamics, ed. S. Jorna. AIP Conference Proceedings, 46. American Institute of Physics. Berry, M.V. (1979a). Diffractals. J. Phys. A12, 781. Berry, M.V. (1979b). Catastrophe and stochasticity in semiclassical quantum mechanics; Catastrophe and fractal regimes in random waves. In Structural Stability in Physics. Springer-Verlag. Berry, M. (1981). Singularities in waves and rays, pp. 456-541. In Les Houches. Session 35: Physics of Defects, ed. R. Balian, M. Kleman & J.-P. Poirier. North-Holland. Billingsley, P. (1965). Ergodic Theory and Information. Wiley. Birkhoff, G.D. (1913). Proof of Poincare's geometric theorem. Trans. Amar. Math. Soc. 14, 14-22. Birkhoff, G.D. (1917). Dynamical systems with two degrees of freedom. Trans. Amer. Math. Soc. 18, 199-300.
Birkhoff, G.D. (1922). Surface transformations and their dynamical application. Acta Math. 43, 1-119. Birkhoff, G.D. (1927). Dynamical Systems. Amer. Math. Soc. Birkhoff, G. (1955). Hydrodynamics. Dover. Birman, J.S. & Williams, R.F. (1983). Knotted periodic orbits in dynamical systems. I: Lorenz's equations. Topology 22, 47-82. Bishop, A., Campbell, D. and Nicolaenko, B. (Eds.) (1981). Nonlinear problems: Present and Future. Math Studies 61. North-Holland. Bishop, A.R. & Schneider, T. (Eds.) (1978). Solitons and Condensed Matter Physics. Springer-Verlag. Bishop, G. (1985). Pattern selection and low-dimensional chaos in systems of coupled nonlinear oscillation. In Dynamical Problems in Soliton Systems, ed. S. Takeno, pp. 250-7. Springer-Verlag. Block, L., Guckenneimer, J., Misiurewicz, M. & Young, L.S. (1980). Periodic points and topological entropy of one dimensional maps. In Global Theory of'Dynamical Systems, eds. A. Dold & B. Eckmann, Lecture Notes in Mathematics, 819, Springer-Verlag. Bluman, G.W. & Cole, J.D. (1974). Similarity Methods for Differential Equations. Springer-Verlag. Blumenfeld, L.A. (1981). Problems of Biological Physics. Springer-Verlag. Bocchieri, P., Scotti, A., Bearzi, B. & Loinger, A. (1970). Anharmonic chain with Lennard-Jones interaction. Phys. Rev. A2, 2013-9. Bogoliubov, N. & Mitropolsky, Y.A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Breach. Bogoliubov, N.N., Mitropoliskii, Ju. A. & Samoilenko, A.M. (1976). Methods of Accelerated Convergence in Nonlinear Mechanics. Springer-Verlag. Bohm, D. (1984). Causality and Chance in Modern Physics, Univ. of Pennsylvania Press. Bohm, D. & Peat, F.D. (1987). Science, Order, and Creativity, Bantam. Bohr, T., Bak, R. & Jensen, M.H. (1984). Transition to chaos by interaction of resonances in dissipative systems. Phys. Rev. A30, 1970-81. Boon, J.P. & Noullez, A. (1986). Development, Growth, and Form in Living Systems. In On Growth
and Form, eds. H.F. Stanley & N. Ostrowsky, pp. 174-83, Martinus Nijhoff Pub. Borrelli, R.L. & Coleman, C.S. (1987). Differential Equations: A Modeling Approach. Prentice-Hall. Bountis, T.C. (1984). A singularity analysis of integrability of chaos in dynamical systems. In Singularities and Dynamical Systems, ed. S. Pneumatikos. North-Holland. Bountis, T. & Segur, H. (1982). Logarithmic singularities and chaotic behavior in Hamiltonian systems. AIP Conference Proceedings, 88, ed. M. Tabor & Y.M. Treve, pp. 279-92. American Institute of Physics. Mathematical methods in hydrodynamics and integrability in dynamical systems.
558
Bibliography
Bountis, T., Segur, H. & Vivaldi, F. (1982). Integrable Hamiltonian systems and the Painleve property. Phys. Rev. A25, 1257-64. Bowen, R. (1975). Methods of Symbolic Dynamics. Lecture Notes in Mathematics, 470. Springer Berlin.
Bowen, R. (1978). On axiom a diffeomorphisms. CBMS Regional Conference Series in Math. 35. Brandstater, A., Swift, J., Swinney, H.L., Wolf, A., Farmer, J.D., Jen, E. & Crutchfield, P.J. (1983). Low-dimensional chaos in a hydrodynamic system. Phys. Rev. Letters 51, 1442-5. Brillouin, L. (1963). Science and Information Theory. Academic Press. Brillouin, L. (1964). Scientific Uncertainty and Information. Academic Press. Brocker, Th. & Janich, K. (1982). Introduction to Differential Topology. Cambridge Univ. Press. Broomhead, D.S. and King, G.P. (1986). Extracting qualitative dynamics from experimental data. Physica 20D, 217-36. Broomhead, D.S. & King, G.P. (1986). On the qualitative analysis of experimental dynamical systems, pp. 113-44. In Nonlinear Phenomena and Chaos, ed. S. Sarkar, Adam Hilger Ltd. Brown, M. & Neumann, W.D. (1977). Proof of the Poincare-Birkhoff fixed point theorem. Mich. Math. J. 24, 21-31. Buck, R.C. (1981). The solutions to a smooth PDE can be dense in C(1). J. Diff. Eq. 41, 239-44. Budinsky, N. & Bountis, T. (1983). Stability of nonlinear modes and chaotic properties of 1D Fermi-Pasta-Ulam lattices. Physica 813, 445-54. Bullard, E. (1978). The disk dynamo. In Topics in Nonlinear Dynamics. AIP Conference Proceedings, ed. S. Jorna, pp. 373-89. American Institute of Physics. Bullough, R.K. & Caudrey, P.J. (Eds.) (1980). Solitons Topics in Current Physics 17. Springer-Verlag. Bullough, R.K. & Dodd, R.K. (1979). Solitons in physics: basic concepts; Solitons in mathematics. In Structural Stability in Physics, ed. W. Guttinger & H. Eikemier, pp. 219-53. Springer-Verlag. Burgers, J.M. (1948). A mathematical model illustrating the theory of turbulence: In Advances in Applied Mechanics 1, ed. R. von Mises & T. von Karman, pp. 171-99. Academic Press. Burks, A.W. (Ed.) (1970). Essays on Cellular Automata. Univ. of Illinois Pres: Urbana. Busse, F.H. (1978). Magnetohydrodynamics of the earth's dynamo. Ann. Rev. Fluid Mech. 10, 435-62. Busse, F.H. (1980). Science 208, 173-5, Convection in a rotating layer. Busse, F.H. (1980). Convections in a rotating layer: a simple case of turbulence. Science 208, 173-5. Cairns-Smith, A.G. (1982). Genetic Takeover and the Mineral Origins of Life. Cambridge Univ. Press. Calogero, F. (1983). Integrable dynamical systems and related mathematical results. In Nonlinear Phenomena. Lecture Notes in Physics 189; ed. K.B. Wolf, pp. 49-95. Springer-Verlag. Campbell, J.E. (1903). Introductory Treatise on Lie's Theory of finite Continuous Transformation Groups. Oxford: Clarendon Press. Reprinted by Chelsea, 1966. Campbell, D. & Rose, H. (Eds.) (1983). Order in chaos. Physica 7D, 1-362. Campbell, D.K., Newell, A.C., Schrieffer, R.J. & Segur, H. (Eds.) (1986). Solitons and coherent structures. Physica 18D, 1-480. Capasso, V. (1981). Periodic solutions for a system of nonlinear differential equation modeling the evolution of oro-faecal diseases. In Nonlinear Differential Equations, eds. P. deMottoni & L. Salvadori. Academic Press. Carrigan, C.R. & Gubbins, D. (1979). The source of the earth's magnetic field. Sci. Amer. 240, (2), 118. Cartwright, M.L. & Littlewood, J.E. (1945). On non-linear differential equations of the second order.
1. The equation i`+k(1 -y2)y+y-b)kcos(At+a),k large. J. Lond. Math. Soc. 20, 180 -9. Casati, G. & Ford, J. (1975). Stochastic transition in the unequal-mass Toda lattice, Phys. Rev. A12, 1702-9.
Casati, G., Chirikov, B.V. & Ford, J. (1980). Marginal local instability of quasi-periodic motion, Phys. Lett. 77A, 91-4.
Bibliography
559
Cesari, L. (1963). Asymptotic Behavior and Stability Problems in Ordinary Differential Equations Academic Press. Cesari, L. (1976). Functional Analysis, Nonlinear Deferential Equations, and the Alternative Method. Lecture Notes in Pure and Applied Mathematics 19, 1-197. Dekker. Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability. Oxford Univ. Press. Chang, S.J. & McCown, J. (1985). Universality behaviors and fractal dimensions associated with M-furcations. Phys. Rev. 31A, 3791-801. Chang, S.J. & Wright, J. (1981). Transitions and distribution functions for chaotic systems. Phys. Rev. A22, 1419-33. Chang, S.J., Wortis, M. & Wright, J.A. (1981). Iterative properties of a one-dimensional quartic map: critical lines and tricritical behavior. Phvs. Rev. A24, 2669-84. Chang, Y.F., Tabor, M. & Weiss, J. (1982). Analytic structure of the Henon-Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Phys. 23, 531-8. Channel], P.J. (1978). An illustrative example of the Chirikov criterion for stochastic instability. In Topics in Nonlinear Dynamics. AIP Conference Proceedings 46, ed. S. Jorna, pp. 248-59. American Institute of Physics. Chapman, S. & Cowling, T.R. (1970). The Mathematical Theory of Nonuniform Gases. Cambridge Univ. Press. Charney, J.G., & DeVore, J.G. (1979). Multiple flow equilibrium in the atmosphere and blocking. J. Atmos. Sci. 36, 1205-16. Chernikov, A.A., Sagdeev, R.Z., Usikov, D.A., Zakharov, M.Yu. & Zaslavsky, G.M. (1987). Minimal chaos and stochastic webs. Nature 326(9), 559-63. Chevalley, C. (1946). Theory of Lie Groups. Princeton Univ. Press. Chillingworth, D.R.J. (1976). Structural stability of mathematical models: the role of the catastrophe method. In Mathematical Modeling, ed. J.G. Andrews & R.R. McLone, pp. 231-58. Butterworths. Chinn, W.G. & Steenrod, N.E. (1966). First Concepts of Topology. M.A.A. Chirikov, B.V. (1979). A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 265-376. Chirikov, By., Keil, E., Seisler, A.M. (1971). Stochasticity in many-dimensional nonlinear oscillator systems. J. Stat. Phys. 3, 307-21. Chirikov, By., Ford, J. & Vivaldi, F. (1980). Some numerical studies of Arnold diffusion in a simple model. In Nonlinear Dynamics and the Beam-Beam Interaction, pp. 323-40, Amer. Inst. Phys. Chirikov, By., lzrainev, F.M., & Tryursky, V.A. (1973). Numerical experiments on the statistical behavior of dynamical systems with a few degrees of freedom. Comp. Phys. Comm. 5, 11-6. Choquet-Bruhat, Y., Dewitt-Morette, C. & Dillard-Bleick, M. (1977). Analysis, Manifolds and Physics. North-Holland. Chow, S.N. & Hale, J.K. (1982). Methods of Bifurcation Theory. Springer-Verlag. Ciliberto, S. & Gollub, J.P. (1984). Pattern competition leads to chaos. Phys. Rev. Letters, 922-5. Clark, J.W., Rafelski, J. & Winston, J.V. (1985). Brain without mind: computer simulation of neutral networks with modifiable neuronal interactions. Phys. Rep. 123(4), 215-73. Clark, J.W., Winston, J.V. & Rafelski, J. (1984). Self-organization of neural networks. Phys. Letters 10A, 207-11. Coodington, E.A. & Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill. Coggeshall, S.V. & Axford, R.A. (1986). Lie group invariance properties of radiation hydrodynamic equations and their associated similarity solutions. Phys. Fluids 28, 2398-420. Cohen, A. (1911). Introduction to the Lie Theory of One-Parameter Groups. D.C. Heath. Cole, J.D. (1968). Perturbation methods in applied mathematics. Blaisdell. Coles, D. (1965). Transition in circular couette flow. J. Fluid Mech. 21, 385-425.
560
Bibliography
Collet, P. & Eckmann, J.-P. (1980). Iterated Maps on the Interval as Dynamic Systems. Birkhauser. Collet, P., Eckmann, J.-P. & Koch, H. (1981). On universality for area-preserving maps of the plane. Physica 3D, 457-67. Colvin, J.T. & Stapleton, H.J. (1985). Fractal and spectral dimensions of biopolymer chains: solvent studies of electron spin relaxation rates in myoglobin azide, J. Chem. Phys. 82, 4699-706. Contopoulous, G. (1963). A classification of the integral of motion. Ap. J. 138, 1297-1305. Cook, A.E. & Roberts, P.H. (1970). The Rikitake two-disc dynamo system. Proc. Camb. Phil. Soc. 68, 547-69. Cooley, J.W. & Dodge, F.A. (1966). Digital computer solutions for excitation and propagation of the nerve impulse. Biophys. J. 6, 583-99. Cooper, L.N. (1973). A possible organization of animal memory and learning. In Proceedings of the Nobel Symposium on Collective Properties of Physical Systems 24, 252-64, ed. B. Lundquist & L. Lundquist, Academic Press. Coven, E.M., Kan, I. & Yorke, J.A. (1986). Pseudo-orbit shadowing in the family of tent maps. Cox, A. (1982). Magnetostratigraphic time scale. In Geologic Time Scale, ed. W.B. Harland et al., p. 63: Cambridge Univ. Press. Crawford, J.D. & Omohundro, S. (1984). On the global structure of period doubling flows. Physica 13D, 161-80. Creveling, H.F., dePaz, J.F., Baladi, J.Y. & Schoenhals, R.J. (1975). Stability characteristics of a single-phase free convection loop. J. Fluid Mech. 65, 67. Cronin, J. (1981). Mathematics of Cell Electrophysiology. Lecture Notes in Applied Mathematics 63. Dekker: New York. Croom, F.H. (1989). Principles of Topology. Sanders. Crutchfield, J.P., Farmer, J.D., Packard, N.H. & Shaw, R.S. (1986). Chaos. Sci. Amer. 255(6), 46-57. Crutchfield, J., Farmer, D., Packard, N., Shaw, R., Jones, G. & Donnelly, J. (1980). Power spectral analysis of a dynamical system. Phys. Letters 76A, 1-4. Cunningham, W.J. (1964). Introduction to Nonlinear Analysis. McGraw-Hill. Curry, J.H. (1978). A generalized Lorenz system. Commun. Math. Phys. 60, 193-204. Curry, J.H. (1979). On the Henon transformation. Commun. Math. Phys. 68, 129-40. Cutland, N. (1980). Computability Cambridge Univ. Press. Cvitanovic, P. (Ed.) (1984). Universality in Chaos. Adam Hilger. Bristol. Davidson, M. (1983). Uncommon Sense. J.P. Tarcher: Los Angeles. Davis, H.T. (1962). Introduction to Nonlinear Differential and Integral Equations. Dover. Davis, R. (1988). The Cosmic Blueprint, Simon and Schuster. Davydov, A.S. (1985). Solitons in Molecular Systems. D. Reidel. Demongeot, J., Goles, E. & Tchuente, M. (Eds.) (1985). Dynamical Systems and Cellular Automata. Academic Press. Denman, H.H. (1965). Invariance and conservation laws in classical mechanics, J. Math. Phvs. 6, 1611-16. De Santo, J.A., Saenz, A.W. & Zachary, W.W. (Eds.) (1980). Mathematical Methods and Applications of Scattering Theory. Lecture Notes in Physics 130. Springer-Verlag. Deutsch, D. (1985). Quantum theory, the Church-Turing principle and universal quantum computer. Proc. R. Soc. Lond. A400, 97-117. De Vogelaere, R. (1958). On the structure of symmetric period solutions of conservative systems, with applications. In Contributions to the Theory of Nonlinear Oscillations, 4, eds. S. Lefshetz, pp. 53-84. Devaney, R.L. (1986). An Introduction to Chaotic Dynamical Systems. W.H. Benjamin. Dickson, L.E. (1924). Differential equations from the group standpoint. Ann. Math. 25, 287. Dinaburg, E.I. (1971). On the relations among various entropy characteristics of dynamical systems. Math. USSR. Izvestija 5, 337-78.
Bibliography
561
Diner, S., Fargue, D. & Lochak, G. (Eds.) (1986). Dynamical Systems. World Scientific. Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. (1982). Solitons and Nonlinear Wave Equations. Academic Press. Dorizzi, B., Grammatieos, B. & Ramani, A. (1983). A new class of integrable systems. J. Math. Phys. 24, 2282-8. Dorizzi, B., Grammaticos, B. & Ramani, A. (1984). Explicit integrability for Hamiltonian systems and the Painleve conjecture. J. Math. Phys. 25, 481-5. Dorodnitsyn, A.A. (1947). Asymptotic solution of the van der Pol equation. Inst. Mech. Acad. Sci. USSR, 11, (Russian). Dragt, A.J. & Finn, J.M. (1976). Insolubility of trapped particle motion in a magnetic dipole field. J. Geophys. Res. 81, 2327-40. Drazin, P.G. & Reid, W.H. (1981). Hydrodynamic Stability. Cambridge Univ. Press. Dresden, M. (1962). A study of models in non-equilibrium statistical mechanics. In Studies in Statistical Mechanics, ed. J. de Boer & G.E. Uhlenbeck, pp. 303-43. North-Holland. Duffin, R.J. (1981). Rubel's universal differential equation. Proc. Nat. Acad. Sci. 78, 4661-2. Duncan, R. & Weston-Smith, M. (Eds.) (1978). The Encyclopaedia of Ignorance. Pergamon. Dym, C.L. & Ivey, E.S. (1980). Principles of Mathematical Modeling. Academic Press. Dyson, F.J. (1982). A model for the origin of life. J. Mol. Evol. 18, 344-50. Dyson, F.J. (1985). Origins of Life. Cambridge Univ. Press. Dziewonski, A.M. (Ed.) (1980). Physics of the Earth's Interior. Proc. International School of Physics Enrico Fermi. Articles by F.H. Busse, Motions in the earth's core and the origin of geomagnetism, pp. 493-507 & J.A. Jacobs. The evolution of the earth's core and geodynamics, pp. 508-30. North-Holland. Ebeling, W. & Perschel, M. (Eds.) (1985). Lotka-Volterra: Approach to Cooperation and Competition in Dynamic Systems. Academic-Verlag: Berlin. Ebert, K.H., Deuflhard, P. & Jager, W. (1981). Modelling of Chemical Reaction Systems. SpringerVerlag.
Eckhaus, W. (1973). Matched Asymptotic Expansions and Singular Perturbaiions. North-Holland. Eckhaus, W. (1977). Formal approximations and singular perturbations. SIAM Review 19. Eckhaus, W. & Van Harten, A. (1981). The Inverse Scattering Transformation and the Theory of Solitons. North-Holland. Eckmann, J.-P. (1981). Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643.
Eckmann, J.-P. & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617-56. Edelman, G.M. (1987). Neural Darwinism. Basic Books. Eglund, J.C., Snapp, R.R. & Schieve, W.E. (1984). Fluctuations, instabilities, and chaos in the laser-driven nonlinear ring cavity. Proy. in Optics 21, ed. E. Wolf, pp. 357-441. North-Holland. Eigen, M. (1971). Self-organization of matter and the evolution of biological macro-molecules. Naturwissenschaften 33a, 465-522. Eigen, M. (1970). Mathematical Models in Cell Biology and Cancer Chemotherapy. Lecture Notes in Biomathematics 3. Springer-Verlag. Eigen, M. (1984). The origin and evolution of life at the molecular level. In Aspects of Chemical Evolution, ed. G. Nicolis, pp. 119-37. Wiley. Eigen, M. (1985). Macromolecular evolution. In Emerging Synthesis in Science, ed. D. Pines, pp. 25-69. Santa Fe Institute. Eigen, M. (1986). The physics of molecular evolution. Chemica Scripta 26B, 13-26. Eigen, M., Gardiner, W.C. & Schuster, P. (1980). Hypercycles and compartments. J. Theor. Biol. 85, 407-11.
562
Bibliography
Eigen, M. & Schuster, P. (1974). The Hypercycle: A Principle of Natural Self-organization. Springer-Verlag. Eigen, M. & Schuster, P. (1979). The Hypercycle: A Principle of Natural Self-organization. Springer-Verlag. Eilenberger, G. (1981). Solitons. Springer-Verlag. Eisenhart, L.P. (1933). Continuous Ground of Transformations. Princeton Univ. Press. Eminhizer, C.H., Helleman, R.H.G. & Montroll, E.W. (1976). On a convergent nonlinear perturbation theory without small denominators or secular terms. J. Math. Phys. 17, 121-. Enns, R.H., Jones, B.L., Miura, R.M. & Rangnekar, S.S. (Eds.) (1981). Nonlinear Phenomena in Physics and Biology. Plenum. Epstein, I.R. (1984). New chemical oscillators. In Non-Equilibrium Dynamics in Chemical Systems, ed. C. Vidal & A. Pacault, pp. 24-34. Springer-Verlag. Erdelyi, A. (1956). Asymptotic Expansions. Dover. Ermentrout, E.B. & Cowan, J.D. (1979a). A mathematical theory of visual hallucination patterns. Biol. Cybernetics 34, 137-50. Ermentrout, E.B. & Cowan, J.D. (1979b). Temporal oscillations in neuronal nets. J. Math. Biol. 7, 265-80. Ermentrout, E.B. & Cowan, J.D. (1980). Large scale spatially organized activity in neural nets. SIAM J. Appl. Math. 38, 1-21. Falconer, K.J. (1985). The Geometry of Fractal Sets, Cambridge Univ. Press. Farmer, D., Crutchfield, J., Froehling, H., Packard, N. & Shaw,'R. (1980). Power spectra and mixing properties of strange attractors. Ann. N.Y. Acad. Sci. 357, 453-72. Farmer, D., Lapedes, A., Packard, N. & Wendroff, B. (Eds.) (1986). Evolution, games and learning; models for adaptation in machines and nature, Physica 22D (Nos. 1-3). Farmer, D., Toffoli, T. & Wolfram, S. (Eds.) (1984). Cellular automata. Physica 10D, 1-246. Farmer, J.D., Ott, E. & Yorke, J.A. (1983). The dimension of chaotic attractors. Physica 7D, 153-80. Fenstermacher, P.R., Swinney, H.L., Benson, S.V. & Gollub, J.P. (1979). Bifurcations to periodic, quasiperiodic and chaotic regimes in rotating and convecting fluids. Ann. N.Y. Acad. Sci. 316, 652-66. Fenstermacher, P.R. Swinney, H.L. & Gollub, J.P. (1979). Dynamic instabilities and the transition to chaotic Taylor vortex flow. J. Fluid Mech. 94, 103-28. Feigenbaum, M.J. (1978). Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25-52. Feigenbaum, M.J. (1979a). The universal metric properties of nonlinear transformations. J. Stat. Phys. 22, 186-223. Feigenbaum, M.J. (1979b). The onset spectrum of turbulence. Phys. Letters 74A, 375-8. Fein, A.P., Heutmaker, M.S. & Gollub, J.P. (1985). Scaling and transition from quasiperiodicity to chaos in a hydrodynamic system. Physica Scripta T9, 79-84. Feit, S.D. (1978). Characteristic exponents and strange attractors. Commun. Math. Phys. 61, 249-60. Fermi, E. (1923). Beweis dass ein mechanisches Normalsystem im Allgemeinen quasi-ergodisch ist, Phys. Zeits. 24, 261-5. Fermi, E., Pasta, J.R. & Ulam, S.M. (1965). Studies of nonlinear problems. I. Los Alamos report LA-1940, May 1955. In Collected Works E. Fermi, Vol. 2, 978-88. Univ. of Chicago Press. Fesser, K., McLaughlin, D.W., Bishop, A.R. & Holian, B.L. (1985). Chaos and nonlinear modes in perturbed Toda chain. Phys. Rev. A31, 2728-31. Feynman, R.P. (1967). The Character of Physical Law, pp. 57-8. M.I.T. Press: Cambridge Mass.
Bibliography
563
Feynman, R.P. (1982). Simulating physics with computers. Int. J. Theor. Phys. 21, 467-88. Feynman, R. P. (1986). Quantum mechanical computers, Foundation Phys. 16, 506-3 1. Field, R.J. & Noyes, R.M. (1974). Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60, 1877-84. Fitzhugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445-64. Fitzhugh, R. (1969). Mathematical models of excitation and propagation in nerves. In Biological Engineering, ed. H.P. Schwan. McGraw-Hill. Flaherty, J.E. & Hoppenstendt, F.C. (1978). Frequency entrainment or a forced Van der Pol oscillator. Studies Appl. Math. 58, 5-15. Flaschka, H. (1975). Discrete and periodic illustrations of some aspects of the inverse method. In Dynamic Systems Theory and Applications, ed. J. Moser, pp. 441-66. Springer-Verlag. Flaschka, H. (1974a). The Toda lattice. I. Existence of integrals. Phys. Rev. B9, 1924-5. Flaschka, H. (1974b). On the Toda lattice. Prog. Theor. Phys. 51, 703-16. Flaschka, H. & McLaughlin, D.W. (Eds.) (1978). Conference on the theory and application of solitons. Rocky Mountain J. Math. 8, 1-428. Flaschka, H. & Newell, A.C. (1975). Integrable systems of nonlinear evolution equations. In Dynamic Systems, Theory and Applications, ed. J. Moser, pp. 355-440. Springer-Verlag. Fokas, A.S. & Ablowitz, M.J. (1983). The inverse scattering transform for multi-dimensional (2 + 1) problems. In Nonlinear Phenomena. Lecture Notes in Physics 189, ed. K.B. Wolf, pp. 139-54. Springer-Verlag. Ford, J. (1961). Equipartition of energy for nonlinear systems. J. Math. Phys. 2, 387-93. Ford, J. (1973). The transition from analytic dynamics to statistical mechanics. Adv. Chem. Phys. 24, 155-85. Ford, J. (1975). The statistical mechanics of classical analytic dynamics. In Fundamental Problems in Statistical Mechanics, ed. E. Cohen, pp. 215-55. North-Holland. Ford, J. (1977). Relevance of computer experiments to the pure mathematics of integrable and ergodic dynamical systems. In Dynamical Systems. I. Warsaw, pp. 75-92. Societe Mathematique de France. Ford, J. (1983). How random is a coin toss?, Physics Today, April, pp. 40-47. Ford, J. (1988). What is chaos, that we should be mindful of it? In The New Physics, ed. P.C. Davis, Cambridge Univ. Press. Ford, J., Stoddard, S.D. & Turner, J.S. (1973). On the integrability of the Toda lattice. Prog. Theor. Phys. 50, 1547-60. Fowler, A.C. & McGuinness, M.J. (1982a). A description of the Lorenz attractor at high Prandtl number. Physica 5D, 149-82. Fowler, A.C. & McGuinness, M.J. (1982b). Hysteresis in the Lorenz equations. Phys. Letters 92A, 103-6. Fowler, D.H. (1972). The Riemann-Hugoniot catastrophe and Van der Waal's equation. Towards a Theoretical Biology 4, 1-7, ed. C.H. Waddintgon. Fox, R.F. (1988). Energy and the Evolution of Life. W.H. Freeman. Fox, S. (1988). The Emergence of Life. Basic Books. Francis, G.K. (1987). A Topological Picturebook, Springer-Verlag. Fraser, S. & Kapral, R. (1982). Analysis of flow hysteresis by a one-dimensional map. Phys. Rev. A25, 3223-33. Frauenfelder, H., Petsko, G.A. & Tsenoglu, D. (1979). Temperature-dependent X-ray diffraction as a probe of protein structural dynamics. Nature 280, 558-63.
564
Bibliography
Frauenthal, J.C. (1979). Introduction to Population Modeling. (EDC/umap/55 Chapel St., Newton Mass.
Fredkin, E. (1982). Digital Information Mechanics (talk given January, 1982, Moskito Island, BVI); Also see: The Atlantic Monthly, April 1988, p. 29; Three scientists and their Gods, by R. Wright, Times Books, 1988. Fredkin, E. & Toffoli, T. (1982). Conservative logic. Int. J. Theor. Phys. 21, 219-53. Freedman, H.I. (1980). Deterministic Mathematical Models in Population Ecology. Dekker. Frisch, U. (1986). Fully developed turbulence: where do we stand? In Dynamical Systems: A Renewal of Mechanism, ed. S. Diner, D. Fargue & G. Lochak, pp. 13-28. World Scientific. Frisch, U., Hasslacher, B. & Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Letters 56, 1505-8. Froehling, H., Crutchfield, J.P., Farmer, D., Packard, N.N. & Shaw, R. (1981). On determining the dimension of chaotic flows. Physica 3D, 605-17. Froeschile, C. (1978). Connectance of dynamical systems with increasing number of degrees of freedom. Phys. Rev. A18, 277-81. Frlyland, J. & Alfsen, K.H. (1984). Lyapunov-exponent spectra for the Lorenz model. Phys. Rev. A29, 2928-31. Furstenberg, H. (1983). Poincare recurrence and number theory. In The Mathematical Heritage of Henri Poincare. Part 2, pp. 193-216. Amer. Math. Soc. Gantmacher, R.R. (1959). The Theory of Matrices. Chelsea. Gardner, C.S., Greene, J.M., Krushal, M.D. & Miura, R.M. (1967, 1974). Method for solving the Korteweg-deVries equation. Phys. Rev. Letters 19, 1095-7. Korteweg-deVries equation and generalization. VI. Methods for exact solution. Commun. Pure Appl. Math. 27, 97-133. Gardner, M. (1970, 1971). Sci. Amer. 223(10), 120; 224(2), 112; 224(4), 117.
Gardner, M. (1976). Knotted Doughnuts and Other Mathematical Entertainments. W.H. Freeman. Gardner, M. (1981). The Laffer curve and other laughs in current economics. Sci. Amer. 245(6), 18. Gardner, M. (1983). Wheels, Life, and Other Mathematical Amusements, Freeman. Gardner, M. (1986). Knotted Doughnuts and Other Mathematical Entertainments, W.H. Freeman. Garrido, L. (Ed.) (1983). Dynamic Systems and Chaos. Lecture Notes in Physics 179. Springer-Verlag. Gazis, D.C. (1967). Mathematical theory of automobile traffic. Science 157, 273-81. Gazis, D.C. (Ed.) (1974). Traffic Science, Wiley. Greene, J.M. (1979). A method of determining a stochastic transition. J. Math. Phys. 20, 1183-1201. Gibbon, J.D., Freeman, N.C. & Johnson, R.S. (1978). Correspondence between the classical gd°. Double and single Sine-Gordan equations for three-dimensional solitons. Phys. Letters 65A, 380-4. Gierer, A. & Meinhardt, H. (1972). A Theory of Biological Pattern Formation, Kybernetik 12, 30-39. Gilmore, R. (1981). Catastrophe Theory for Scientists and Engineers. Wiley. Gilpin, M.E. (1973). Do hares eat lynx? Amer. Nat. 107, 727-30. Glansdorff, P. & Prigogine, 1. (1971). Thermodynamic Theory of Structure Stability and Fluctuations. Wiley-Interscience.
Glass, L. & Mackey, M.C. (1988). From Clocks to Chaos; The Rythms of Life. Princeton University Press.
Glass, L. & Perez, R. (1982). Fine structure of phase locking. Phys. Rev. Letters 48, 1772-5. Goldanskii, V.I., Anikin, S.A., Avetisov, V.A. & Kuz'min, V.V. (1987). On the decisive role of chiral purity in the organization of the biosphere and its possible destruction. Comments. Mol. Cell. Biphys. 4, 79-98. Goel, N.S., & Thompson, R.L. (1988). Computer Simulations of Self-Organization in Biological Systems. Macmillan Pub.
Bibliography
565
Goel, N.S., Maitra, S.C. & Montroll, E.W. (1971). On the Volterra and other non-linear models of interacting populations. Rev. Mod. Phys. 43, 231-76. Gollub, J.P. & Benson, S.V. (1978). Chaotic response to period perturbation of convecting fluid. Phys. Rev. Letters 41, 948.
Gollub, J.P. & Freilich, M.H. (1974). Optical heterodyne study of the Taylor instability in a rotating fluid. Phys. Rev. Letters 33, 1465-8. Gollub, J.P. & Meyer, C.W. (1983). Symmetry-breaking instabilities on a fluid surface, Physica 6D, 337-46. Gollub, J.P. & Swinney, H.L. (1975). Onset of turbulence in a rotating fluid. Phys. Rev. Letters 35, 927-30. Gollub, J.P. & Benson, S.V. (1978). Chaotic response to periodic perturbation of convecting fluid. Phys. Rev. Letters 41, 948. Gollub, J.P. & Meyer, C.W. (1983). Symmetry-breaking instabilities on a fluid surface. Physica 6D, 337-46. Golubev, V.V. (1953). Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point. State Publishing House; Moscow. Goursat, E. (1964). A Course in Mathematical Analysis. Dover. Grabert, H. (1982). Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer Tracts in Modern Physics 95. Springer-Verlag. Grasman, J. & Veling, E.J.M. (1979). Asymptotic Methods for the Volterra-Lotka Equation. Lecture notes in Mathematics 11, ed. F. Verhulst. Springer-Verlag. Grassberger, P. & Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica 9D, 189-202.
Grebogi, C., Ott, E. & Yorke, J.A. (1983). Fractal basin boundaries, long-lived chaotic and unstableunstable pair bifurcations, Phys. Rev. Letters 50, 935-8. Greene, J.M. (1979). A method of determining a stochastic transition. J. Math. Phys. 20, 1183-1201. Greene, J.M. (1986). How a swing behaves. Physica 18D, 427-47. Greene, J.M., MacKay, R.S., Vivaldi, F. & Feigenbaum, M.J. (1981). Universal behavior in families of area-preserving maps. Physica 3D, 468-86. Greenspan, D. (1980). Arithmetic Applied Mathematics. Pergamon. de Groot, S.R. (1961). Thermodynamics of Irreversible Processes. North-Holland. Gu, Y., Tung, M., Yuan, J.-M., Feng, D.H. & Narducci, L.M. (1984). Crises and hysteresis in coupled logistic maps. Phys. Rev. Letters 52, 701-4. Gubbins, D. (1974). Theories of the geomagnetic and solar dynamos. Rev. Geophys. Sp. Phys. 12, 137-54.
Guckenheimer, J. (1977). On the bifurcation of maps of the internal. Inventiones Math. 39, 165-78. Guckenheimer, J. (1979). The bifurcation of quadratic functions. In Bifurcation Theory, Ann. N.Y. Acad. Sci. 316, pp. 78-85. Guckenheimer, J. (1979). Sensitive dependence to initial conditions for one dimensional maps. Commun. Math. Phys. 70, 133-60. Guckenheimer, J. & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields. Springer-Verlag. Guckenheimer, J., Moser, J. & Newhouse, S.E. (1980). Dyanmical Systems. Birkhauser. Guckenheimer, J. Oster, G. & Ipaktchi, A. (1977). The dynamics of density dependent population models. J. Math. Biol. 4, 101-47. Guevara, M.R. & Shrier, A. (1983). Bifurcation and chaos in periodically stimulated cardiac oscillator. Physica 7D, 89-101.
566
Bibliography
Guillemin, V. & Pollack, A. (1974). Differential Topology. Prentice-Hall. Gurel, G. & Rossler, O.E. (Eds.) (1979). Bifurcation theory and applications in scientific disciplines. Ann. N.Y. Acad. Sci. 316. Guttinger, W. (1986). Bifurcation geometry in physics. In Frontiers of Nonequilibrium Statistical Physics, G.T. Moore & M.O. Scully, pp. 57-82. Plenum. Guttinger, W. & Eikemeier, H. (Eds.) (1979). Structural Stability in Physics. Springer-Verlag. Hagihara, Y. (1974). Celestial Mechanics. Vol. 1-Vol. V. Japan Soc. Promotion of Science, Tokyo Press.
Haken, H. (1977). Synergetics: An Introduction. Vol. 2. A Workshop. Springer-Verlag. Haken, H. (Ed.) (1981). Chaos and Order in Nature. Springer-Verlag. Haken, H. (1983). Advanced Synergetics. Springer-Verlag. Hale, J. (1977). Theory of Functional Differential Equations. Springer-Verlag. Hale, J.K. (1980). Ordinary Differential Equations. R.E. Krieger. Hasley, T.C., Jensen, M.H., Kadanoff, L.P., Procassia, I. & Shraiman, B.I. (1986). Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A33, 1141-51. Hammel, S.M., Yorke, J.A. & Grebogi, C. (1987). Do numerical orbits of chaotic dynamical processes represent true orbits? J. Complexity 3, 136-45. Hansen, A.G. (1964). Similarity Analyses of Boundary Value Problems in Engineering. Prentice-Hall. Hardy, G.E. (1916). Weierstrass's non-differentiable function. Trans. Amer. Math. Soc. 17, 301-25. Harmon, L.D. (1961). Studies with artificial neurons, I Properties and functions of an artificial neuron, Kyhernetik 1, 89-101. Hartman, P. (1964). Ordinary Differential Equations. Wiley. Reprinted by Hartman, 1973. Hasegawa, A. (1985). Self-organization processes in continuous media. Adv. in Phys. 34, 1-42. Hastings, S.P. & Murray, I.D. (1975). The existence of oscillatory solutions in the Field-Noyes model of the Belousov-Zhabotinskii reactions. J. Appl. Math. 28, 678-88. Hawking, S.W. (1988). A Brief History of Time, Bantam. Hayashi, C. (1985). Nonlinear Oscillations in Physical Systems. Princeton Univ. Press. Reims, S.J. (1984). John Von Neumann and Norbert Wiener, MIT Press. Heileman, R.H.G. (1977). On the iterative solution of a stochastic mapping. In Statistical Methods, Theory and Applications, ed. U. Landman, Plenum. Heileman, R.H.G. (1980). Self-generated chaotic behavior in nonlinear mechanics. In Fundamental Problems in Statistical Mechanics, ed. E.G.D. Cohen, pp. 165-233. North-Holland. Heileman, R.H.G. (Ed.) (1980). Nonlinear dynamics. Ann. N.Y. Acad. Sci. 257. Henon, H. (1969). Numerical study of quadratic area-preserving mappings. Quart. Appl. Math. 27, 291-312. Henon, H. (1974). Integrals of the Toda lattice. Phys. Rev. B9, 1921-3. Henon, M. (1976). A two dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69-77. Henon, M. (1982). On the numerical computation of Poincare maps. Physica 5D, 412-44. Henon, M. & Heiles, C. (1964). The applicability of the third integral of motion: some numerical experiments. Astrophys. J. 69, 73-9. Herman, M.R. (1977). Measure DeLebesque et nombre de rotation, pp. 271-93. In Geometry and Topology, ed. J. Palis. Springer-Verlag. Hibberd, F.H. (1979). The origin of the earth's magnetic field. Proc. R. Soc. Lond. A369, 31-45. Hill, E.L. (1951). Hamilton's Principle and the conservation theorems of mathematical physics. Rev. Mod. Phvs. 23, 253-60. Hill, J.M. (1982). Solution of Differential Equations by Means of One-Parameter Groups. Pitman.
Bibliography
567
Hirooka, H., Saito, N. & Ford, J. (1984). Chaos around hyperbolic fixed points. J. Phys. Soc. Japan 53, 895-98. Hirota, R. (1971). Exact solution of the Korteweg-de Vreies equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192-4. Hirota, R. (1976). Direct methods of finding exact solutions of nonlinear evolution equations. In Backlund Transformations, ed. R.M. Miura. Lecture Notes in mathematics 515. Springer-Verlag. Hirota, R. (1985). Fundamental properties of the binary operators in soliton theory and their generalization. In Dynamic Problems in Soliton Systems, ed. S. Takeno. pp.42-9. Springer-Verlag. Hirota, R. (1980). Direct methods in soliton theory. In Solitons. Topics of Modern Physics, ed. R.K. Bullough & P.J. Candrey. Springer-Verlag. Hirsch, M.W. (1984). The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11, 1-64. Hirsch, M.W. & Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra. Academic Press. Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerves. J. Physiol. 117, 500-44. Hofbauer, J. (1981). On the occurance of limit cycles in the Volterra-Lotka equation. Nonlinear Analysis T.M.A. 5, 1003-7. Hogg, T. & Huberman, B.A. (1984). Generic behavior of coupled oscillators. Phys. Rev. A29, 275-81. Holden, A.V. (1986). Chaos. Princeton Univ. Press. Holmes, P.J. (1980). Averaging and chaotic motions. In Forced Oscillations. SIAM J. Appl. Math. 38, 65-80. (Correction in New Approaches to Nonlinear Problems in Dynamics, ed. P.J. Holmes. SIAM. Holmes, P. & Marsden, J.E. (1979). Qualitative techniques for bifurcation analysis of complex systems. Ann. NY Acad. Sci. 316, 608-22. Holmes, P.J. & Moon, F.C. (1983). Strange attractors and chaos in nonlinear mechanics. J. Appl. Mech. 50, 1021-132. Holmes, P.J. & Rand, D.A. (1976). The bifurcations of Duffing's Equation: an application of catastrophe theory. J. Sound and Vib. 44, 237-53. Holmes, P.J. & Whitley, D.C. (1984). Bifurcations of one- and two-dimensional maps. Phil. Trans. R. Soc. Lond. A311, 43-102; Erratum, ibid., 601. Holmes, P. & Williams, R.F. (1985). Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences, Arch. Ration. Mech. Anal. 90, 115-94. Hoover, W.G. (1986). Molecular Dynamics. Lecture Notes in Physics 258. Springer-Verlag.
Hopf, E. (1948). A mathematical example displaying features of turbulence. Commun. Pure Appl. Math. 1, 303-22. Hopf, F.A. & Hopf, F.W. (1986). Darwinian evolution in physics and biology: the dilemma of missing links. In Frontiers of Nonequilibrium Statistical Physics, ed. G.T. Moore & M.O. pp. 411-9. Scully, Plenum.
Hopfield, J.J. (1984). Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. 79, 2554-8. Hopfield, J.J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. 81, 3088-92. Hoppensteadt, F.C. Mathematical Theories of Populations: Demographics, Genetics, and Epidemics. SIAM.
Hoppensteadt, F.C. (Ed.) (1979). Nonlinear Oscillations in Biology. Lectures in Applied Mathematics 17, Amer. Math. Soc.
568
Bibliography
Hoppensteadt, F.C. (Ed.) (1981). Mathematical Aspects of Physiology. Lectures in Applied Mathematics 19, Amer. Math. Soc. Houtappel, R.M.F. Van Dam, H. & Wigner, E.P. (1965). The conceptual basis and use of the geometric invariance principles. Rev. Mod. Phvs. 37, 595-632. Hsu, C.S. (1987). Cell-To-Cell Mapping, Springer-Verlag. Hunt, E.R. (1982). Comment on a driven nonlinear oscillator. Phys. Rev. Letters 49, 1054. Testa, Perez, and Jeffries respond. Phys. Rev. Letters. Hurewicz, W. & Wallman, H. (1948). Dimension Theory. Princeton Univ. Press. Ichikawa, Y.H. & Ino, K.H. (1985). Lax-pair operators for squared-sum and squared-difference Eigen-functions. J. Math. Phys. 26, 1976-8. Ichikawa, Y.H., Kamimura, T. & Karney, C.F.F. (1983). Stochastic motion of particles in tandom mirror devices. Physica 6D, 233-40. Ince, E.L. (1956). Ordinary Differential Equations. Dover. Inglis, D.R. (1981). Dynamo theory of the earth's varying magnetic field. Rev. Mod. Phys. 53, 481-96. Inselberg, A. (1972). Superpositions for nonlinear operators. I. Strong super-positions and linearizability. J. Math. Anal. Appl. 40, 494-508. Iooss, G., Helleman, R.H.G. & Stora, R. (Eds.) (1983). Chaotic behavior of Deterministic Systems. North-Holland. Iooss, G. & Joseph, D.D. (1980). Elementary Stability and Bifurcation Theory. Springer-Verlag. Irwin, M.C. (1980). Smooth Dynamical Systems. Academic Press. Iyanaga. S. & Kawada, Y. (Eds.) (1980). Encyclopedic Dictionary of Mathematics. Math. Soc. Japan. M.I.T. Press. Jackson, E.A. (1963). Nonlinear coupled oscillators. I. Perturbation theory: ergodic problem. J. Math. Phys. 4, 551-8; Nonlinear coupled oscillators. II. Comparison of theory with computer solutions. J. Math. Phys. 4, 686-700. Jackson, E.A. (1978). Nonlinearity and irreversibility in lattice dynamics. Rocky Mountain J. Math. 8, 127-46. Jackson, E.A. (1984). Radiation reaction dynamics in an electromagnetic wave and constant magnetic field, J. Math. Phys., 1584-91. Jackson, E.A. (1985). The lorenz system. I. The global structure of its stable manifolds. Physica Scripta 32, 469-75; The lorenz system. II. The homoclinic convolution of the stable manifolds. Physica Scripta 32, 476-81. Jackson, E. A. & Mistriotis, A. D. (1989). Thermal conductivity of one- and two-dimensional lattices. J. Phys.: Condens. Matter 1, 1223-38. Jackson, E.A., Pasta, J.R. & Waters, J.F. (1968). Thermal conductivity of one-dimensional lattices, J. Comput. Phys. 2, 207-27. Jacobs, A.J. (1984). Reversals of the Earth's Magnetic Field. Heyden Philadelphia. Jeffries, C.D. (1985). Chaotic dynamics of instabilities in solids. Physica Scripta T9, 11-26. Jeffrey, A. & Taniuti, T. (1964). Nonlinear Wave Propagation. Academic Press. Jen, E. (1986). Global properties of cellular automata. J. Stat. Phys. 43, 219-42; Strings and pattern recognizing properties of one-dimensional cellular automata. J. Stat. Phys. 43, 243-65. Jensen, M.H., Kadanoff, L.P. Libchaber, A. Procaccia, I. & Stavans, J. (1985). Global universality at the onset of chaos: results of a forced Rayleigh-Benard experiment. Phys. Rev. Letters 55, 2798-801. Jensen, R.V. & Oberman, C.R. (1982). Statistical properties of dynamical systems. Physica 4D, 183-96. Jorna, S. (Ed.) (1978). Topics in nonlinear dynamics. AIP Conference Proceedings, 46. American Institute of Physics. Joseph, D. (1976). Nonlinear Stability of Fluid Motions. Springer-Verlag.
Bibliography
569
Jowett, J.M., Month, M. & Turner, S. (1985). Nonlinear Dynamics Aspects of Particle Accelerators. Lecture Notes in Physics 247, Springer-Verlag. Kac, M. (1959). Probability and Related Topics in Physical Sciences. Interscience Pub. Kadomtsev, B.B. & Petviashvili, V.I. (1970). Sov. Phys. Dokl. 15, 539-41. Kai, T. & Tomita, K. (1979). Stroboscopic phase portrait of a forced nonlinear oscillator. Prog. Theor. Phys. 61, 54-73. Kako, F. & Yajima, N. (1980). Interaction of ion-acoustic solitons in two-dimensional space. J. Phys. Soc. Japan. 49, 2063-71. Kalia, R.K. & Vashista, P. (Eds.) (1982). Melting, Localization, and Chaos. North-Holland. Kaneko, K. (1983). Transition from torus to chaos accompanied by frequency lockings with symmetry breaking. Prog. Theor. Phys. 69, 1427-42. Kaneko, K. (1984). Period-doubling of kink-antikink patterns. Prog. Theor. Phys. 72, 480-6. Kaplan, J.L. & Yorke, J.A. (1979). Preturbulence: a regime observed in fluid flow model of lorenz. Commun. Math. Phys. 67, 93-108. Kaplan, J.L. & Yorke, J.A. (1979b). Chaotic behavior of multidimensional difference equations, pp.204-27. In Functional Differential Equations, ed. H.O. Peitgen & H.D. Waither. Lecture Notes in Mathematics 730, Springer-Verlag. Kaplan, J.L. & Yorke, J.A. (1979c). The onset of chaos in a fluid flow model of lorenz. Ann. N.Y. Acad. Sci. 316, 400-7. Kaplan, J.L., Mallet-Paret, J. & Yorke, J.A. (1984). The Lyapunov Dimension of a Nowhere Differentiable Attracting Torus, Ergod. Th. and Dynam. Sys. 4, 261-81. Kapral, R. (1985). Pattern formation in two-dimensional arrays of coupled, discrete time oscillators. Phys. Rev. A31, 3868-79. Karney, C.F.F. (1983). Long-time correlations in the stochastic regime, Physica 8D, 360-80. Kaup, D.J. (1976). The three-wave interaction: a nondispersive phenomena. Studies Appl. Math. 55, 9-44. Kay, I. & Moses, HE. (1982). Inverse Scattering Papers 1955-1963, 12, Lie Groups, History Frontiers and Applications, Mathematical Science. Keeler, J.D. & Farmer, J.D. (1986). Robust space-time intermittency and 1/f noise. Physica 23D, 413-35. Keller, J.B. & Antman, S. (Eds.) (1969). Bifurcation Theory and Nonlinear Eigenvalue Problems. W.A. Benjamin. Khinchin, A.I. (1949). Mathematical Foundations of Statistical Mechanics, Dover. Kruskal, M. (1982). The Korteweg-deVries Equation. In Laser-Plasma Interaction, eds. R. Balian & J.C. Adam, pp. 788-808, North-Holland. Kolata, G.B. (1977). Catastrophe theory: the emperor has no clothes. Science 196, 287. Kolata, G. (1984). Order out of chaos in computers. Science 223. 917-9. Kolmogoroff, A. (1936). Sulla teoria di volterra della lotta per 1'esistenza. Giornale dell'Istituto Italiano degli Attuari 7, 74-80. Kolmogorov, AN. (1954). The General Theory of Dynamical Systems and Classical Mechanics (English translation in Foundations of Mechanics by R. Abraham & J.E. Marsden.) Kolmogorov, A.N. (1958). A new invariant of transitive dynamical systems, DokI. Akad. Nauk. SSSR 119, 861-65. Kolmogorov, AN. (1984). Automata and life. In Cybernetics Today, ed. I.M. Makarov, pp. 20-41. MIR Pub. Moscow. Kolmogorov, AN. & Uspensky, V.A. (1987). Algorithms and randomness. In Proceedings of the Ist
570
Bibliography
World Congress of the Bernoulli Society, eds. Yu.A. Prohorov & V.V. Sazonov, pp. 3-53, VNU Science Press.
Korteweg, D.J. & deVries, G. (1895). On the change of form of long waves advancing in a rectangular canal, and a new type of long stationary waves. Phil. May. 39, 422-43. Koschmieder, E.L. (1981). Experimental aspects of hydrodynamic instabilities. In Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, ed. G. Nicolis, G. Dewel & J.W. Turner, pp. 159-88. Wiley. Kozak, J.J. (1979). Nonlinear Problems in the Theory of Phase Transitions. Adv. Chem. Phys. 40, 229-368. Interscience Publ. Krinskii, V.I., Pertsov, A.M. & Reshetilov, AN. (1972). Investigation of one mechanism of origin of the etopic focus of excitation in modified Hodgkin-Huxley equations. Biofizika 17, 271-7. Kruskal, M. (1962). Asymptotic theory of Hamiltonian systems. J. Math. Phys. 3, 806-28. Kruskal, M.D., Miura, R.M., Gardner, C.S. & Zabusky, N.J. (1970). Uniqueness & nonexistence of polynomial conservation laws. J. Math. Phys. 11, 952-60. Kruskal, M.D. & Zabusky, N.J. (1964). Stroboscopic perturbation procedure for treating a class of nonlinear wave equations. J. Math. Phys. 5, 231-44. Kryloff, N. & Bogliuboff, N. (1947). Introduction to Non-Linear Mechanics. Annals of Math. Studies. Princeton Univ. Press. Kuramoto, Y. (1983). Chemical Oscillations, Waves & Turbulence. Springer-Verlag. Kusnirenko, A.G., Kotok, A.B. & Alekseev, V.M. (1981). Three Papers on Dynamical Systems. Amer. Math. Soc. Kuznetsov, S.P. (1986). Universality and scaling in the behavior of coupled feigenbaum systems. Radiophys. Quantum Electronics 28, 681-95. Lamb, G.L. Jr. (1971). Analytical descriptions to ultrashort optical pulse propagation in resonant media. Rev. Mod. Phys. 43, 99-124. Lamb, G.L. Jr. (1974). Backlund transformations for certain nonlinear evolution equations. J. Math. Phys. 15, 2157. Lamb, G.L. Jr. (1980). Elements of Soliton Theory. Wiley-Interscience. Lamb, H. (1945). Hydrodynamics. Dover. Lancaster, P. (1985). Theory of Matrices. Academic Press. Landau, L.D. & Lifshitz, E.M. (1959). Fluid Mechanics. Addison-Wesley. Lanford, O.E. (1977). In Turbulence Seminar. Lecture Notes in Mathematics 615, ed. P. Bernard & T. Rativ, p. 114, Springer-Verlag. Lanford, O.E. (1987). Circle Mappings. In Recent Developments in Mathematical Physics, ed. H. Mitter, & L. Pittner. pp. 1-17, Springer-Verlag. Langley, P., Simon, H.A., Bradshaw, G.L. & Zythow, J.M. (1987). Scientific Discovery, MIT Press. Langton, C.G. (1984). Self-reproduction in cellular automata. Physica 110D, 135-44. Langton, C.G. (1986). Studying artificial life with cellular automata, Physica 22D, 120-49. Langton, C.G. (Ed.) (1989). Artificial Life (Vol. VI, Santa Fe Institute). Addison-Wesley. LaSalle, J. (1949). Relaxation oscillations. Quart. J. App. Math. 7, 1. LaSalle, J.P. & Lefschetz, S. (Eds.) (1963). Nonlinear Differential Equations & Nonlinear Mechanics. Academic Press. Laslow, E. (1987). Evolution: The Grand Synthesis. Shambhala Pub. Boston. Lasota, A. (1977). Ergodic problems in biology. In Dynamical Systems 2, Warsaw, 239-250. Societe Mathematique de France. Lauterborn, W. & Crames, E. (1981). Subharmonic route to chaos observed in acoustics. Phys. Rev. Letters 47, 1445-48.
Bibliography
571
Lax, P.D. (1968). Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467-90. Lax, P.D. (1986). Mathematics & computing. J. Stat. Phys. 43, 749-56. Lebowitz, N.R. & Schaar, R. (1975). Exchange of stabilities in autonomous systems. Studies Appl. Math. 54, 229-60. Lee, T.D. (1985). Discrete mechanics. In How Far Are We From the Gauge Forces?, ed. A. Zichichi, pp. 15-144, Plenum Press. Lee, T.D. (1986). Solutions of discrete mechanics near the continuum limit, in Rationale of Beings, eds K. Ishikawa, Y. Kawazoe, Y. Matsuzaki, K. Takahashi, p. 324, World Scientific. Ledrappier, F. & Young, L.-S. (1985). The metric entropy of diffeomorphisms Part I: Characterization of Measures Satisfying Pesin's entropy Formula, Ann. Math. 122, 509-39; Part II: Relations between entropy, exponents, and dimension, ibid., 540-74. Lefever, R. (1968). Stabilite des structures dissipatives, Bull. Classe Sci., Acad. Roy. Belg. 54, 712-19. Lefschetz, S. (1959). Differential Equations: Geometric Theory. Interscience Pub. Leggett, A.J. (1987). The Problems of Physics, Oxford Univ. Press. Leisegang, S. (1953). Zum Astigmatismus von Electronenlinsen. Optik 10, 5-14. Levi, M. (1981). Qualitative analysis of the periodically forced relaxation oscillations. Memoirs Amer. Math. Soc. 32, 244, 1-145. Levinson, N. (1949). A second order differential equation with singular solutions. Ann. Math. 50, 127-53.
Levinson, N. & Smith, O.K. (1942). A general equation for relaxation oscillations, Duke Math. J. 9, 287-403. Li, T.Y. & Yorke, J.A. (1975). Period three implies chaos. Amer. Math. Monthly 82, 983-92. Lichtenberg, A.J. & Lieberman, M.A. (1983). Regular and Stochastic Motion. Springer-Verlag. Lie, S. (1893). Vorlesungen ueber Continuierliche Gruppen mit Geometrischen and Anderen Anwendugen. B.C. Teubner: Leipzig. Linkens, D.A., Biological Systems (1980). In Modelling of Dynamical Systems, Vol. 1 ed. N. Nicholson, pp. 193-223. Peregrinus. Linsay, P.S. (1981). Period doubling and chaotic behavior in a driven anharmonic oscillator. Phys. Rev. Letters 47, 1349-52. Little W.A. (1974). The existence of persistent states in the brain. Math. Biosci. 19, 101-20. Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani, A. (1983). Relaxation to different stationary states in the Fermi-Pasta-Ulam model. Phys. Rev. A28, 3544-52. Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani, A. (1985). Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model. Phys. Rev. A31, 1039-45. Longren, K & Scott, A. (Eds.) (1975). Solitons in Action. Academic Press. Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130-41; The mechanics of vacillation. J. Atmos. Sci. 20, 448-64. Lorenz, E.N. (1964). The problem of deducing the climate from the governing equations. Tellus 16, 1-11.
Lorenz, E.N. (1979). On the prevalence of aperiodicity in simple systems. In Global Analysis, ed. M. Grmela & J.E. Marsden. Lecture Notes in Mathematics 755, pp. 53-75. Springer-Verlag. Lorenz, E.N. (1980). Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37, 1685-99. Lorenz, E.N. (1984). The local structure of a chaotic attractor in four dimensions. Physica 13D, 90-104. Lotka-Volterra (1985). Approach to Cooperation & Competition in Dynamic Systems. Ebeling, W. & Peschel, M. (Eds.) Academic-Verlag: Berlin.
572
Bibliography
Lu, Y.C. (1976). Singularity Theory & an Introduction of Catastrophy Theory. Springer-Verlag. Lugiato, L.A. (1984). Theory of Optical Bistability. Progress in Optics 21, ed. Wolf. E. pp.71-211. North-Holland. Lundquist, S. (Ed.) (1985). The physics of chaos and related problems, Physica Scripta T9, 1-219. Lunsford, G.H. & Ford, J. (1972). On the stability of periodic orbits for nonlinear oscillator systems in regions exhibiting stochastic behavior. J. Math. Phys. 13, 700-5. Macdonald, K.C. & Luyendyk, B.P. (1981). The crest of the East Pacific rise. Sci. Amer. 244(5), 100-16. MacDonald, R.A. & Tasi, D.H. (1978). Molecular dynamics calculations of energg transport in crystalline solids. Phys. Rep. 46, 1-48. MacKay, R.S. (1986). Transition to chaos in area-preserving maps. Nonlinear Dynamics Aspects of Particle Accelerators, ed. J.M. Jowett, M. Month. & S. Turner. pp. 390-454. Springer-Verlag. MacKay, R.S. & Percival, I.C. (1985). Converse KAM: theory and practice. Commun. Math. Phys. 98, 469-512. Mackay, R.S. & Tresser, C. (1986). Transitions to topological chaos for circle maps. Physica 19D, 206-37. Makarov, I.M. (Ed.) (1984). Cybernetics Today. Mir Pub. Makhankov. V.G. (1978). Dynamics of classical solitons (in non-integrable systems). Phys. Rep. 35, 1. (See under Backlund transformations.) Mammerville, P. & Pomeau, Y. (1979). Intermittency and the Lorenz model. Phys. Letters 75A, 1-2. Manakov, S.V. (1974). Complete integrability and stochasticity of discrete dynamical systems. Sov. Phys. JETP 40, 269-74. Manakov, S.V. (1974). Nonlinear fraunhofer diffraction. Sov. Phys. JETP 38, 693-6. Manakov, S.V. (1975). Complete integrability and stochastization of discrete dynamical systems. Sov. Phys. JETP 40, 269-74. Manakov, S.V. & Zakhorov, V.E. (Eds.) (1981). Soliton Theory. Physica 3D, No. 1 & 2. Mandelbrot, B.B. (1977). Fractals: Form, Chance & Dimension. W.H. Freeman. Mandelbrov. B.B. (1983). The Fractal Geometry of Nature. W.H. Freeman. Manneville, P. & Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamic systems. Physica 1D, 219-26. Maradudin, A.A., Montroll, E.W., Weiss, G.H. & Ipatova, I.P. (1971). Theory of Lattice Dynamics in the Harmonic Approximation, 2nd edn. Academic Press. Margolus, N. (1984). Physic-like models of computation, Physica 10D, 81-95. Margolus, N., Toffoli, T. & Vichniac, G. (1986). Cellular-automata supercomputers for fluiddynamics modeling. Phys. Rev. Letters 56, 1694-6. Markley, N.G., Martin, J.C. & Perrizo, W. (1978). The Structure of Attractors in Dynamical Systems. Springer-Verlag. Marotto, F.R. (1978). Snap-back repellers imply chaos in R". J. Math. Anal. App!. 63, 199-223. Marsden, J.E. & McCracken, M. (1976). The Hopf Bifurcation & its Applications. Applied Math. Sci. 19, Springer-Verlag. Martin, P.C. (1975). The onset of turbulence: a review of recent developments in theory and experiment. In Proceedings of the International Conference on Statistical Physics, Budapest 1975, ed. L. Pal. & P. Szepfalusy. pp. 69-96. North-Holland. Martin, 0., Oldzko, A.M. & Wolfram, S. (1984). Algebraic properties of cellular automata. Comm. Math. Phys. 93, 219-58. Martinez, H.M. (1972). Morphogenesis & chemical dissipative structures. J. Theor. Biol. 36, 479-501. Martini, R. (Ed.) (1980). Geometrical Approaches to Differential Equations. Lecture Notes in Mathematics 810. Springer-Verlag.
Bibliography
573
Maselko, J. & Swinney, H.L. (1985). A complex transition sequence in the Belousov-Zhabotinskii reaction, Physica Scripta T9, 35-9. Mather, J.N. (1984). Non-existence of invariant circles. Ergod. Th. & Dynam. Sys. 4, 301-9. Mayer-Kress, G. & Haken, H. (1987). An explicit construction of a class of suspensions and autonomous differential equations for diffeomorphisms in the plane. Commun. Math. Phys. 111, 63-74. May, R.M. (1976). Simple mathematical models with very complicated dynamics. Nature 261, 459-67. (Review article.) Matsuno, K., Dose, K., Harada, K. & Rohlfing (Eds.) (1984). Molecular Evolution and Protobiology, Plenum Press. May, R.M. (1973). Stability and Complexity in Model Ecosystems, Princeton University Press. Mayr, E. (1988). Toward A New Philosophy of Biology, Belknap Press of Harvard Univ. Press. Maynard Smith, J. (1982). Evolutionary Game Theory. Cambridge Univ. Press. Maynard Smith, J. (1986). The Problems of Biology. Oxford Univ. Press. McCauley, J.L. Jr. & Palmore, J.I. (1986). Computable chaotic orbits. Phys. Letters A 15, 433 McDonald, S.W., Grebogi, C., Ott, E. & Yorke, J.A. (1985). Fractal basin boundaries. Physica 7D, 125-53. McGuiness, M.J. (1985). A computation of the limit capacity of the Lorenz attractor. Physica 16D, 265-75. McLachlan, N.W. (1964). Theory and Application of Mathieu Functions. Dover. McLaughlin, D.W. (1975). Four examples of the inverse method as a canonical transformation. J. Math. Phys. 16, 96-9. McLaughlin, J. (1976). Successive bifurcations leading to stochastic behavior. J. Stat. Phys. 15, 307-26. McLaughlin, J.B. & Martin, P.C. (1975). Transition to turbulence in a statistically stressed fluid system. Phys. Rev. A12, 186-203. Meadows, D.L. et al. (1973). Toward Global Equilibrium. Wright-Allen Press: Cambridge, Mass. Meadows, D.L. et al. (1974a). Dynamics of Growth in a Finite World. Wright-Allen Press: Cambridge, Mass.
Meadows D.L. et al. (1974b). The Limits to Growth. Universe Books: NY. Meinhardt, H. (1982). Models of Biological Pattern Formation. Academic Press. Melekhin, V.N. (1976). Phase dynamics of particles in a microtron & the problem of stochastic instability of nonlinear systems. Sov. Phys. JETP 41, 803-8. Melnikov, V.K. (1963). On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, (1), 1-57. Merrill, R.T. & McElminny, M.W. (1983). The Earth's Magnetic Field: Its History, Origin & Planetary Perspective International Geophysics Series, 32. Academic Press. Metropolis, N., Stein, M. & Stein, P. (1973). On finite limit sets for transformations on the unit interval. J. Combinatorial Theory 15, 25-44. Meyer, R.E. (Ed.) (1981). Transition and Turbulence. Academic Press. Meyer, R.E. & Parter, S.V. (Eds.) (1980). Singular Perturbations & Asymptotics. Academic Press. Minorsky, N. (1947). Introduction to Nonlinear Mechanics. J.W. Edwards: Ann Arbor, Mich. Minorsky, N. (1962). Non-Linear Oscillations. Van Nostrand. Misiurewicz, M. (1976). Topological conditiond entropy, Studia Mathematica 55, 175-200. Misiurewicz, M. & Szlenk, W. (1980). Entropy of Piecewise Monotone Mappings, Studia Mathematica 57, 45-63. Mistriotis, M.D. (1986). From determinism to stochasticity, Thesis, Dept. Phys., Univ. Illinois, Urbana.
574
Bibliography
Mistriotis, A.D. & Jackson, E.A. (1987). Transition to stochasticity for periodically perturbed area preserving system. Phvsica Scripta 35, 97-104. Mitchell, K.E. & Dutton, J.A. (1981). Bifurcations from stationary to periodic solutions in a loworder mode of forced dissipative barotropic flow. J. Atmos. Sci. 38, 690-716. Miura, R.M. (1968). A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202-4. Miura, R.M. (1976). The Korteweg-deVries equation: a survey of results. SIAM Review 18, 412-59. Miura, R.M. (Ed.) (1976). Backlund Transformations. The Inverse Scattering Method, Solitons & Their Applications: Lecture Notes in Mathematics 515. Springer-Verlag. Miura, R.M., Gardner, C.S., Kruskal, M.D. (1968). Existence of conservation laws & constants of motion. J. Math. Phys. 9, 1204-9. Mo, K.C. (1972). Theoretical prediction for the onset of widespread instability in conservative nonlinear oscillator systems. Physica 57, 445-54. Moffat, H.K. (1977). Six lectures on general fluid dynamics & two on hydromagnetic dynamo theory. In Fluid Dynamics, ed. R. Balian & J.L. Peube, pp. 151-233. Gordon & Breach. Moffat, H.K. (1978). Magnetic Field Generation in Electrically Conducting Fluids. Cambridge Univ. Press.
Mokross, F. & Buttner, H. (1983). Thermal conductivity in the diatomic Toda lattice, J. Phys. C16, 4539-4546. Montroll, E.W. (1978). On some mathematical models of social phenomena. In Nonlinear Equations in Abstract Spaces, ed. V. Lakshmikantham, pp. 161-216. Academic Press. Moon, F.C. (1980). Experimental models for strange attractor vibrations in elastic systems. In New Approach to Nonlinear Problems in Dynamics, ed. P.J. Holmes. pp.487-95. SIAM. Moon, F.C. (1984). Magneto-Solid Mechanics. Wiley. Moon, F.C. (1987). Chaotic Vibrations, Wiley. Moore-Ede, M.C., Sulzman, F.M. & Fuller, C.A. (1982). The Clocks that Time US. Harvard Univ. Press.
Morozov, A.D. (1976). A complete qualitative investigation of duffing's equation. Diff. Equations 12, 164-74. Morse, P.M. & Feshbach, H. (1953). Methods of Theoretical Physics, McGraw-Hill. Moser, J. (1962). On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Wiss. Gottingen Math. Phys. K1. 2, 1-20. Moser, J. (1967). Convergent series expansions for quasi-periodic motions. Math. Annal. 169, 136-76. Moser, J. (1968). Lectures on Hamiltonian systems. Memoirs Amer. Math. Soc. 81, 1-60. Moser, J. (1973). Stable & Random Motions in Dynamical Systems. Ann. Math. Studies, 77. Princeton Univ. Press. Moser, J. (1975). Three integrable Hamiltonian systems connected with isospectral deformations. Adv. in Math. 16, 197-220. Moser, J. (1978). Nearly Integrable and Integrable Systems. In Topics in Nonlinear Dynamics, ed. S. Jorna, pp. 1-15, Amer. Inst. Phys. Moser, J. (1981). Integrable Hamiltonian Systems & Spectral Theory. Academia Nazionale dei Lincei: Pisa.
Muncaster, R.G. & Zinnes, D.A. (1984). Phase portraits, singularities, and the dynamics of hostility in international relations. SIAM J. Appl. Math. Musha, T. & Higuchi, H. (1977). Proceedings of the Symposium on 1/f Fluctuations. p. 189. Inst. Elect. Engineers: Tokyo. Nagumo, J. & Sato, S. (1972). On a response characteristic of a mathematical neuron model, Kybernetik 10, 155-64.
Bibliography
575
Nakamura, A. (1979). A direct method of calculating periodic wave solutions to nonlinear evolution equations. 1. Exact two-periodic wave solution. J. Phys. Soc. Japan 47, 170. Nakazaea, H. (1970). On the lattice thermal conduction. Prog. Theor. Phys. Suppl. 45, 231-62. Nash, C. & Sen, S. (1983). Topology and Geometry for Physicists. Academic Press. Nayfeh, A.H. (1986). Perturbation methods in nonlinear dynamics. In Nonlinear Dynamics Aspects of Accelerators. Lecture Notes in Physics 247, ed. J.M. Jowett, M. Month, & S. Turner, pp. 238-314. Springer-Verlag. Nayfeh, A.H. & Mook, D.T. (1979). Nonlinear oscillations. Wiley. Nekhoroshev, N.N. (1971). Behavior of hamiltonian systems close to integrable, Funct. Anal. Appl. 5, 338-9. Nekhoroshev, N.N. (1977). An exponential estimate of the time of stability of near-integrable Hamiltonian systems, Russ. Math. Survey 32, 1-65. Nemytskii, V.V. & Stepanov, V.V. (1960). Qualitative Theory of Differential Equations. Princeton Univ. Press. von Neumann, J. (1966). Theory of Self-Reproducing Automata, ed. and completed by A.W. Burks. Univ. of Illinois Press: Urbana. Newell, A.C. (1985). Soliton in Mathematics and Physics. SIAM. Newell, A.C. & Redekopp, L.G. (1977). Breakdown of Zakharov-Shabat theory and soliton creation, Phys. Rev. Lett. 38, 377-80. Newhouse, S., Ruelle, D. & Takens, F. (1978). Occurrence of strange axiom A attractors near quasi-periodic flow on T'°, m > 3. Commun. Math. Phys. 64, 35-40. Newton, R.G. (1978). Three-dimensional solitons. J. Math. Phys. 19, 1068-73. Nicolis, C. & Nicolis, G. (1984). Is there a climatic attractor? Nature 311, 529-32. Nicolis, G. (1971). Stability and dissipative structures in open systems for iron equilibrium. Adv. Chem. Phys. 19, 209. Nicolis, G. (Ed.) (1985). Aspects of Chemical Evolution. Adv. Chem. Phys. 55, Wiley. Nicolis, G. & Prigogine, 1. (1978). Set-Organization in Nonequilibrium Systems. Wiley-Interscience. Nicolis, J.S. (1986). Dynamics of Hierarchical Systems. Springer-Verlag. Niesert, U., Harnasch, & Bresch, C. (1981). Origin of life between seylla and charybolis. J. Mol. Evol. 17, 348-53. Nitecki, Z. (1971). Differential Dynamics. M.I.T. Press. Noid, D.W., Koszykowski, M.L. & Marcus, R.A. (1981). Quasiperiodicity and stochastic behavior in molecular, Ann. Rev. Phys. Chem. 31, 267-309.
Normand, C., Pomeau, Y. & Velarde, M.G. (1977). Convective instability: a physicist's approach. Rev. Mod. Phys. 49, 581-624. Northcote, R.S. & Potts, R.B. (1964). Energy sharing and equilibrium for nonlinear systems. J. Math. Phys. 5, 383.
Nose, S. (1984). A molecular dynamics method for simulations in the canonical ensemble. Molecular Phys. 52, 255-68. Noyes, R.M. (1976). Oscillations in chemical systems. Corrected stoichiometry of the oregonator. J. Chem. Phys. 65, 848-9. Olds, C.D. (1963). Continued Fractions. Random House. O'Malley, R.E. Jr. (1974). Introduction to Singular Perturbations. Academic Press. O'Molley, R.E. Jr. (1976). Phase plane solutions to some singular perturbation problems. J. Math. Anal. Appl. 54, 449-66. O'Malley, R.E. Jr. (1976). Asymtotic Methods and Singular Peturbations. SIAM-AMS Proceedings 10. Amer. Math. Soc.
576
Bibliography
O'Malley, R.E. Jr. (1977). Singular Perturbation Analysis for Ordinary Differential Equations. Comm. Math. Institute Rijksuniversiteit: Utrecht. Oono, Y. (1978). A Heuristic approach to Kolmogorov entropy as a disorder parameter, Prog. Theor. Phys. 60, 1944-46. Oono, Y. & Kohmoto, M. (1985). Discrete model of chemical turbulence. Ph}s. Rev. Letters 55, 2927-31. Oono, Y. & Osikawa, M. (1980). Chaos in nonlinear difference equations. I. Prog. Theor. Phys. 64, 54-67. Oono, Y. & Yeung, C. (1987). A cell dynamical system model of chemical turbulence. J. Stat. Phys. 48, 593-644. Orszag, S.A. (1977). Lectures on the statistical theory of turbulence. In Fluid Dynamics, ed. R. Balian & J.-L. Peube, pp. 237-368. Gordon and Breach. Orszag, S.A. & Kells, L.C. (1980). Transition to turbulence in plane poiseuille and plane couette flow. J. Fluid Mech. 96, 159-205. Orszag, S.A. & Patera, A.T. (1981). Subcritical transition to turbulence in plane shear flows. In Transition and Turbulence, ed. R.E. Meyer. pp. 127-46. Academic Press. Orszag, S. & Yakhot, V. (1986). Reynolds number scaling of cellular-automation hydrodynamics. Phys. Rev. Letters 56, 1691-3. Osborne, A. & Stuart, A.E.G. (1978). On the separability of the Sine-Gordon equation and similar quasilinear partial differential equations. J. Math. Phys. 19, 1573-8. Oseledec, V.I. (1968). A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197. Osikawa, M. & Oono, Y. (1981). Chaos in C -endomorphism of interval. Res. Inst. Math. Sci. Kyoto Univ. 17, 165-77. Ostlund, S., Rand, D., Sethna, J. & Siggia, E. (1983). Universal properties of the transition from quasi-periodicity to chaos in dissipative systems. Physica 8D, 303-42. Othmer, N.G. & Scriven, L.E. (1971). Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32, 507-37. Ovsjanikov (1958). Group and group invariant solutions of differential equations. Dokl. Akad. Nauk. USSR 3, 439. Pacault, A. & Vidal, C. (Eds.) (1979). Synergetics: From Equilibrium. Springer-Verlag. Packard, N.H. & Wolfram, S. (1985). Two-dimensional cellular automata, J. Stat. Phys. 38, 901-46. Packard, N.H., Crutchfield, J.P., Farmer, J.D. & Shaw, R.S. (1980). Geometry from a time series, Phys. Rev. Letters 45, 712-6. Paladin, G. & Vulpiani, A. (1987). Anomalous scaling laws in multifractral objects, Phys. Reports 156, 147-225. Palmore, J.I. & McCauley, J.L. (1987). Shadowing by computable chaotic orbits. Phys. Letters A122, 399-402. Park, J.K., Steglitz, K. & Thurston, W.D. (1986). Soliton-like behavior in automata, Physica 19D, 423-32. Parker, E.N. (1975). The dynamo mechanism for the generation of large-scale magnetic fields. Ann. N.Y. Acad. Sci. 252, 141-55. Pars, L.A. (1983). A Treatise on Analytical Dynamics. Wiley. Paulos, J.A. (1981). Mathematics and Humor. Univ. of Chicago Press. E. Paulre, E. (Ed.) (1981). System Dynamics and the Analysis of Change. North-Holland. (Economic, medical, war, urban, etc., models.) Pauwelussen, J.P. (1981). Nerve impulse propagation in a branching nerve system: a simple model. Physica 4D, 67-88. Pavelle, R., Rothstein, M. & Fitch, J. (1981). Computer algebra. Sci. Amer. 245 (6), 136.
Bibliography
577
Peitgen, H.-O. & Richter, P.H. (1986). The Beauty of Fractals, Springer-Verlag. Peixoto, M.M. (Ed.) (1973). Dynamical Systems. Academic Press. Peliti, L. & Vulpiani, A. (Eds.) (1988). Measures of Complexity. Springer-Verlag. Percival, I. (1982). Introduction to Dynamics. Cambridge Univ. Press. Perlmutter, A. & Scott, L.F. (Eds.) (1977). The Significance of Nonlinearity in the Natural Sciences. Plenum.
Perring, J.K. & Skyrme, T.H.R. (1962). A model unified field equation. Nucl. Phys. 31, 550-5. Peschel, J. & Mende, W. (1986). The Predator-Prey Model. Springer-Verlag. Petrovsky. I.G. (1954). Lectures on Partial Differential Equations. Interscience Pub. Petrosky, P. & Prigogine, 1. (1982). The KAM invariant and Poincare's theorem. In Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics, ed. V. Szebehely, pp. 185-99, Reidel. Pietronero, L. & Tosatti, E. (Eds.) (1986). Fractals in Physics, North-Holland. Pimbley, G.H. Jr. (1969). Eigenfunction Branches of Nonlinear Operators and Their Bifurcation. Lecture Notes in Mathematics 104. Springer-Verlag. Pines, D. (Ed.) (1985). Emerging Syntheses in Science. Santa Fe Institute. Pippard, A.B. (1978). The Physics of Vibration. Cambridge Univ. Press. Pippard, A.B. (1985). Response and Stability. Cambridge Univ. Press. Pnevmatikos, S.N. (Ed.) (1985). Singularities and Dynamical Systems. Math. Studies 103, North-Holland. Pnevmatikos, S.N. (1985). Solitons in nonlinear atomic chains. In Singularities and Dynamical Systems, ed. S.N. Pnevmatikos, pp. 397-437. North-Holland. Poineare, H. (1892, 1893, 1894). Les Methodes Nouvelles de la Mecanique Celeste. Vols. 1-3. Gauthier-Villars: Paris. Reprinted by Dover, 1957. (Translation: New Methods of Celestial Mechanics. NASA, 1967.) Poincare, J.H. (1929). The Foundations of Science, Science Press. Popper, K. (1968). The Logic of Scientific Discovery. Harper & Row. Poston, T. & Stewart, IN. (1976). Taylor Expansions and Catastrophes. Pitman. Poston, T. & Stewart, I. (1978). Catastrophe Theory and Its Applications. Pitman. Poston, T. & Woodcock, A.E.R. (1973). Zeeman's catastrophe machine. Proc. Camb. Phil. Soc. 74, 217.
Poundstone, W. (1985). The Recursive Universe. Contemporary Books. Press, W.H., Flannery, B.F., Teukolsky, S.A. & Vetterling, W.T. (1986). Numerical Recipes. Cambridge Univ. Press. Preston, C. (1983). Iterates of Maps on an Interval. Springer-Verlag. Prigogine, I. (1962). Non-Equilibrium Statistical Mechanics, Interscience. Prigogine, 1. (1980). From Being to Becoming. Freeman. Prigogine, I. & Lefever, R. (1968). Symmetry breaking instabilities in dissipative systems, II. J. Chem. Phys. 48, 1695-700. Prigogine, I. & Stengers, 1. (1984). Order Out of Chaos. Bantam Books. Rabinowitz, P.H. (Ed.) (1977). Applications of Bifurcation Theory. Academic Press. Rajaraman, A. (1975). Some non-perturbative semi-classical methods in quantum field theory. Phys. Rep. 21C, 227-313. Ramajaran, R. (1982). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland. Ramani, A., Dorizzi, B., Grammaticos, B. & Bountis, T. (1984). Integrability and the painleve property for low-dimensional systems. J. Math. Phys. 25, 878-83. Rannou, F. (1974). Numerical study of discrete plane area-preserving map. Astron. Astrophys. 31, 289-301. Rashevsky, N. (1938). Mathematical Biophysics. Univ. of Chicago Press. Reprint, Dover, 1960.
578
Bibliography
Raty, R., Isomaki, H.M. & von Boehm, J. (1984). Chaotic motion of a classical anharmonic oscillator. Acta Polytechnica Scandinavica. Mech. Eng. Ser. 85, 1-30. Rebbi, C. (1979). Solitons. Sci. Amer. 240(2), 92-116. Richtmyer, R.D. (1986). A study of the Lorenz attractor, science and computers, Adv. Math. Suppl. Studies, 10, 207-19. Risset, J.C. (1982). Stochastic processes in music and art. In Stochastic Processes in Quantum Theory and Statistical Physics. Lecture Notes in Physics 173, ed. S. Albeverio, Ph. Combe, & M. Sirugue-Collin. Springer-Verlag. Robbins, K.A. (1977). A new approach to subcritical instability and turbulent transitions in a simple Dynamo. Math. Proc. Camb. Phil. Soc. 82, 309-25. Robbins, K.A. (1979). Periodic solutions and bifurcation structure at high R in the Lorenz model. J. Appl. Math. 36, 457-71. Roberts, P.H. (1971). Dynamo theory. In Lectures in Applied Mathematics, ed. W.H. Reid, pp. 129-206. Amer. Math. Soc. Roederer. J. (1979). Introduction to the Physics and Psychophysics of Music. Springer-Verlag. Rogers, C. & Shadwick, W.F. (1982). Backlund Transformations and Their Applications. Academic Press.
Rohlin, V.A. (1964). Exact endomorphisms of a Lebesque space. Trans. Amer. Math. Soc. 39, 1-36. Rohlin, V.A. (1967). Lectures on the Entropy Theory of Measure-Preserving Transformations, Russian Math Survey 22(5), 1-52. Rolfsen, D. (1976). Knots and Links. Publish or Perish: Berkeley. Rollins, R.W. & Hunt, E.R. (1982). Exactly solvable model of a physical system exhibiting universal chaotic behavior. Phys. Rev. Letters 49, 1295-8. Rosen, R. (1985). Anticipatory Systems, Pergamon. Rosenbluth. M.N. (1972). Superadiabaticity in mirror machines. Phys. Rev. Letters 29, 408-10. Rossler, O.E. (1972). A principle for chemical multivibration. J. Theor. Biol. 36, 413-7. Rossler, O.E. (1977). Continuous chaos. In Synergetics: A Workshop, ed. H. Haken, pp. 184-97. Springer-Verlag. Rossler, O.E. (I979a). Continuous chaos: four prototype equations. Ann. N.Y. Acad. Sci. 316, 376-92. Rossler, O.E. (1979b). Chaos. In Structural Stability in Physics, ed. W. Guttinger & H.E. Kemeier, pp.290-309. Springer-Verlag. Rossler, O.E. (1983). The chaotic hierarchy. Z. Naturforsch. 38a, 788-801. Roux, J.C., Simoyi, R.H. & Swinney, H.L. (1983). Observation of a strange attractor. Physica 8D, 257-66. Rubel, L.A. (1981). A universal differential equation. Bull. Amer. Math. Soc. New Sevies. 4, 345-49. Rubel, L.A., Digital simulation of analog computation and Church's thesis, J. Symbolic Logic (to appear). Rubinstein, J. (1970). Sine-Gordon equation. J. Math. Phys. 11, 258-66. Rudakov, L.I. & Tsytovich, V.N. (1978). Strong langmuir turbulence. Phys. Rep. 40, 1-73. Ruelle, D. (1978). Dynamical systems with turbulent behavior. In Mathematical Problems in Theoretical Physics, ed. G. Del'Atonio, S. Doplicher & G. Jona-Lasinio, pp. 341-60. Lecture Notes in Physics 80. Springer-Verlag. Ruelle, D. (1983). Small random perturbations and the definition of attractors. In Geometric dynamics. Lecture Notes in Mathematics, Eds. A Doid & B. Eckmann, pp. 663-76. Springer-Verlag. Ruelle, D. & Takens, F. (1971). On the nature of turbulence. Commun. Math. Phys. 20, 167-92, 23, 343-4.
Bibliography
579
Russell, D.A., Hanson, J.D. & Ott, E. (1980). Dimension of strange attractors. Phys. Rev. Letters 45, 1175-8. Russell, J.S. (1844). Report on waves. In Report of 14th Meeting, British Assoc. Adv. Sci. pp.311-90. Marray: London. Ryutov, D.D. & Stupakov, G.V. (1978). Diffusion of resonance particles in ambipolar plasma traps. Sov. Phys. Dokl. 23, 412-4. Saaty, T. L., Alexander, J.M. (1981). Thinking with Models. Pergamon. Saari, D.G. (1984). The ultimate of chaos resulting from weighted voting systems. Adv. Appl. Math. 5, 286-308. Sagdeev, R.Z. (Ed.) (1984). Nonlinear and Turbulent Processes in Physics, Vols. 1-3. Harwood Acad. Pub. Sagdeev, R.Z., Usikov, D.A., & Zaslavsky, G.M. (1988). Nonlinear Physics. Harwood Academic Pub.. Sait6, N., Ooyama, N. & Aizawa, Y. (1970). Computer experiments on ergodicity problems in anharmonic lattice vibrations. Prog. Theor. Phys. Suppl. 45, 209-30. Saito, N., Hirotomi, N. & Ichimura, A. (1975). The induction phenomenon and ergodicity in the anharmonic lattice vibration. J. Phys. Soc. Japan 39, 1431-8. Saltzman, B. (1962). Finite amplitude free convection as an initial value problem I. J. Atmos. Sci. 19, 329-41. Sanders, J.A. & Verhulst, F. (1984). Averaging Methods in Nonlinear Dynamical Systems. Appl. Math. Sciences. Springer-Verlag. Schmidt, G. & Tondl, A. (1986). Non-Linear Vibrations, Cambridge Univ. Press. Schmutz M. & Rueff, M. (1984). Bifurcations schemes of the Lorenz model. Physica 11D, 167-78. Schrodinger, E. (1914). Zur dynamik elastisch gekoppelter punktsysteme. Ann. d. Phys. 44, 916-34. Schult, R.L., Creamer, D.B., Henyey, F.S. & Wright, J.A. (1987). Symmetric and nonsymmetric coupled logistic maps, Phys. Rev. A35, 3115-18. Schuster, H.G. (1984). Deterministic Chaos. Physik-Verlag: Weinheim. Schuster, P. & Sigmund, K. (1983). Replicator dynamics. J. Theor. Biol. 100, 533-8. Schuur, P.C. (1986). Asymptotic Analysis of Soliton Problems. Springer-Verlag. Scott, A.C. (1970). Active and Nonlinear Wave Propagation in Electronics. Wiley. Scott, A.C., Chu, F.Y.F. & McLaughlin, D.W. (1973). The soliton: a new concept in applied science. Proc. IEEE 61, 1443-83. Sedov, L.I. (1959). Similarity and Dimensional Methods in Mechanics. Academic Press. Segur, H. (1986). Some open problems, Physica 18D, 1-12. Shannon, C.E. (1948). A mathemational theory of communication. Bell. Tech. J. 27, 623-56. Sharkovsky, A.N. (1964). Coexistence of the cycles of a continuous mapping of the line into itself. Ukr. Mat. Zh. 16(1), 61-71. Shaw, R. (1981). Strange attractors, chaotic behavior, and information flow. Z. Naturforsch. 36a, 80-112. Shaw, R. (1984). The Dripping Faucet as a Model Chaotic System. Aerial Press. Shimada, 1. & Nagashima, T. (1979). A numerical approach to ergodic problems of dissipative dynamical systems. Prog. Theor. Phys. 61, 1605. Siegel, C.L. & Moser, J.K. (1971). Lectures on Celestial Mechanics. Springer-Verlag. Sigmund, K. (1983). The maximum principle for replicator equations. J. Theor. Biol. 100, 63-72. Simoyi, R.H., Wolf, A. & Swinney, H.L. (1982). One-dimensional dynamics in a multi-component chemical reaction. Phys. Rev. Letters 49, 245-8. Simo, C. (1979). On the Henon-Pomeau attractor, J. Stat. Phys. 21, 465-94. Sinai, J.G. (1959). On the concept of entropy of a dynamical system. Dokl. Akad. Nauk. SSSR 124, 768-76. (Russian.)
580
Bibliography
Sinai, Ya.G. (1977). Introduction to Ergodic Theory. Princeton Univ. Press. Sinclair, R.M., Hosea, J.C. & Sheffield, G.V. (1970). A method for mapping a toroidal magnetic field by storage of phase stabilized electrons. Rev. Sci. Inst. 41, 1552-9. Sinhah, D.K. (Ed.) (1981). Catastrophe Theory and Applications. Wiley. Singer, D. (1978). Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 2607.
Smale, S. (1963). Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology, ed. S.S. Cairns, pp. 63-80. Princeton Univ. Press. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747-817. Smale, S. (1974). A mathematical model of two cells via Turing's equation. Lecture Notes in Applied Mathematics 6, 15-26. Amer. Math. Soc. Smale, S. (1976). On the differential equations of species in competition. J. Math. Biol. 3, 5-7. Smale, S. (1977). Dynamical systems and turbulence. In Turbulence Seminar. Lecture Notes in Matehmatics 615, 48-70. Springer-Verlag. Smale, S. (1978). Review of catastrophe theory, by E.C. Zeeman. Bull. Amer. Math. Soc. 84, 1360-8. Smale, S. (1981). Essays on Dynamical Systems, Economic Processes, and Related Topics. Springer-Verlag. Smale, S. (1981). The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. New Series 4, 1-36. Smale, S. & Williams, R. (1976). The quantitative analysis of a difference equation of population growth. J. Math. Biol. 3, 1-4. Smart, D.R. (1974). Fixed Point Theorems. Cambridge Univ. Press. Sparrow, C. (1982). Bifurcations in the Lorenz Equation. Lecture Notes in Applied Mathematics. Springer-Verlag. Stanley, H.E. & Ostrowsky, N. (Eds.) (1986). On Growth and Form. Nijhoff. Stapleton, H.J., Allen, J.P., Flynn, C.P., Stinson, D.G. & Kurtz, S.R. (1980). Fractal form of proteins. Phys. Rev. Letters 45, 1456-9. Stefan, O. (1977). A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Commun. Math. Phys. 54, 237-48. Stewart, 1. (1982). Catastrophe theory in physics. Rep. Prog. Phys. 45, 185-221. Stine, J.R. & Noid, D.W. (1983). Method to determine the number of constants of the motion for multidimensional systems, J. Phys. Chem. 87, 3039-42. Stoker, J.J. (1950). Nonlinear Vibrations. Interscience Pub. Stoker, J.J. (1957). Water Waves. Interscience Pub. (Appendix: On the Derivation of the Shallow Water theory, by K.O. Friedrichs.) Stormer, C. (1955). The Polar Aurora. Clarendon Press: Oxford. Stoutemyer, D.R. & Yun, D.Y.Y. (1980). Symbolic mathematical computation. In Encyclopedia of Computer Science and Technology 15, Supplement, ed. J. Belzer, A.G. Holzman & A. Kent. Dekker. Straffin, P.D. Jr. (1978). Periodic points of continuous maps. Math. Mag. 51, 99-105. Strangway, D.W. (1970). History of the Earth's Magnetic Field. McGraw-Hill. Strauss, W.A. (1977). Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149-62. Stravrondis, ON. (1972). The Optics of Rays, Wavefronts and Caustics. Academic Press. Stuart, J.T. (1971). Nonlinear stability theory. Ann. Rev. Fluid Mech. 3, 347-70. Su, C.H. & Gardner, C.S. (1969). Derivation of the KdV equation and Burger's equation. J. Math. Phys. 10, 536-9.
Bibliography
581
Sudarshan, E.C.G. & Mukunda, N. (1974). Classical Dynamics: A Modern Perspective. Wiley. Sussman, H.J. (1977). A skeptic. Science 197, 821. Swindale, N.V. (1980). A model for the formation of ocular dominance stripes. Proc. R. Soc. Lond. B208, 243-64. Swinney, H.L. & Gollub, J.P. (1978). The transition to turbulence. Phys. Today 31(8), 41. Swinney, H.L. & Gollub, J.P. (Eds.) (1981). Hydrodynamic Instabilities and the Transition to Turbulence. Topics in Applied Physics 45, Springer-Verlag. Swinney, H.W. & Gollub, J.P. (Eds.) (1985). Hydrodynamic Instabilities and the Transition to Turbulence. Topics in Applied Physics 45. Springer-Verlag. Sverdlove, R. (1977). Inverse problems for dynamical systems in the plane. In Dynamical Systems, eds A.R. Bednarek & L. Cesi, pp. 499-502, Academic Press. Szebehely, V. (Ed.) (1982). Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics. D. Reidel. Szebehely, V. (1984). Review of concepts of stability. Celestial Mech. 34, 49-64. Szebehely, V. (1986). New nondeterministic celestial mechanics. In Space Dynamics and Celestial Mechanics, ed. K.B. Bhatnagar, pp.3-14. D. Reidel. Szlenk, W. (1984). An Introduction to the Theory of Smooth Dynamical Systems. Wiley. Tabor, M. & Weiss, J. (1981). Analytic structure of the Lorenz system, Phys. Rev. A24, 2157-67. Takeno, S. (Ed.) (1985). Dynamical Problems in Soliton Systems. Springer-Verlag. Takens, F. (1973). Introduction to Global Analysis. Comm. Math. Inst., Rijksuniversiteit Utrecht. Takens, F. (1981). Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics 898, ed. D.A. Rand & L.-S. Young. Springer-Verlag. Takens, F. (1985). On the numerical determination of the dimensions of an attractor. In Dynamical Systems and Bifurcations, eds. B.L.J. Braaksma, H.W. Broer & F. Takens, pp. 99-106. Lect. Notes Math. 1125, Springer-Verlag. Testa, J., Perez, J. & Jeffries, C. (1982). Evidence of universal chaotic behavior of a driven nonlinear oscillator. Phys. Rev. Letters 48, 714-7. Thirring, W. (1978). Classical Dynamical Systems. Springer-Verlag. Thom, R. (1969). Topological models in biology. Topology 8, 313-35. Thom, R. (1975). Structural Stability and Morphogenesis. W.A. Benjamin. Thom, R. (1977). Structural stability, catastrophe theory and applied mathematics. SIAM Review, April.
Thompson, J.M.T. & Hunt, G.W. (1973). A General Theory of Elastic Stability. Wiley. Thompson, J.M.T. & Hunt, G.W. (1975). Towards a unified bifurcation theory. J. Appl. Math. Phys. 26.
Thompson, J.M.T. & Stewart, H.B. (1986). Nonlinear Dynamics and Chaos. Wiley. Thornhill, S.G. & Haar, D. (1978). Langmuir turbulence and modulational instability. Phys. Rep. 43, 43.
Toda, M. (1967). J. Phys. Soc. Japan 22, 431-6; 23, 501-6. Toda, M. (1971). Waves in nonlinear lattice. Prog. Theor. Phys. Suppl. 45, 174. Toda, M. (1975). Studies of a non-linear lattice. Phys. Rep. 18C, 1-124. Toda, M. (1981). Theory of Nonlinear Lattices. Springer-Verlag. Toffoli, T. (1977). Cellular Automata Mechanics. Technical Report 208. Computer and Communication Sciences Department, Univ. of Michigan. Toffoli, T. (1982). Physics and computation. Int. J. Theor. Phys. 21, 165-75. Toffoli, T. (1984). Cellular automata as an alternative to (rather than an approximation of) differential equations modeling physics. Physica 10D, 117-27.
582
Bibliography
Toffoli, T. & Margolus, N. (1987). Cellular Automata: A New Environment. for Modeling, MIT Press. Tomita, K. (1982). Chaotic response of nonlinear oscillators. Phys. Rep. 86, 113-67. Treve, Y.M. (1975). Theory of chaotic motion with application to controlled fusion research. In Topics in Nonlinear Dynamics, ed. S. Jorna, pp. 147-220. North-Holland. Tricomi, F.G. (1957). Integral Equations. Interscience Pub.
Troy, W. (1977). A threshold phenomenon in the Field-Noyes model of the Belousov-Zhabotinskii reaction. J. Math. Anal. Appl. 58, 233-48. Tuck, J.L. & Menzel, M.T. (1972). The superperiod of the nonlinear weighted string (FPU) problem. Adv. Math. 9, 339-407. Tucker, A.W. & Bailey, H.S. Jr. (1950). Topology. Sci. Amer. 182(1), 18-24. Turing, A.M. (1952). The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B237, 37-72. Turner, J.S., Roux, J.C., McCormick, W.D. & Swinney, H.L. (1981). Alternating periodic and chaotic regimes in chemical reaction-experiment and theory. Phys. Letters 85A, 9-12. Tyson, J.J. (1979). Oscillations, bistability, and echo waves. In Models of the Belousov-Zhobotinskii rgaction, ed. O. Gurrel & O.E. Rossler. Ann. Y.Y. Acad. Sci. 316, 279-95. Ueda, Y. (1979). Randomly transitional phenomena in systems governed by Duffing's equation. J. Stat. Phys. 20, 181-96. Uezu, T. (1983). Topology in dynamical systems, Phys. Letters 93A, 161-6. Uezu, T. & Aizawa, Y. (1982). Topological character of a periodic solution in three-dimensional ordinary differential equation systems. Prog. Theor. Phys. 68, 1907-16. Ulam, S.M. (1960). A Collection of Mathematical Problems, Wiley. Ulam, S.M. (1965). Introduction to Studies of Nonlinear Problems by E. Fermi, J. Pasta & S.M. Ulam. In Collected Papers of Enrico Fermi, Vol. 2, Univ. Chicago Press. Ulam, S.M. (1982). Transformations, Iterations and mixing flows. In Dynamical Systems II, eds. A.R. Bednarek and L. Cesari, pp. 419-26, Academic Press. Ulam, S.M. (1983). Adventures of a Mathematician, Scribner. Ulam, Stanislaw (1909-1984); Los Alamos Science, Number 15, Special Issue (1987). Umberger, D.K. & Farmer, J.D. (1985). Fat fractals on the energy surface, Phys. Rev. Letters 55, 66164.
Vainberg, M.M. & Trenogin, V.A. (1974). Theory of Branching of Solutions of Nonlinear Equations. Noordhoff: Leyden. van Bendegem, J.P. (1987). Finite Empirical Mathematics; Outline of a Model. Rijksuniversitent Gent. Van der Blij, F. (1978). Some details of the History of the Korteweg-deVries equation, Nieu Archief VoorWiskunde(3) 26, 54-64. van der Pol, B. & van der Mark, J. (1927). Frequency demultiplication. Nature 120, 363-4. VanVelsen, J.F.C. (1978). On linear response theory and area preserving mappings. Phys. Rep. 41C, 135-90. Vendrik, M.C.M. (1979). A classification of phase diagrams by means of "elementary" catastrophe theory (ECT). 1. Physica 99A, 103-44. Verhulst, F. (Ed.) (1979). Asymptotic Analysis. Lecture Notes in Mathematics 711. Springer-Verlag. Verhulst, F. (Ed.) (1983). Asymptotic Analysis of Hamiltonian Systems. Lecture Notes in Mathematics 985. Springer-Verlag. Vichniac, G.Y. (1984a). Simulating physics with cellular automata. Physica 10D, 96-116. Vichniac, G.Y. (1984b). Instability in discrete algorithms and exact reversibility. SIAM J. Alg. Disc. Meth. 5. Vidal, C. & Pacault, A. (1982). Spatial chemical Structures: chemical waves. A. Review. In Evolution of Order and Chaos, ed. H. Haken, pp. 74-99. Springer-Verlag.
Bibliography
583
Vidal, C. & Pacault, A. (Eds.) (1984). Nonequilibrium Dynamics in Chemical Systems. Springer-Verlag. Vidyasaga, M. (1978). Nonlinear Systems Analysis. Prentice-Hall. Visscher, W.M. (1976). Computer Studies of Transport Properties in Simple Models of Solids, Meth. Comput. Phys. 15, ed. G. Gilat, pp. 371-408. Academic Press. Von Karmen, T. (1940). The engineer grapples with nonlinear problems (fifteenth Josiah Willard Gibbs lecture). Bull. Amer. Math. Soc. 46, 615-83. Wadati, M. (1972). The exact solution of the modified Korteweg-deVries equation. J. Phys. Soc. Japan 32, 1681. Wadati, M. (1975). Wave propagation in nonlinear lattice. J. Phys. Soc. Japan 38, 673-80, 682-6. Wahlquist, H.D. & Estabrook, F.B. (1973). Backlund transformations for solutions of the Kortewig-de Vries equation. Phys. Rev. Letters 31, 1386. Walker, G.H. & Ford, J. (1969). Amplitude instability and ergodic behavior for conservative nonlinear oscillator systems. Phys. Rev. 188, 416-32. Walker, J. (1978). Chemical systems that oscillate between one color and another. Sci. Amer. 239, 152-60. Waller, 1. & Kapral, R. (1984). Spatial and temporal structure in systems of coupled nonlinear oscillators. Phys. Rev. A30, 2047-55. Wasow, W. (1965). Asymptotic Expansion for Ordinary Differential Equations. Interscience. Weiland, J. & Wilhemsson, H. (1977). Coherent Non-Linear Interaction of Waves in Plasmas. Pergamon. Weinreich, G. (1979). The coupled motions of piano strings, Scientific American 240(l), 118-27. Weiss, L. (Ed.) (1972). Ordinary Differential Equations. Academic Press. Wesfreid, J.E. & Zaleski, S. (Eds.) (1984). Cellular Structures in Instabilities. Lecture Notes in Physics 210. Springer-Verlag. Whiteman, K.J. (1977). Invariants and stability in classical mechanics. Rep. Prog. Phys. 40, 1033-69. Whitham, G.B. (1974). Linear and Nonlinear Waves. Wiley. Whitney, H. (1936). Differentiable manifolds. Ann. Math. 37, 645. Whitney, H. (1944). The self-intersections of a smooth n-manifold in 2n-space. Ann. Math. 45, 220-46. The singularities of a smooth n-manifold in (2n-1)-space. Ann. Math. 45, 247-93. Whittaker, E.T. (1944). Analytical Dynamics of Particles and Rigid Bodies. Dover. Wicken, J.S. (1987). Evolution, Thermodynamics, and Information, Oxford Univ. Press. Wiener, N. & Rosenblueth, A. (1946). The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Arch. Inst. Cardiol. Mexico 16, 205-65. Wilhelmsson, H. (Ed.) (1979). Solitons in Physics, Physica Scripta 20, 289-560. Williams, R.F. (1979). The Structure of the Lorenz Attractor, Pbl. Math. IHES 50, 73-99. Winfree, A.T. (1980). The Geometry of Biological Time. Lecture Notes in Biomathematics 8. Springer-Verlag. Winfree, AT. (1987). When Time Breaks Down. Princeton Univ. Press. Winternitz, P. (1983). Lie groups and solutions of nonlinear differential equations. In Nonlinear Phenomena. Lecture Notes in Physics 189, ed. K.B. Wolf, pp. 265-325. Springer-Verlag. Wintner, A. (1947). The Analytic Foundations of Celestial Mechanics. Princeton Univ. Press.
Wolf, A., Swift, J.B., Swinney, H.L. & Vastano, J.A. (1985). Determining Lyapunov exponents from a time series. Physica 16D, 285-317. Wolfram, S. (1982). Physics and computation. Int. J. Theor. Phys. 21, 165-75. Wolfram, S. (1983). Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601. Wolfram, S. (1984). Universality and complexity in cellular automata, Physica 10D, 1-35.
584
Bibliography
Wolfram, S. (1985). Twenty problems in the theory of cellular automata. Physica Scripta T9, 107-83. Wolfram, S. (1986). Theory and Applications of'Cellular Automata. World Scientific. Wolfram, S. (1988). Mathematica, Wolfram Research Inc., Champaign, Illinois. Wright, R. (1988). Three Scientists and Their Gods, Times Books. Yahata, H. (1978). Temporal development of the Taylor vortices in a rotating fluid. Prog. Theor. Phys. Suppl. 64, 165-85. Yajima, N. (1985). Soliton resonance in plasmas. In Dynamical Problems in Soliton Systems, ed. S. Takeno, pp. 144-52. Springer-Verlag. Yajima, N., Oikawa, M. & Satsuma, J. (1978). Interaction of ion-acoustic solitons in threedimensional space. J. Phys. Soc. Japan 44, 1711-4. Yakubovich, V.A. & Starzhinskii, V.M. (1975). Linear Differential Equations with Periodic Coefficients, Vols. 1 and 2. Keter Publishing House: Jerusalem. Yamada, T. & Fujisaka, H. (1983). Stability theory of synchronized motion in couple-oscillator systems II. Prog. Theor. Phys. 70, 1240-8. Yan-Qian, Y. et al. (1986). Theory of Limit Cycles. Amer. Math. Soc. Translation of Math. Monographs 66. Yeung, C. (1988). Some problems on spatial patterns in nonequilibrium systems. Thesis, University of Illinois.
Yorke, J.A. & Yorke, E.D. (1979). Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys. 21, 263-77. Yorke, J.A. & Yorke, E.D. (1981). Chaotic behavior and fluid dynamics. In Hydrodynamic Instabilities and the Transition to Turbulence, ed. H.L. Swinney & J.P. Gollub, pp. 77-94. Springgr-Verlag.
Zabusky, N.J. (1969). Nonlinear lattice dynamics and energy sharing. J. Phys. Soc. Japan 26, Suppl. Zabusky, N.J. (1981). Computational synergetics and mathematical innovation. J. Comput. Phys. 43, 195-249.
Zabusky, N.J. & Kruskal, M.D. (1965). Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys. Rev. Letters 15, 240-3. Zahler, R.S. & Sussman, H.J. (1977). Claims and accomplishments of applied catastrophe theory. Nature 269, 759-63. Review article. Zakharov, V.E. (1972). Collapse of Langmuir waves. Soc. Phys. JETP 35, 908-14. Zakharov, V.E. (1974). On stochastization of one-dimensional chains of nonlinear oscillators. Sov. Phys. JETP 38, 108-110. Zakharov, V.E. & Manakov, S.V. (1977). Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method. Sov. Phys. JETP 44, 106-112. Zakharov, V.E. & Shabat, A.B. (1972). Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62-9. Zaslavsky, G.M. (1984). Chaos in Dynamic Systems. Harwood Academic Press. Zeeman, E.C. (1977). Catastrophe Theory. Addison-Wesley. (Selected papers, 1972-1977.) Zeeman, E.C. (1980). Population dynamics from game theory. In Global Theory of Dynamical Systems. Lecture Notes in Mathematics 814, ed. Z. Nitecki & G. Robinson. Springer-Verlag. Zehnder, E. (1973). Homoclinic points near elliptic fixed points. Commun. Pure Appl. Math. 26, 131-92. Zhabotinskii, A.M. (1967). Oscillatory Processes in Biological and Chemical Systems. Science Pub.: Moscow.
References by topics
TOPICS Backlund transformations Bifurcation, branching Biological, ecological, and genetic systems Catastrophe theory Cellular automata Chaotic dynamics (reviews; collections) Chaotic dynamics (Theory) Chaotic dynamics, experimental Chemical oscillations, turbulence Chemical systems (also see, Biological systems) Circle map Classical mechanics Computer computations Computer dynamics Computer studies - nonchaotic dynamics Computer studies of chaos, integrability, irreversibility, etc. Coupled maps (patterns) Difference equations (maps) Dynamic systems Dynamic entropies and information production (also see, chaotic dynamics) Dynamo theory; Earth's magnetic field Explosive dynamics Fractals Hydrodynamics Integrable systems Invariance properties and conservation laws Inverse scattering method (also see, solitons) Korteweg-deVries equation Lattice dynamics Lie theory of continuous groups Lorenz equations Lyapunov exponents
Mathematical definitions and terminology Mathematics Mechanical and electrical systems (also see, classical mechanics) Meteorology Modeling methodology Nerve conduction Neural nets Nonlinear superposition (also see, Lie-Backlund transformations) Nonsoliton asymptotic part Optical phenomena Painleve conjecture Patterns Pattern dynamics (also see, Chemical turbulence, and coupled maps) Related topics Rossler models Similarity analysis Sine Gordon equation Singular perturbation, asymptotic expansions Social phenomena (see references on modeling methodology) Solitons (also see, Korteweg-deVries, Backlund, Lattice dynamics and inverse scattering) Solitons, direct method (Hirota) Solitons, higher dimensional Standard and area-preserving maps; microtron instability Surveys Synergetics Turbulent-like dynamics (also see, chemical systems, and explosive dynamics) Turbulence, onset
586
References by topics
Backlund transformations Anderson, R.L. & Ibraginov, N.H. (1979). Lie-Backlund Transformations in Applications, SIAM contains historical references. Lamb, G.L. Jr. (1974). Backlund transformations for certain nonlinear evolution equations, J. Math. Phys. 15, 2157. Martini, R. (Ed.) (1980). Geometrical approaches to differential equations, Lecture Notes in Mathematics 810, Springer-Verlag. Miura, R.M. (Ed.) (1976). Backlund Transformations, Transformations, The Inverse Scattering Method, Solitons, and Their Applications, Lecture Notes in Mathematics, 515, Springer-Verlag. Rogers, C. & Shadwick, W.F. (1980). Backlund Transformations and Their Applications, Academic Press.
Wahlquist, H.D. & Estabrook, F.B. (1973). Backlund transformations for solutions of the KortewegdeVries equation, Phys. Rev. Lett. 31, 1386.
Bifurcation, branching Amann, H., Bazley, N. & Kirchgassner, K., (Eds.) (1981). Applications of Nonlinear Analysis in the Physical Sciences, Pitman. Andronov, A.A., Leontovich, E.A., Gordon, I.I. & Maier, A.G. (1971). Theory of Bifurcations of Dynamic Systems on a Plane, US Dept. Commerce; Israel Program for Scientific Translations. Arnold, V.I. (1972). Lectures on bifurcations in versal families, Russian Math Survey, 27, 54-123. Bardos, C. & Bessis, D. (Eds.) (1980). Bifurcation Phenomena in Mathematical Physics and Related Topics, D. Reidel Pub. Co. Chow, S.N. & Hale, J.K. (1982). Methods of Bifurcation Theory, Springer-Verlag. Crawford, J.D. & Omohundro, S. (1984). On the Global Structure of Period Doubling flows, Physica 13D, 161-80. Gurel, G. & Rossler, O.E. (Eds.) (1979). Bifurcation theory and applications in scientific disciplines, Ann. NYAcademy Science, 316. Gattinger, S. (1986). Bifurcation geometry in physics, in Frontiers of Nonequilibrium Statistical Physics, eds. G.T. Moore and M.O. Scully, pp. 57-82, Plenum Press. Holmes, P. & Williams, R.F. (1985). Knotted periodic orbits in suspensions of Smale's horseshoe: torus knots and bifurcation sequences, Arch. Ration. Mech. Anal. 90, 115-94. Iooss, G. & Joseph, D.D. (1980). Elementary Stability and Bifurcation Theory, Springer-Verlag. Keller, J. B. & Antman, S. (Eds.) (1969). Bifurcation Theory and Nonlinear Eigenvalue Problems, W.A. Benjamin. Marsden, J.E. & McCracken, M. (1976). The Hopf Bifurcation and Its Applications, Applied Math. Sci. 19, Springer-Verlag. Pimbley, G.H. Jr. (1969). Eigenfunction Branches of Nonlinear Operators and Their Bifurcation, Lecture Notes in Mathematics, 104, Springer-Verlag. Rabinowitz, P.H. (Ed.) (1977). Applications of Bifurcation Theory, Academic Press. Takens, F. (1973). Introduction to Global Analysis, Comm. Math. Inst., Rijksuniversiteit Utrecht. Thompson, J.M.T. & Hunt, G.W. (1975). Towards a unified bifurcation theory. J. Appl. Math. Phys. 26.
Uezu, T. and Aizawa, Y. (1982). Topological character of a periodic solution in three-dimensional ordinary differential equation system, Prog. Theor. Phys. 68, 1907-16. Uezu, T. (1983). Topology in dynamical systems, Phys. Letters 93A, 161-6. Vainberg, M.M. & Trenogin, V.A. (1974). Theory of Branching of Solutions of Nonlinear equations, Noordhoff International Pub., Leyden.
References by topics
587
Biological, ecological, and genetic systems Albrecht, F., Gatzke, H., Haddad, A. & Wax, N. (1974). The dynamics of two interacting populations, J. Math. Anal. and Appl. 46, 658-70. Aubin, J.P., Saari, D. & Sigmund, K. (Eds.) (1985). Dynamics of Macrosystems, Notes in Economic and Mathematical Systems, 257, Springer-Verlag. Babloyantz, A. (1986). Molecules, Dynamics and Life, Wiley. Blumenfeld, L.A. (1981). Problems of Biological Physics, Springer-Verlag. Capasso, V. (1981). Periodic solutions for a system of nonlinear differential equations modeling the evolution of oro-faecal diseases. In Nonlinear Differential Equations, eds. P. deMottoni and L. Salvadori, Academic Press. Cronin, J. (1981). Mathematics of Cell Electrophysiology, Lecture Notes in Applied Math., 63, Marcel Dekker Inc., NY. Dyson, F.J. (1982). A Model for the origin of life, J. Mol. Evol. 18, 344-50. Dyson, F.J. (1985). Origins of Life, Cambridge Univ. Press. Ebeling, W. & Perschel, M. (Eds.) (1985). Lotka-Volterra-Approach to Cooperation and Competition in Dynamic Systems, Math. Res. 23, Akademe-Verlag. Eigen, M. (1971). Self-organization of matter and the evolution of biological macro-molecules, Naturwissenschaften 33a, 465-522. Eigen, M. & Schuster, P. (1979) The Hypercycle: A Principle of Natural Self-organization, SpringerVerlag.
Eigen, M. (1985). Macromolecular Evolution, In Emerging Synthesis in Science ed. D. Pines, pp. 25-69. The Santa Fe Institute. Eigen, M., Gardiner, W.C. & Schuster, P. (1980). Hypercycles and compartments, J. Theor. Biol. 85, 407-411. Eigen, M., (1984). The Origin and Evolution of Life at the Molecular Level. In Aspects of Chemical Evolution, ed. G. Nicolis, Wiley, pp. 119-37. Eigen, M. (1986). The Physics of Molecular Evolution, Chemica Scripta 26B, 13-26. Enns, R.H., Jones, B.L., Miura, R.M. & Rangnekar, S.S. (Eds.) (1981). Nonlinear Phenomena in Physics and Biology, Plenum Press. Frauenthal, J.C. (1979). Introduction to Population Modeling, EDC/umap/55 Chapel St., Newton, Mass.
Freedman, H.I. (1980). Deterministic Mathematical Models in Population Ecology, Dekker. Gilpin, M.E. (1973). Do hares eat lynx? Amer. Nat. 107, 727-30. Goel, N.S., Maitra, S.C. & Montroll, E.W. (1971). On the Volterra and other non-linear models of interacting populations, Rev. Mod. Phys. 43, 231-76. Goldanskii, V.I., Anikin, S.A., Avetisov, V.A. & Kuz.min, V.V. (1987). On the decisive role of chiral purity in the organization of the biosphere and its possible distruction, Comments Mol. Cell. Biophys. 4, 79-88. Grasman, J. & Veling, E.J.M. (1979). Asymptotic Methods for the Volterra-Lotka Equation, Lecture Notes in Mathematics, 711, ed. F. Verhulst, Springer-Verlag. Hofbauer, J. (1981). On the occurance of limit cycles in the Volterra-Lotka equation, Nonlinear Analysis T M.A. 5, 1003-7. Hopf, F.A. & Hopf, F.W. (1986). Darwinian evolution in physics and biology: the dilemma of missing links. In Frontiers of Nonequilibrium Statistical Physics, eds. G.T. Moore & M.O. Scully, pp. 41119, Plenum Press. Hoppe, W., Lohmann, W., Markl, H. & Ziegler, H. (Eds.) (1983). Biophysics, Springer-Verlag.
588
References by topics
Hoppensteadt, F.C. (Ed.) (1979). Nonlinear Oscillations in Biology, Lectures in Applied Mathematics 17, AMS. Hoppensteadt, F.C., Mathematical Theories of Populations: Demographics, Genetics, and Epidemics, SIAM. Hoppensteadt, F.C. (Ed.) (1981). Mathematical Aspects of Physiology, Lectures in Applied Mathematics 19, AMS. Kolmogoroff, A. (1936). Sulla teoria di Volterra della lotta per 1'esistenza, Giornale dell'Istituto Italiano degli Attuari 7, 74-80. Linkens, D.A., Biological systems. In Modelling of Dynamical Systems, Vol. 1, ed. N. Nicholson, pp. 193-223, P. Peregrinus. Maynard Smith, J. (1986). The Problems of Biology, Oxford Univ. Press. Maynard Smith, J. (1982). Evolutionary Game Theory, Cambridge Univ. Press. Mayr, E. (1988). Toward A New Philosophy of Biology, Belknap Press of Harvard Univ. Press. Meinhardt, H. (1982). Models of Biological Pattern Formation, (Academic Press. Nicolis, G. (Ed.) (1985). Aspects of Chemical Evolution, Adv. Chem. Phys., 55, Wiley. Niesert, U., Harnasch, D. & Bresch, C. (1981). Origin of life between seylla and charybolis, J. Mol. Evol. 17, 348-353. Othmer, N.G. & Scriven, L.E. (1971). Instability and dynamic pattern in cellular networks, J. Theor. Biol. 32, 507-37. Peschel, M. & Mende, W. (1986). The Predator-Prey Model, Springer-Verlag. Prigogine, I. & Lefever, R. (1968). Symmetry Breaking Instabilities in Dissipative Systems, II, J. Chem. Phys. 48, 1695-700. Rashevsky, N. (1938). Mathematical Biophysics, Univ. of Chicago Press, Reprint, Dover, 1960. Schuster, P. & Sigmund, K. (1983). Replicaotr dynamics, J. Theor. Biol. 100, 533-8. Sigmund, K. (1983). The maximum principle for replicator equations, J. Theor. Biol. 100, 63-72. Smale, S. (1976). On the differential equations of species in competition, J. Math. Biol. 3, 5-7. Turing, A.M. (1952). The chemical bases of morphogenesis, Phil. Trans. Roy. Soc. (London) B237, 37-72. Wicken, J.S. (1987). Evolution, Thermodynamics, and Information, Oxford Univ. Press. Winfree, A.T. (1980). The Geometry of Biological Time, Biomathematics, 8, Springer-Verlag. Winfree, AT. (1987). When Time Breaks Down, Princeton Univ. Press. Zeeman, E.C. (1980). Population dynamics from game theory. In Global Theory of Dynamical Systems; Lecture Notes in Mathematics 814, eds. Z. Nitecki & G. Robinson, Springer-Verlag.
Catastrophe theory Arnold, V.I. (1984). Catastrophe Theory, Springer-Verlag. Berry, M.V. Waves and Thom's Theorem, Adv. Phys. 25, 1-26. Berry, M.V. (1979). Catastrophe and stochasticity in semiclassical quantum mechanics; also, Catastrophe and fractal regimes in random waves. In Structural Stability in Physics, SpringerVerlag.
Berry, M. (1981). Singularities in Waves and Rays. In Les Houches, Session XXXV, Physics of Defects, eds. R. Balian, M. Kleman & J.-P. Poirier pp. 456-541, North Holland. Chillingworth, D.R.J. (1976). Structural stability of mathematical models: the role of the catastrophe method. In Mathematical Modelling, eds. J.G. Andrews & R.R. McLone, pp. 231-58, Butterworths. De Alfaro, V. & Rasetti, M. (1978). Structural stability theory and phases transition models, Fortschritte der Physik 26, 143-73. Fowler, D.H. (1972). The Riemann-Hugoniot Catastrophe and Van der Waal's Equation, Towards a Theoretical Biology 4, 1-7, ed. C.H. Waddington.
References by topics
589
Gilmore, R. (1981). Catastrophe Theory for Scientists and Engineers, John Wiley & Sons. Guttinger, W. & Eikemeier. H. (Eds.) (1979). Structural Stability in Physics, Springer-Verlag. Holmes, P.J. & Rand, D.A. (1976). The bifurcations of Duffing's equation: an application of catastrophe theory, J. Sound and Vib. 44, 237-53. Kolata, G.B. (1977). Catastrophe theory: the emperor has no clothes, Science 196, 287. Lu, Y.C. (1976). Singularity Theory & an Introduction of Catastrophy Theory, Springer-Verlag. Paulos, J.A. (1981). Mathematics and Humor, University of Chicago Press. Poston, T. & Stewart, 1. (1978). Catastrophe Theory and Its Applications, Pitman. Poston, T. & Stewart, I.N. (1976). Taylor Expansions and Catastrophes, Pitman. Poston, T. & Woodcock, A.E.R. (1973). Zeeman's catastrophe machine, Proc. Camb. Phil. Soc. 74, 217. Russell, D.A., Hanson, J.D. & Ott, E. (1980). Dimension of strange attractors, Phys. Rev. Letters 45, 1175-8. Sinhah, D.K. (Ed.) (1981). Catastrophe Theory and Applications, John Wiley. Smale, S (1978). Review of Catastrophe Theory, by E.C. Zeeman, Bull. Amer. Math. Soc. 84, 1360-8. Stewart, 1. (1982). Catastrophe theory in physics, Rep. Prog. Phys. 45, 185-221. Sussman, H. J. (1977). A skeptic, Science 197, 821. Thom, R. (1975). Structural Stability and Morphogenesis, Benjamin. Thom, R. (1969). Topological models in biology, Topology 8, 313-35. Thom, R. (1977). Structural Stability. Catastrophe Theor .v and Applied Mathematics, SIAM Rev., April. Thompson, J.M.T. & Hunt, G.W. (1973). A. General theory of elastic stability, Wiley. Vendrik, M.C.M. (1979). A classification of phase diagrams by means of "elementary" catastrophe theory (ECT) I. Physica 99A, 103-44. Zahler, R.S. & Sussman, H.J. (1977). Claims and accomplishments of applied catastrophe theory (review article), Nature 269, 759-63. Zeeman, E.C. (1977). Catastrophe Theory (selected papers, 1972-1977), Addison-Wesley.
Cellular automata Aizawa, Y. & Nishikawa, 1. (1986). Toward the classification of the patterns generated by onedimensional cell automata. In Dynamical Systems and Nonlinear Oscillations, ed. G. Ikegami, pp. 210-22, World Scientific. Berlekamp, E.R., Conway, J.H. & Guy, R.K. (1982). Winning Ways, for Your Mathematical Plays, Vol. 2, Chap. 25, Academic Press. Bennett, C.H. (1986). On the nature and origin of Complexity in discrete, homogeneous, locallyinteracting systems, Foundation Physics. 16, 585-92. Burks, A.W. (Ed.) (1970). Essays on Cellular Automata, Univ. of Illinois Press, Urbana. Complex Systems 4, 545-839 (1987) (Lattice Gas Models). Demongeot, J., Goles, E. & Tchuente, M. (Eds.) (1985). Dynamical Systems and Cellular Automata, Academic Press. Dresden, M., A study of models in non-equilibrium statistical mechanics. In Studies in Statistical Mechanics, J. de Boer and G.E. Uhlenbeck, pp. 303-43, North Holland. Farmer, D., Toffoli, T. & Wolfram, S. (Eds.) (1984). Cellular automata, Physica D, 10, 1-246. Feynman, R.P. (1967). The Character of Physical Law, MIT Press, Cambridge, pp. 57-8. Feynman, R.P. (1982). Simulating physics with computers, Int. J. Theor. Phys. 21, 467-8. Fredkin, E. (1982). Digital Information Mechanics (talk given January, 1982, Moskito Island, BVI); Also see: The Atlantic Monthly, April, p. 29, 1988; Three Scientists and their Gods, by R. Wright, Times Books, 1988.
590
References by topics
Fredkin, E. & Toffoli, T. (1982). Conservative Logic, Int. J. Theor. Phys. 21, 219-53. Frisch, U, Hasslacher, B. & Pomeau, Y. (1986). Lattice-gas automata for the Navier-Stokes equation, Phys. Rev. Letters 56, 1505-8. Gardner, M. (1970, 1971). Scientific American 223(10) 120; 224(2) 112; 224(4) 117. Gardner, M. (1983). Wheels, Life, and Other Mathematical Amusements, Freeman. Jen, E. (1986). Global Properties of Cellular Automata, J. Stat. Phys. 43, 219-42; Strings and pattern-recognizing properties of one-dimensional cellular automata, ibid. 243-65. Kac, M. (1959). Probability and Related Topics in Physical Sciences, Interscience Pub. Kolmogorov, A.N. (1984). Automata and Life. In Cybernetics Today, ed. I.M. Makarov, pp. 20-41, MIR Pub., Moscow. Langton, C.G. (1984). Self-reproduction in cellular automata, Physica 10D, 135-44. Langton, C.G. (1986). Studying Artificial Life with Cellular Automata, Physica 22D, 120-49. Martin, 0., Oldzko, A.M. & Wolfram, S. (1984). Algebraic properties of cellular automata, Comm. Math. Phys. 93, 219-58. Margolus, N. (1984). Physic-like models of computation, Physica 10D, 81-95. Margolus, N., Toffoli, T. & Vichniac, G. (1986). Cellular-automata supercomputers for fluid-dynamics modeling, Phys. Rev. Letters 56, 1694-6. Oono, Y. and Kohmoto, M. (1985). Discrete model of chemical turbulence, Phys. Rev. Letters 55, 2947-51. Oono, Y. & Yeung, C. (1987). A cell dynamical system model of chemical turbulence, J. Stat. Phys. 48, 593-644. Orszag, S. & Yakhot, V. (1986). Reynolds number scaling of cellular-automation hydrodynamics, Phys. Rev. Letters 56, 1691-3. Packard, N.H. & Wolfram, S. (1985). Two-dimensional cellular automata, J. Stat. Phys. 38, 901-46. Park, J.K., Steglitz, K. & Thurston, W.D. (1986). Soliton-like behaviour in automata, Physica 19D, 423-32. Poundstone, W. (1985). The Recursive Universe, Contemporary Books. Toffoli, T. (1977). Cellular Automata Mechanics, Technical Report No. 208, Computer and Communication Sciences Department, Univ. of Michigan. Toffoli, T. (1982). Physics and Computation, Int. J. Theor. Phys. 21, 165-75. Toffoli, T. (1984). Cellular automata as an alternative to (rather than an approximation of) differential equations modeling physics, Physica 10D, 117-27. Toffoli, T. & Margolus, N. Cellular Automata: A New Environment for Modeling, MIT Press. Vichniac, G. Y. (1984). Simulating Physics with Cellular Automata, Physics 10D, 96-116. Vichniac, G.Y. (1984). Instability in discrete algorithms and exact reversibility, SIAM J. Alg. Disc. Meth. 5. Von Neumann, J. (1966). Theory of Self-Reproducing Automata, edited and completed by A.W. Burks, Univ. of Illinois Press, Urbana. Wolfram, S. (1982). Physics and computation, Int. J. Theor. Phys. 21, 165-75. Wolfram, S. (1983). Statistical Mechanics of Cellular Automata, Rev. Mod. Phys. 55, 601-44. Wolfram S. (1985). Universality and complexity in cellular automata, Physica 10D, 1-35. Wolfram, S. (1985). Twenty problems in the theory of cellular automata, Physica Scripta T9, 107-83. Wolfram, S. (1986). Theory and Applications of Cellular Automata, World Scientific Pub. Co.
Chaotic Dynamics (reviews; collections) Abraham, N.B., Gollub, J.P. & Swinney, H.L. (1984). Testing nonlinear dynamics, Physica 11D, 252-64. Bai-lin, H. (1984). Chaos, World Scientific, Singapore.
References by topics
591
Barnsley, M.F. & Demko, S.G. (Eds.) (1986). Chaotic Dynamics and Fractals, Academic Press. Berge, P., Pomeau, Y., & Vidal, C. (1984). Order Within Chaos; Towards a Deterministic Approach to Turbulence Wiley. Berry, M.V. (1978). Regular and Irregular Motion. In Topics in Nonlinear Dynamics, ed. S. Jorna,
AIP46. Bishop, A., Campbell, D. & Nicolaenko, B. (Eds.) (1981). Nonlinear Problems: Present and Future, North-Holland, Math. Studies 61. Campbell, D. & Rose, H. (Eds.) (1983). Order in chaos, Physica 7D, 1-362. Crutchfield, J.P., Farmer, J.D., Packard, N.H., & Shaw, R.S. (1986). Chaos, Sci. Amer. 255(6), 46-57. Cvitanovic, P. (Ed.) (1984). Universality in Chaos, Adam Hilger Ltd., Bristol. Devaney, R.L. (986). An Introduction to Chaotic Dynamical Systems, Benjamin. Eckmann, J.-P. & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617-56. Farmer, J.D., Ott, E. & Yorke, J.A. (1983). The dimension of chaotic attractors, Physica 7D, 153- 80. Ford, J. (1983). How random is a coin toss?, Physics Today, April, 40-47. Ford, J. (1989). What is chaos, that we should be mindful of it? In The New Physics, ed. P.C. Davis, Cambridge Univ. Press. Garrido, L. (Ed.) (1983). Dynamical Systems and Chaos, Lect. Notes Physics 179, Springer-Verlag. Helleman, R.H.G. (1980). Self-generated chaotic behaviour in nonlinear mechanics. In Fundamental Problems in Statistical Mechanics, ed. E.G.D. Cohen, pp. 165-233, North-Holland. Holden, A.V. (1986). Chaos, Princeton Univ. Press. Holmes, R.J. & Moon, F.C. (1983). Strange attractors and chaos in nonlinear mechanics, J. Appl. Mech. 50, 1021-32. Iooss, G., Helleman, R.H.G. & Stora, R. (Eds.) (1983). Chaotic Behaviour of Deterministic Systems, North-Holland. Moon, F.C. (1987). Chaotic Vibrations, Wiley. Ruelle, D. (1980). Strange attractors, Math. Intelligencer, 126-37. Schuster, H.G. (1984). Deterministic Chaos, Physik-Verlag, Weinheim. Shaw, R. (1981). Strange attractors, chaotic behaviour, and information flow, Z. Naturforsch 36a, 80112.
Thompson, J.M.T. & Stewart, H.B. (1986). Nonlinear Dynamics and Chaos, Wiley. Tomita, K. (1982). Chaotic response of nonlinear oscillators, Phys. Reports 86, 113-67. Zaslavsky, G.M. (1984). Chaos in Dynamic Systems, Harwood Academic Press.
Chaotic dynamics (theory) Balazs, N.L. & Voros, A. (1986). Chaos on the pseudosphere, Phys.Reports 143, No. 3 109-240. Barnsley, M.F. (1986). Making chaotic dynamical systems to order. In Chaotic Dynamics, eds. M.F. Barnsley & S.G. Demko, pp. 55-68. Academic Press. Barnsley, M.F. & Demko, S. (1985). Iterated function system and the global construction of fractals, Proc. R. Soc. Lond. A399, 243-75. Barrow, J.D. (1982). Chaotic Behaviour in General Relativity, Phys. Reports, 85, No. 1. Benzinger, H.E., Burns, S.A. & Palmore, J.I. (1987). Chaotic complex dynamics and Newton's method, Phys. Letters A119, 441-6. Cartwright, M.L. & Littlewood, J.E. (1945). On non-linear differential equations of the second order: 1. The equation y + k(l - y2) y + y = bAk cos (2t + a), k large, J. London Math. Soc. 20, 180-9. Dragt, A.J. & Finn, J.M. (1976). Insolubility of trapped particle motion in a magnetic dipole field, J. Geophys. Res. 81, 2327-40.
592
References by topics
Farmer, J.D., Ott, E. & Yorke, J.A. (1983). The dimension of chaotic attractors, Physica 7D, 153-80. Froehling, H., Crutchfield, J.P., Farmer, D., Packard, N.N. & Shaw, R. (1981). On determining the dimension of chaotic flows, Physica 3D, 605-17. Grassberger, P. & Procaccia, I. (1983). Measuring the strangeness of strange attractors, Physica 9D, 189-202.
Hammel, S.M., Yorke, J.A. & Grebogi, C. (1987). Do numerical orbits of chaotic dynamical processes represent true orbits? J. Complexity 3, 136-45. Holmes, P.J. (1980). Averaging and chaotic motions in forced oscillations, SIAM J. Appl. Math 38, 65-80 (correction in New Approaches to Nonlinear Problems in Dynamics, ed. P.J. Holmes, SIAM 1980.
Hirooka, H., Saito, N. & Ford, J. (1984). Chaos around hyperbolic fixed points, J. Phys. Soc. Japan 53, 895-8. Jensen, R.V. & Oberman, C.R. (1982). Statistical properties of dynamical systems, Physica 4D, 18396.
Levi, M. (1981). Qualitative analysis of the periodically forced relaxation oscillations, Memoirs Amer. Math. Soc. 32, 244, 1-145. Levinson, N. (1949). A second order differential equation with singular solutions, Ann. Math. 50, 127-53. Lichtenberg, A.J. & Lieberman, M.A. (1983). Regular and Stochastic Motion, Springer-Verlag. Lorenz, E.N. (1984). The local structure of a chaotic attractors in four dimensions, Physica 13D, 90104.
McDonald, S.W., Grebogi, C., Ott, E. & Yorke, J.A. (1985). Fractal basin boundaries, Physica D, 17D, 125-53. Melnikov, V.K. (1963). On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12(1), 1-57. Mistriotis, A.D. & Jackson, E.A. (1987). Transition to stochasticity for periodically perturbed area preserving system, Physica Scripta 35, 97-104. Motosov, A.D. (1976). A complete qualitative investigation of Duffing's equation, Differential Eqs. 12, 164-74.
Newhouse, S.E. (1979). The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S. 50, 101-51. Oono, Y. & Osikawa, M. (1980). Chaos in nonlinear difference equations. I., Prog. Theor. Phys. 64, 54-67. Osikawa, M. & Oono, Y. (1981). Chaos in C°-endomorphism of interval, Res. Inst. Math. Sci., Kyoto Univ., 17, 165-77. Osipov, A.V. (1976). Properties of solutions of Levinson's equations, Diff. Eq., 1401-7. Saari, D.G. (1984). The ultimate of chaos resulting from weighted voting systems, Adv. Appl. Math. 5, 286-308. Takens, F. (1985). On the numerical determination of the dimensions of an attractor. In Dynamical Systems and Bifurcations, eds. Lecture Notes in Mathematics 1125, eds. B.L.J. Braaksma, H. Broer & F. Takens, pp. 99-106, Springer-Verlag.
Chaotic dynamics, experimental (also see turbulence) Brandstater, A., Swift, J., Swinney, H.L., Wolf, A., Farmer, J.D., Jen, E. & Crutchfield, P.J. (1983). Low-dimensional chaos in a hydrodynamic system, Phys. Rev. Letters 51, 1442-5. Creveling, H.F., dePaz, J.F., Baladi, J.Y. & Schoenhals, R.J. (1975). Stability characteristics of a single-phase free convection loop, J. Fluid Mech. 65, 67.
References by topics
593
Haken, H. (Ed.) (1981). Chaos and Order in Nature, Springer-Verlag. Hunt, E.R. (1982). Comment on a driven nonlinear oscillator, Phys. Rev. Letters 49, 1054 (1982); Testa, J., Perez, J., & Jeffries, C. Respond, ibid., 1055. Jeffries, C.D. (1985). Chaotic dynamics of instabilities in solids, Physica Scripta T9, 11-26. Jensen, M.H., Kadanoff, L.P., Libchaber, A., Procaccia, I. & Stavans, J. (1985). Global universality at the onset of chaos: results of a forced Rayleigh-Benard experiment, Phys. Rev. Letters 55, 2798801.
Kalia, R.K. & Vashista, P. (Eds.) (1982). Melting, Localization, and Chaos, North-Holland. Lauterborn, W. & Crames, E. (1981). Subharmonic route to chaos observed in acoustics, Phys. Rev. Letters 47, 1445-8. Linsay, P.S. (1981). Period doubling and chaotic behaviour in a driven anharmonic oscillator, Phys. Rev. Letters 47, 1349-52. Moon, F.C. (1980). Experimental models for strange attractors vibrations in elastic systems. In New Approach to Nonlinear Problems in Dynamics, ed. P.J. Holmes, pp. 487--95, SIAM. Nicolis, C. & Nicolis, G. (1984). Is there a climatic attractor?, Nature 311, 529-32. Rollins, R.W. & Hunt, E.R. (1982). Exactly solvable model of a physical system exhibiting universal chaotic behavior, Phys. Rev. Letters 49, 1295-8. Roux, J.C., Simoyi, R.H. & Swinney, H.L. (1983). Observation of a strange attractor, Physica 8D, 257-66. Shaw, R. (1984). The Dripping Faucet as a Model Chaotic system, Aerial Press. Testa, J., Perez, J. & Jeffries, C. (1982). Evidence of universal chaotic behaviour of a driven nonlinear oscillator, Phys. Rev. Letters 48, 714-7. van der Pol, B. & van der Mark, J. (1927). Frequency demultiplication, Nature 120, 363-4. Chemical oscillations, turbulence Belousov, B.P. (1959). Sb. Ref. Radiats. Med., 1958, Medgiz, Moscow 145. Epstein, I.R. (1984). New chemical oscillators, In Non-Equilibrium Dynamics in Chemical Systems, eds. C. Vidal & A. Pacault, pp. 24-34, Springer-Verlag. Field, R.J. & Noyes, R.M. (1974). Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys. 60, 1877-84. Glandsdorff, P. & Prigogine, I. (1971). Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley.
Hastings, S.P. & Murray, J.D. (1975). The existence of oscillatory solutions in the field-noyes model of the Belousov-Zhabotinskii reactions, J. Appl. Math. 28, 678-88. Kuramoto, Y. (1983). Chemical Oscillations, Waves and Turbulence, Springer-Verlag. Lefever, R. (1968). Stabilite des structures dissipatives, Bull. Classe Sci., Acad. Roy. Belg. 54, 712-19. Maselko, J. & Swinney, H. L. (1985). A complex transition sequence in the Belousov-Zhabotinskii reaction, Physica Scripta T9, 35-9. Nicolis, G. (1971). Stability and dissipative structures in open systems far from equilibrium, Adv. Chem. Phys. 19, 209-324. Noyes, R.M. (1976). Oscillations in chemical systems. XIV. Corrected stoichiometry of the oregonator, J. Chem. Phys. 65, 848-9. Oono, Y. & Yeung, C. (1987). A. Cell dynamical system model of chemical turbulence, J. Stat. Phys. 48, 593-644. Rossler, O.E. (1972). A. Principle for chemical multivibration, J. Theor. Biol. 36, 413-17. Simayi, R.H., Wolf, A., & Swinney, H.L. (1982). One-dimensional dynamics in a multi-component chemical reaction, Phys. Rev. Letters 49, 245-8.
594
References by topics
Troy, S. (1977). A threshold phenomenon in the Field-Noyes model of the Belousov-Zhabotinskii reactions, J. Math. Anal. Appl. 58, 233-48. Turner, J.S., Roux, J.C., McCormick, W.D. & Swinney, H.L. (1981). Alternating periodic and chaotic regimes in chemical reaction-experiment and theory, Phys. Letters 85A, 9-12. Tyson, J.J. (1979). Oscillations, bistability, and echo waves in models of the Belousov-Zhabotinskii reaction, Ann. NY Acad. Sci. 316, 279-95, eds. O. Gurel & O.E. Rossler. Vidal, C. & Pacault, A. (Eds.) (1984). Nonequilibrium dynamics in chemical systems, Springer-Verlag. Walker, J. (1978). Chemical systems that oscillate between one color and another, Scientific American 239, 152-60. Zhabotinskii, A.M. (1967). Oscillatory processes in biological and chemical systems, Science Publ., Moscow.
Chemical systems (also see, biological systems) Ebert, K.H., Deufihard, P. & Jager, W. (1981). Modelling of Chemical Reaction Systems, SpringerVerlag.
Vidal, C. & Pacault, A. (1982). Spatial chemical structures, chemical waves. A reivew. In Evolution of Order and Chaos, eds. H. Haken, Springer-Verlag.
Circle map Arnold, V.I. (1965). Small denominators. I. Mappings of the circumference onto itself, Transi. Amer. Math. Soc. 2nd Series 46, 213-84. Aronson, D.G., Chory, M.A., Hall, G.R. & McGehee, R.P. (1982). Bifurcations from an invariant circle for two-parameter families of maps of the plane, Commun. Math. Phys. 83, 303-54. Belair, J. & Glass, L. (1985). Universality and self-similarity in the bifurcations of the circle maps, Physica 16D (No. 2), 143-54. Bohr, T., Bak, R. & Jensen, M.H. (1984). Transition to chaos by interaction of resonances in dissipative systems, Phys. Rev. A30, 1970-81. Fein, A.P., Heutmaker, M.S. & Gollub, J.P. (1985). Scaling and transition from quasiperiodicity to chaos in ahydrodynamic system, Physica Scripta T9, 79-84.
Classical mechanics Abraham, R. & Marsden, J.E. (1978). Foundations of Mechanics, second edition, Benjamin/Cummings - contains: A.N. Kolmogorov's 1954 address to the International Congress of Mathematicians: The General Theory of Dynamical Systems and Classical Mechanics'. Arnold, V.I. & Avez, A. (1968). Ergodic Problem of Classical Mechanics, Benjamin. Arnold, V.I. (1978). Mathematical Methods of Classical Mechanics, Springer-Verlag. Arnold, V.I., Kozlov, V.V. & Neishtadt, A.I. (1988). Mathematical aspects of classical and celestial mechanics. In Dynamical Systems, III, ed. V.I. Arnold Springer-Verlag. Bartlett, J.H. (1975). Classical and Modern Mechanics, Univ. of Alabama Press. Benettin, G., Galgani, L. & Giorgilli, A. (1987). Realization of holonomic constraints and freezing of high frequency degrees of freedom in light of classical perturbation theory, Commun. Math. Phys. 113, 87-103. Birkhoff, G.D. (1917). Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18, 199-300. Birkhoff, G.D. (1927). Dynamical Systems, Amer. Math. Soc. Contopoulous, G. (1963). A classification of the integral of motion, Ap. J. 138, 1297-305.
References by topics
595
Fermi, E. (1923). Beweis dass ein mechanisches Normalsystem in Allgemeinen quasi-ergodisch ist, Phys. Zeits. 24, 261-265. Hagihara, Y. (1974). Celestial Mechanics, Vols I-V, Japan Soc. Promotion of Science, Tokyo Press. Hoover, W.G. (1986). Molecular Dynamics, Lecture Notes in Physics 258, Springer-Verlag. Moser, J. (1968). Lectures on Hamiltonian systems, Amer. Math. Soc. Memoirs, No. 81, 1-60, Amer. Math. Soc. Nose, S. (1984). A molecular dynamics method for simulations in the canonical ensemble, Molecular Phys. 52, 255-68. Pars, L.A. (1968). A Treatise on Analytical Dynamics, J. Wiley. Petrosky, P. & Prigogine, I. (1982). The KAM invariant and Poincare's theorem. In Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics, ed. V. Szebehely pp. 185-99, D. Reidel.
Poincare, H. (1892, 1893, 1894). Les Methodes Nouvelles de la Mecanique Celeste, Vols. 1-3, Gauthier-Villars, Paris, Reprinted by Dover, 1957 (Translation: New Methods of Celestial Mechanics, NASA, 1967. Siegel, C.L. & Moser, J.K. (1971). Lectures on Celestial Mechanics, Springer-Verlag. Sudarshan, E.C.G. & Mukunda, N. (1974). Classical Dynamics: A Modern Perspective, Wiley. Szebehely, V. (Ed.) (1982). Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics, D. Reidel. Szebehely, V. (1984). Review of concepts of stability, Celestial Mech. 34, 49-64. Szebehely, V. (1968). New Nondeterministic Celestial Mechanics. In Space Dynamics and Celestial Mechanics, eds. K.B. Bhatnagar, Reidel Pub. Co. Thirring, W. (1978). Classical Dynamical Systems, Springer-Verlag. Whittaker, E.T. (1944). Analytical Dynamics of Particles and Rigid Bodies, Dover. Whiteman, K.J. (1977). Invariants and stability in classical mechanics, Rep. Prog. Phys. 40, 1033-69. Wintner, A. (1947). The Analytic Foundations of Celestial Mechanics, Princeton Univ. Press. Computer computations Pavelle, R., Rothstein, M. & Fitch, J. (1981). Computer algebra, Sci. Amer. 245, No. 6, 136. Press, W.H., Flannery, B.F., Teukolsky, S.A. & Vetterling, W.T. (1986). Numerical Recipes, Cambridge University Press. Stoutemyer, D.R. & Yun, D.Y.Y. (1980). Symbolic mathematical computation In Encyclopedia of Computer Science and Technology 15: Supplement, eds. J. Belzer, A.G. Holzman and A. Kent, Marcel Dekker. Symbolic and Algebraic Computation; Eurosam '79, an International Symposium on Symbolic and Algebraic Manipulation. Lecture Notes in Computer Science 72. Springer-Verlag, 1979. Wang, P.S. (Ed.) (1981). Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation. Association for Computer Machinery.
Computer dynamics Benettin, G.A., Casartelli, G.M. Galgani, L., Giorgilli, A. & Streleyn, J.M. (1978). On the reliability of numerical studies of stochasticity. Nuovo Cimento 44B, 183-5. Cutland, N. (1980). Computability, Cambridge Univ. Press. Deutsch, D. (1985). Quantum theory, the Church-Turing principle and universal quantum computer, Proc. R. Soc. Lond. A400, 97-117. Feynman, R.P. (1982). Simulating physics with computers, Int. J. Theor. Phys. 21, 467-88. Fredkin, E. & Toffoli, T. (1983). Conservative logic, Int. J. Theor. Phys. 21, 219-53.
596
References by topics
Lax, P.D. (1986). Mathematics and computing, J. Stat. Phys. 43, 749-56. Makarov, I.M. (Ed.) (1984). Cybernetics Today, MIR Publishers, Moscow. McCauley, J.L. Jr. & Palmore, J.I. (1986). Computable Chaotic Orbits, Phys. Letters A 15, 433 Toffoli, T. (1982). Physics and computation, Int. J. Theor. Phys. 21, 165-75.
Computer studies - nonchaotic dynamics Flaherty, J.R. & Hoppenstendt, F.C. (1978). Frequency entrainment or a forced Van der Pol oscillator, Studies Applied. Math 58, 5-15. Greenspan, D. (1980). Arithmetic Applied Mathematics, Pergamon Press. Mistriotis, A.D. & Jackson, E.A. (1987). Transition to Stochasticity for a Periodically Perturbed Area Preserving System, Physics Scripta 35, 97-104. Stormer, C. (1955). The Polar Aurora, Oxford, Clarendon Press. Zabusky, N.J. (1981). Computational Synergetics and Mathematical Innovation, J. Comput. Phys. 43, 195-249.
Computer studies of chaos, integrability, irreversibility, etc. Abraham, R. & Simo, C. (1985). Bifurcations and Chaos in Forced van der Pol Systems. In Singularities and Dynamical Systems, ed. S.N. Pnevmatikos, pp. 313-44, Elsevier Sci. Pub. Channell, P.J. (1978). An illustrative example of the Chirikov criterion for stochastic instability. In Topics in Nonlinear Dynamics, AIP Conf No. 46, ed. S. Jorna, pp. 248-59, AIP. Chirikov, B.V. (1979). A universal instability of many-dimensional oscillator systems Phys Reports 52, 265-376. Chirikov, By., Keil, E. & Seisler, A.M. (1971). Stochasticity in many-dimensional Nonlinear Oscillator Systems, J. Statistical Phys. 3, 307-21. Chirikov, By., Izrailev, F.M. & Tryursky, V.A. (1973). Numerical experiments on the statistical behavior of dynamical systems with a few degrees of freedom, Comp. Phys. Comm. 5, 11-16. Chirikov, B.V., Ford, J. & Vivaldi, F. (1980). Some numerical studies of Arnold diffusion in a simple model. In Nonlinear Dynamics and the Beam-Beam Interaction, p. 323-40, AIP. Ford, J. (1977). Relevance of Computer Experiments to the Pure Mathematics of Integrable and Ergodic Dynamical Systems. In Dynamical Systems I-Warsaw, pp. 75-92, Societe Math. de France. Ford, J. (1973). The transition from analytic dynamics to statistical mechanics Adv. Chem. Phys. 24, 155-85.
Ford, J. (1975). The statistical mechanics of classical analytic dynamics. In Fundamental Problems in Statistical Mechanics, ed. E. Cohen, pp. 215-55, North-Holland. Heileman, R.H.G. (1977). On the Iterative Solution of a Stochastic mapping. In Statistical Methods, Theory and Applications, ed. U. Landman, Plenum. Henon, M. & Heiles, C. (1964). The applicability of the third integral of motion: some numerical experiments, Astrophys. J. 69, 73-9. Holmes, P. & Marsden, J.E. (1979). Qualitative techniques for bifurcation analysis of complex systems, Ann. NYAcad. Sci., 316, 608-22. Kai, T. & Tomita, K. (1979). Stroboscopic phase protrait of a forced nonlinear oscillator, Prog. Theor. Phys. 61, 54-73. Lunsford, G.H. & Ford, J. (1972). On the stability of periodic orbits for nonlinear oscillator systems in regions exhibiting stochastic behavior, J. Math. Phys. 13, 700-5. Mo, K.C. (1972). Theoretical prediction for the onset of widespread instability in conservative nonlinear oscillator systems, Physica 57, 445-54.
References by topics
597
Morozov, A.D. (1976). A. Complete qualitative investigation of Duffing's equation, Diff. Equations 12, 164-74. Raty, R., Isomaki, H.M. & von Boehm, J. (1984). Chaotic motion of a classical anharmonic oscillator, Acta Polytechnica Scandinavica, Mech. Eng. Ser. 85, 1-30. Rossler, O.E. (1979). Chaos. In Structural Stability in Physics, eds. W. Guttinger & H.E. Kemeier, pp. 290-319, Springer-Verlag. Simb, C. (1979). On the Henon-Pomeau Attractor, J. Stat. Phys. 21, 465-94. Ueda, Y. (1979). Randomly Transitional Phenomena in Systems Governed by Duffing's Equation, J. Stat. Phys. 20, 181-96.
Coupled maps (patterns) Gu, Y., Tung, M., Yuan, J.-M., Feng, D.H. & Narducci, L.M. (1984). Crises and hysteresis in coupled logistic maps, Phys. Rev. Letters 52, 701-4. Hogg, T. & Huberman, B.A. (1984). Generic behavior of coupled oscillators, Phys. Rev. A29, 275-81. Kaneko, K. (1983). Transition from torus to chaos accompanied by frequency lockings with symmetry breaking, Prog. Theor. Phys. 69, 1427-42. Kaneko, K. (1984). Period-doubling of kink-antikink patterns, Prog. Theor. Phys. 72, 480-6. Keeler, J.D. & Farmer, J.D. (1986). Robust space-time intermittency and 1/f noise, Physica D23, 413-35. Waller, I. & Kapral, R. (1984). Spatial and temporal structure in systems of coupled nonlinear oscillators, Phys. Rev. A30, 2047-55. Yamada, T. & Fujisaka, H. (1983). Stability theory of synchronized motion in couple-oscillator systems II, Prog. Theor. Phys. 70, 1240-8.
Difference equations (maps) Aubry, S. (1983). The twist map, the extended Frenkel-Kontrova model and the devil's staircase, Physica 7D, 240-58. Beau, W., Metzler, W. & Ueberla, A. (1987). The Route to Chaos of Two Coupled Logistic Maps, Univ. of Kassel. Bowen, R. (1978). On Axiom A Diffeomorphisms, CBMS Regional Conference Series in Mathematics No. 35. Chang, S.J. & McCown, J. (1985). Universality behaviors and fractal dimensions associated with Mfurcations, Phys. Rev. A31, 3791-801. Chernikov, A.A., Sagdeev, R.Z., Usikov, D.A., Zakharov, M.Yu. & Zaslavsky, G.M. (1987). Minimal chaos and stochastic webs, Nature 326 (9), 559-63. Collet, P. & Eckmann, J-P. (1980) Iterated Maps on the Interval as Dynamic Systems, Birkhauser. Coven, E.M., Kan, I. & Yorke, J.A. (1986). Pseudo-orbit shadowing in the family of tent maps. Curry, J.H. (1979). On the Henon transformation, Commun. Math. Phys. 68, 129-40. Feigenbaum, M.J. (1978). Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19, 25-52. Feigenbaum, M.J. (1979). The universal metric properties of nonlinear transformations, J. Stat. Phys. 21, 186-223. Guckenheimer, J. (1977). On the bifurcation of maps of the internal, Inventiones Math. 39, 165-78. Guckenheimer, J., Oster, G. & Ipaktchi, A. (1977). The dynamics of density dependent population models, J. Math. Biol. 4, 101-47. Guckenheimer, J. (1979). The bifurcation of quadratic functions. In Bifurcation Theory, NY Acad. Sciences 316, pp. 78-85.
598
References by topics
Guckenheimer, J. (1979). Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys. 70, 133-60. Henon, M. (1976). A two dimensional mapping with a strange attractor, Comm. Math. Phys. 50, 6977.
Henon, M. (1969). Numerical study of quadratic area-preserving mappings, Quart. Appl. Math. 27, 291-312. Henon, M. (1982). On the Numerical Computation of Poincare Maps, Physica 5D, 412-14. Holmes, P.J. & Whitley, D.C. (1984). Bifurcations of one- and two-dimensional maps, Phil. Trans. R. Soc. Lond. A311, 43-102; Erratum, ibid., 601. Hsu, C.S. (1987). Cell-To-Cell Mapping, Springer-Verlag. Ichikawa, Y.H., Kamimura, T. & Karney, C.F.F. (1983). Stochastic motion of particles in tandom mirror devices, Physica 6D, 233-40. Kaplan, J.L. & Yorke, J.A. (1979). Chaotic behavior of multidimensional difference equations. In Functional Differential Equations, eds. H.O. Peitgen & H.D.Waither, Lecture Notes in Mathematics 730, pp. 204-27, Springer-Verlag. Karney, C.F.F. (1983). Long-time correlations in the stochastic regime, Physica 8D, 360-80. Kuznetsov, S.P. (1986). Universality and scaling in the behavior of coupled Feigenbaum systems, Radiophys. Quantum Electronics 28, 681-95. Lasota, A. (1977). Ergodic Problems in Biology, Dynamical Systems 11-Warsaw, pp. 239-50, Societe Mathematique de Franc. Li, T.Y. & Yorke, J.A. (1975). Period three implies chaos, Amer, Math. Monthly 82, 983-92. MacKay, R.S. (1986). Transition to chaos for area-preserving maps. In Nonlinear Dynamics Aspects of Particle Accelerators, eds. J.M. Jowett, M. Month & S. Turner, pp. 390-454, Springer. Marotto, F.R. (1978). Snap-back repellers imply chaos in R", J. Math. Anal. Appl. 63, 199-223. May, R.M. (1976). Simple mathematical models with very complicated dynamics, Nature 261, 459-67 (review article).
Metropolis, N., Stein, M. & Stein, P. (1973). On finite limit sets for transformations on the unit interval, J. Combinatorial Theory 15, 25-44. Osikawa, M. & Oono, Y., (1981). Chaos in C°-endomorphism of interval. Inst. Math. Studies, Kyoto Univ., 17, 165-177 (1981).
Preston, C. (1983). Iterates of Maps on an Interval, Springer-Verlag. Rannou, F. (1974). Numerical study of discrete plane area-preserving map, Astron. Astrophys. 31, 289-301. Ryutov, D.D. & Stupakov, G.V. (1978). Diffusion of resonance particles in ambipolar plasma traps, Sov. Phys. Dokl. 23, 412-14. Schult, R.L., Creamer, D.B., Henyey, F.S. & Wright, J.A. (1987). Symmetric and nonsymmetric coupled logistic maps, Phys. Rev. A35, 3115-18. Sharkovsky, A.N. (1964). Coexistence of the cycles of a continuous mapping of the line into itself, UKr. Mat. Zh. 16(1), 61-71. Singer, D. (1978). Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35, 2607.
Smale, S. & Williams, R. (1976). The quantitative analysis of a difference equation of population growth, J. Math. Biol. 3, 1-4. Stefan, O. (1977). A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Commun. Math. Phys. 54, 237-48. Straffin, P.D. Jr. (1978). Periodic points of continuous maps, Math. Mag. 51, 99-105.
References by topics
599
Dynamic systems Abraham, R.H. & Shaw, C.D. (1982). Dynamics, The Geometry of behavior; Part One: Periodic Behavior; (1983) Part Two: Chaotic Behavior (1984); Part Three: Global Behavior, Aerial Press. Alekseev, V.M. & Yakobson, M.V. (1981). Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep. 75, 287-325. Benettin, G., Galgani, L. & Giorgilla, A. (1985). A proof of Nekhoroshev's theorem for the stability times in nearly-integrable Hamiltonian systems, Celest. Mech. 37, 1-25. Bowen, R. (1975). Methods of symbolic dynamics, Lecture Notes in Mathematics 470, Springer-Verlag. Broomhead, D.S. & King, G.P. (1986). Extracting qualitative dynamics from experimental data, Physica 20D, 217-36. Diner, S., Fargue, D. & Lochak, G. (Eds.) (1986). Dynamical Systems, World Scientific. Garrido, L. (Ed.) (1982). Dynamic Systems and Chaos, Lecture Notes in Physics. 179, Springer-Verlag. Golubev, V.V. (1953). Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, State Publishing House Moscow. Guckenheimer, J. & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag. Guckenheimer, J., Moser, J. & Newhouse, S.E. (1980). Dynamical Systems, Birkhaser. Helleman, R.H.G. (Ed.) (1980). Nonlinear dynamics, Ann. NYAcad. Sciences 257. Hirsch, M.W. & Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra, Academic Press. Hirsch, M.W. (1984). The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. 11, 1-64. Irwin, M.C. (1980). Smooth Dynamical Systems, Academic Press. Jorna, S. (Ed.) (1978). Topics in Nonlinear Dynamics, AIP Conference Proceedings, 46, AIP. Kolmogorov, A.N. (1954). The General Theory of Dynamical Systems and Classical Mechanics (English translation in Foundations of Mechanics by R. Abraham & J.E. Marsden). Lichtenberg, A.J. & Lieberman, M.A. (1983). Regular and Stochastic Motion, Springer-Verlag. Markley, N.G., Martin, J.C. & Perrizo, W. (1978). The Structure of Attractors in Dynamical Systems, Springer-Verlag. Moser, J. (1973). Stable and Random Motions in Dynamical Systems, Ann. Math. Studies, 77, Princeton Univ. Press. Nekhoroshev, N.N. (1971). Behavior of Hamiltonian systems close to integrable, Funct. Anal. Appl. 5, 338-9. Nekhoroshev, N.N. (1977). An exponential estimate of the time of stability of near-integrable Hamiltonian systems, Russ. Math. Survey 32, 1-65. Nicolis, J.S. (1986). Dynamics of hierarchical systems, Springer-Verlag. Nitecki, Z. (1971). Differential Dynamics, MIT Press. Packard, N.H., Crutchfield, J.P., Farmer, J.D. & Shaw, R.S. (1980). Geometry from a time series, Phys. Rev. Letters 45, 712-16. Peixoto, M.V. (Ed.) (1973). Dynamical Systems, Academic Press. Percival, I. (1982). Introduction to Dynamics, Cambridge Univ. Press. Pnevmatikos, S.N. (Ed.) (1985). Singularities and Dynamical Systems, Math. Studies, 103, NorthHolland. Ruelle, D. (1983). Small random perturbations and the definition of attractors. In Geometric Dynamics, Lecture Notes in Mathematics, eds., A. Doid & B. Eckmann, pp. 663-76, SpringerVerlag.
600
References by topics
Sinai, Ya. G. (1977). Introduction to Ergodic Theory, Princeton Univ. Press. Smale, S. (1963). Diffeomorphisms with many periodic points. In Differential and Combinatorial Topology, ed. S.S. Cairns, Princeton Univ. Press, pp. 63-80. Smale, S. (1967). Differentiable dynamical System, Bull. Amer. Math. Soc. 73, 747-817. Smale, S. (1981). Essays on Dynamical Systems, Economic Processes, and Related Topics, SpringerVerlag.
Szlenk, W. (1984). An Introduction to the Theory of Smooth Dynamical Systems, Wiley. Sverdlove, R. (1977). Inverse problems for dynamical systems in the plane. In Dynamical Systems, eds. A.R. Bednarek & L. Cesi, pp. 499-502, Academic Press. Three papers on dynamical systems: A.G. Kusnirenko, Problems in the general theory of dynamical systems on a manifold (1-42); A. B. Katok, Dynamical systems with hyperbolic structure (43-96); V.M. Alekseev, Quasirandom oscillations and qualitative questions in celestial mechanics (97-169); AMS Translations 116, 1981.
Dynamic entropies and information production (also see, chaotic dynamics) Adler, R.L., Konheim, A.G. & McAndrew, (1965). Topological entropy, Trans. Amer. Math. Soc. 114, 309-19. Block, L., Guckenheimer, J., Misiurewicz, M. & Young, L.S. (1980). Periodic points and topological entropy of one dimensional maps. In Global Theory of Dynamical Systems, eds. A. Dold & B. Eckmann, Lecture Notes in Mathematics, 819, Springer-Verlag. Dinaburg, E.I. (1971). On the relations among various entropy characteristics of dynamical systems, Math. USSR Izvestija 5, 337-78. Kolmogorov, A.N. (1958). A new invariant of transitive dynamical systems. Dokl. Akad. Nauk SSSR 119, 861-5. Ledrappier, F. & Young, L.-S. (1985). The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula, Ann. Math. 122, 509-39; Part II: Relations between entropy, exponents, and dimension, ibid, 540-74. Misiurewicz, M. (1976). Topological conditional entropy, Studio Mathematica 55, 175-200. Misiurewicz, M. and Szlenk, W. (1980). Entropy of piecewise monotone mappings, Studia Mathematica 57, 45-63. Oono, Y. (1978). A. Heuristic approach to Kolmogorov entropy as a disorder parameter, Prog. Theor. Phys. 60, 1944-6. Rohlin, V.A. (1967). Lectures on the entropy theory of measure-preserving transformations, Russian Math Survey 22 (5), 1-52. Shaw, R. (1981). Strange attractors, chaotic behavior, and information flow, Z. Naturforsch. 36a, 80112.
Sinai, Ja. G. (1959). On the concept of entropy of a dynamical system, Dokl. Akad. Nauk. SSSR 124, 768-76 (Russian).
Dynamo theory: Earth's magnetic field Bullard, E. (1978). The disk dynamo. In Topics in Nonlinear Dynamics, AIP Conference Proceedings, ed. S. Jorna, p. 373-89, AT P. Busse, F.H. (1978). Magnetohydrodynamics of the Earth's dynamo, Ann. Rev. Fluid Mech. 10, 43562.
Carrigan, C.R. & Gubbins, D. (1979). The source of the Earth's magnetic field, Sci. Amer. 240 (2), 118.
References by topics
601
Cook, A.E. & Roberts, P.H. (1970). The Rikitake two-disc dynamo system, Proc. Camb. Phil. Soc. 68, 547-69. Cox, A. (1982). Magnetostratigraphic time scale. In Geologic Time Scale, eds. W.B. Harland et al. Cambridge Univ. Press. p. 63. Oziewonski, A.M. (Ed.) (1980). Physics of the Earths Interior, Proc. International School of Physics 'Enrico Ferni' North-Holland; articles by F.H. Busse (Motions in the Earth's core and the origin of geomagnesium pp. 493-507) and J.A. Jacobs (The evolution of the Earth's core and Geodynamics, pp. 508-30). Gubbins, D. (1974). Theories of the geomagnetic and solar dynamos, Rev. Geophys. Sp. Phys. 12, 137-54.
Hibberd, F.H. (1979). The origin of the Earth's magnetic field, Proc. R. Soc. Lond. A369, 31-45. Hoffman, K.A. (1988). Ancient magnetic reversals: clues to the geodynamics, Sci. Amer. 258(5), 76-83. Inglis, D.R. (1981). Dynamo theory of the Earth's varying magnetic field, Rev. Mod. Phys. 53, 481-96. Jacobs, A.J. (1984). Reversals of the Earth's Magnetic Field, Heyden & Son, Philadelphia. Macdonald, K.C. & Luyendyk, B.P. (1981). The Crest of the East Pacific Rise, Sci. Amer. 244(5), 10016.
Merrill, R.T. & McElhinny, M.W. (1983). The Earth's Magnetic Field, Its History, Origin and Planetary Perspective, International Geophysics Series, 32, Academic Press. Moffat, N.K. (1978). Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Univ. Press.
Moffat, H.K. (1977). Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory. In Fluid Dynamics, eds. R. Balian and J.L. Peube, p. 151-233, Gordon & Breach. Parker, E.N. (1975). The dynamo mechanism for the generation of large-scale magnetic fields, NY Acad. Sci. 252, 141-55. Robbins, K.A. (1977). A new approach to subcritical instability and turbulent transitions in a simple dynamo, Math. Proc. Camb. Phil. Soc. 82, 309-25. Roberts, P.H. (1971). Dynamo theory. In Lectures in Applied Mathematics, ed. W.H. Reid, pp. 129206 American Mathematical Society. Strangway, D.W. (1970). History of the Earth's magnetic field, McGraw-Hill.
Explosive dynamics Muncaster, R.G. & Zinnes, D.A. (1984). Phase portraits, singularities, and dynamics of hostility in international relations, SIAM J. Applied Math. Rudakov, L.I. & Tsytovich, V.N. (1978). Strong Langmuir turbulence, Phys. Reports 40, 1-73. Thornhill, S.G. & ter Haar, D. (1978). Langmuir turbulence and modulational instability, Phys. Reports 43, 43. Weiland, J. & Wilhelmsson, H. (1977). Coherent Non-linear Interaction of Waves in Plasmas, Pergamon Press. Zakharov, V.E. (1972). Collapse of Langmuir waves, Soviet Physics JETP 35, 908-14.
Fractals Berry, M.V. (1979). Diffractals, J. Phys. A12, 781. Colvin, J.T. & Stapleton, H.J. (1985). Fractal and spectral dimensions of biopolymer chains: solvent studies of electron spin relaxation rates in myoglobin azide, J. Chem. Phys. 82, 4699-706. Falconer, K.J. (1985). The Geometry of Fractal Sets, Cambridge Univ. Press. Frauenfelder, H., Petsko, G.A. & Tsenoglu, (1979). Temperature-dependent X-ray diffraction as a probe of protein structural dynamics, Nature (London) 280, 558-63.
602
References by topics
Grebogi, C., Ott, E. & Yorke, J.A. (1983). Fractal basin boundaries, long-lived chaotic and unstableunstable pair bifurcations, Phys. Rev. Letters 50, 935-8. Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procassia, I. & Shraiman, B.I. (1986). Fractal measures and their singularities: the characterization of strange sets, Phys. Rev. A33, 1141-51. Jackson, E.A. (1985). The Lorenz System: II. The homoclinic convolution of the stable manifolds, Physica Scripta 32, 476-81. Kaplan, J.L., Mallet-Paret, J. & Yorke, J.A. (1984). The Lyapunov dimension of a nowhere differentiable attracting torus, Ergod. Th. and Dynam. Sys. 4, 261-81. Mandelbrot, B.B. (1977). Fractals (Form, Chance and Dimension), W.H. Freeman. Mandelbrot, B.B. (1983). The Fractal Geometry of Nature. W.H. Freeman. Peitgen, H.-O. & Richter, P.H. (1986). The Beauty of Fractals, Springer-Verlag. Pietronero, L. & Tosatti, E. (Eds.) (1986). Fractals in Physics, North-Holland. Stapleton, H.J., Allen, J.P., Flynn, C.P. Stinson, D.G. & Kurtz, S.R. (1980). Fractal form of proteins, Phys. Rev. Letters 45, 1456-9. Umberger, D.K. & Farmer, J.D. (1985). Fat fractals on the energy surface, Phys. Rev. Letters 55, 6614.
Hydrodynamics Balian, R. & Peube, J.L. (Eds.) (1977). Fluid Dynamics, Gordon and Breach. Berge, P. & Dubois, M. (1984). Rayleigh-Benard convection, Contemp. Phys. 25, 535-82. Birkhoff, G. (1955). Hydrodynamics, Dover, Inc. Burgers, J.M. (1948). A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics, 1, eds. R. von Mises and T. von Karman, pp. 171-99, Academic Press. Chandrasekhar, S. (1961). Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press. Drazin, P.G. & Reid, W.H. (1981) Hydrodynamic Stability, Cambridge Univ. Press. Jeffrey, A. & Taniuti, T. (1964). Nonlinear Wave Propagation, Academic Press. Joseph, D. (1976). Nonlinear Stability of Fluid Motions, Springer-Verlag. Lamb, H. (1945). Hydrodynamics Dover, Inc. Landau, L.D. & Lifshitz, E.M. (1959). Fluid Mechancs, Addison-Wesley Publishing Co. Saltzman, B. (1962). Finite amplitude free convection as an initial value problem I, J. Atoms. Sci. 19, 329-41. Stoker, J.J. (1957). Water Waves, Interscience Publishers. (Appendix: On the Derivation of the Shallow Water theory, by K. O. Friedrichs.) Swinney, H.W. & Gollub, J.P. (Eds.) (1985). Hydrodynamic Instabilities and the Transition to Turbulence, Topics Appl. Phys. 45, Springer-Verlag. Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley.
Integrable systems Calogero, F. (1983). Integrable dynamical systems and related mathematical results. In Nonlinear Phenomena Lecture Notes Physics 189, ed. K.B. Wolf, Springer-Verlag, pp. 49-95. Casati, G., Chirikov, B.V. & Ford, J. (1980). Marginal local instability of quasi-periodic motion, Phys. Lett. 77A, 91-4. Moser, J. (1975). Three integrable Hamiltonian systems connected with isospectral deformations, Adv. in Math. 16, 197-220. Moser, J. (1978). Nearly Integrable and integrable systems. In Topics in Nonlinear Dynamics, ed. S. Jorna., Amer. Inst. Phys. Moser, J. (1981). Integrable Hamiltonian Systems and Spectral Theory, Academia Nazionale dei Lincei, Pisa.
References by topics
603
Invariance properties and conservation laws Denman, H.H. (1965). Invariance and conservation laws in classical mechanics, J. Math. Phys. 6, 1611-16.
Hill, E.L. (1951). Hamilton's principle and the conservation theorems of mathematical physics, Rev. Mod. Phys. 23, 253-60. Houtappel, R.M.F., Van Dam, H. & Wigner, E.P. (1965). The conceptual basis and use of the geometric invariance principles, Rev. Mod. Phys. 37, 595-632.
Inverse scattering method (also see, solitons) Ablowitz, M.J., Kaup, D.K., Newell, A.C. & Segur, H. (1974). The inverse scattering transform-fourier analysis for nonlinear problems, Studies Appl. Math. 53, 249-315. Ablowitz, M.J., Kaup, D.J. & Newell, A.C. (1974). Coherent pulse propagation, A dispersive irreversible phenomena, J. Math. Phys. 15, 1852-8. Ablowitz, M.J. & Segur, H. (1981). Solitons and the Inverse Scattering Transform SIAM. Ablowitz, M.J. & Fokas, A.S. (1983). Comments on the inverse scattering transform and related nonlinear evolution equations. In Nonlinear phenomena, Lecture Notes Physics 189, ed. K.B. Wolf, Springer-Verlag, pp. 4-23. Bednar, J.B. et al. (Eds.) (1983). Conference on Inverse Scattering: Theory and Applications, SIAM. Conference on the theory and application of solitons. In Rocky Mountain J. Math. 8, 1-428 (1978) eds. H. Flaschka and D.W. McLaughlin. De Santo, J.A., Saenz, A.W. & Zacharg, W.W. (Eds.) (1980). Mathematical Methods and Applications of Scattering Theory, Lecture Notes in Physics 130, Springer-Verlag. Eckhaus, W. & Van Harten, A. (1981). The Inverse Scattering Transformation and the Theory of Solitons,, North-Holland. Flaschka, H. (1976). On the Toda Lattice, Prog. Theor. Phys. 51, 703-16. Flaschka, H., Discrete & periodic illustrations of some aspects of the inverse method, ibid, pp. 44166.
Flaschka, H. & Newell, A.C. (1975). Integrable systems of nonlinear evolution equations. In Dynamic Systems, Theory and Applications, ed. J. Moser, Springer-Verlag, pp. 355-440. Gardner, C.S., Greene, J.M., Krushal, Krushal, M.D. & Miura, R.M. (1967, 1974). Method for solving the Korteweg-de Veries equation, Phys. Rev. Letters 19, 1095-7; Korteweg-deVries equation and generalization. VI. methods for exact solution, Comm. Pure Appl. Math, 27, 97-133. Kaup, D.J. (1976). The Three-Wave Interaction-A Nondispersive Phenomena, Studies Appl. Math. 55, 9-44. Kay, 1. & Moses, H.E. (1982). Inverse Scattering Papers 1955-1963, Vol. XII, Lie Groups, History Frontiers and Applications, Mathematical Science. Manakov, S.V. (1975). Complete integrability and stochastization of discrete dynamical systems, Sov. Phys. JETP, 40, 269-74. McLaughlin, D.W. (1975). Four examples of the Inverse method as a canonical transformation, J. Math. Phys. 16, 96-9. Miura, R.M. (1976). The Korteweg-deVries equation: a survey of results, SIAM Review 18, 412-59. Perlmutter, A. & Scott, L.F. (Eds.) (1977). The significance of Nonlinearity in the Natural Sciences, Plenum Press. Ramajaran, R. (1982). Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland. Scott, A.C., Chu, F.Y.F. & McLaughlin, D.W. (1973). The soliton: a new concept in applied science, Proc. IEEE 61, 1443-83.
604
References by topics
Zakharov, V.E. & Shabat, A.B. (1972). Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34, 62-9. Zakharov, V.E. (1974). On stochastization of one-dimensional chains of nonlinear oscillators, Sov. Phys. JETP 38, 108-10.
Korteweg-deVries equation Gardner, C.S., Greene, J.M., Kruskal, M.D. & Miura, R.M. (1967). A method for solving the Korteweg-deVries equation, Phys. Rer. Letters 19, 1095-7. Korteweg, D.J. & deVries, G. (1895). On the change of form of long wves advancing in a rectangular canal, and a new type of long stationary waves, Phil. May. 39, 422-43. Kruskal, M. (1982). The Korteweg-deVries equation. In Laser-Plasma Interaction, eds. R. Balian & J.C. Adam, pp. 788-808, North-Holland. Kruskal, M.D., Miura, R.M., Gardner, C.S. & Zabusky, N.J. (1970). V. Uniqueness and nonexistence of polynomial conservation laws, J. Math. Phys. 11, 952-60. Lax, P.D. (1968). Integrals of Nonlinear Equations of Evolution and Solitary Waves, Comm. Pure Appl. Math. 21, 467-90. Miura, R.M. (1968). I.A. remarkable explicit nonlinear transformation, J. Math. Phys. 9, 1202-4. Miura, R.M., Gardner, C.S. & Kruskal, M.D. (1968). 11. Existence of conservation laws and constants of motion, J. Math. Phys. 9, 1204-9. Su., C.H. & Gardner, C.S. (1969). III. Derivation of the KdV equation and Burgers equation, J. Math. Phys. 10, 536-9. Van der Blij, F. (1978). Some details of the history of the Korteweg-deVries equation, Nieu Archief VoorWiskunde (3) 26, 54-64.
Wadati, M. (1972). The exact solution of the modified Korteweg-deVries equation, J. Phys. Soc. Japan 32, 1681. Wadati, M. & Toda, M. (1972). The exact N-soliton solution of the Korteweg-deVries equation, J. Phys. Soc. Japan 32, 1403-11. Zabusky, N.J. & Kruskal, M.D. (1965). Interaction of `solitons' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Letters 15, 240-3.
Lattice dynamics Bocchieri, P., Scotti, A., Bearzi, B. & Loinger, A. (1970). Anharmonic chain with Lennard-Jones interaction, Phys. Rev. A2, 2013-19. Budinsky, N. & Bountis, T. (1983). Stability of Nonlinear Modes and Chaotic Properties of 1D Fermi-Pasta-Ulam Lattices, Physica 8D, 445-54. Casati, G. & Ford, J. (1975). Stochastic transition in the unequal-mass Toda lattice, Phys. Rev. A12, 1702-9.
Fermi, E., Pasta, J.R. & Ulam S.M. (1965). Studies of nonlinear problems, 1. Los Alamos Report LA1940, May 1955. In Collected Works of E. Fermi, Vol. 2, pp. 978-88, University of Chicago Press. Fesser, K., McLaughlin, D.W., Bishop, A.R. & Holian, B.L. (1985). Chaos and nonlinear modes in perturbed Toda chain, Phys. Rev. A31, 2728-31 (1985). Flaschka, H. (1974). The Toda lattice I. Existence of integrals, Phys. Rev. B9, 1924-5. Ford, J. (1961). Equipartition of energy for nonlinear systems, J. Math. Phys. 2, 387-93. Ford, J., Stoddard, S.D. & Turner, J.S. (1973). On the integrability of the Toda lattice, Prog. Theor. Phys. 50, 1547-60. Froeschle, C. (1978). Connectance of dynamical systems with increasing number of degrees of freedom, Phys. Rev. A18, 277. Henon, H. (1974). Integrals of the Toda lattice, Phys. Rev. B9, 1921-3.
References by topics
605
Jackson, E.A. (1963). Nonlinear coupled oscillators. 1. Perturbation theory: ergodic problem, J. Math. Phys. 4, 551-8; Nonlinear coupled oscillators. 11. Comparison of theory with computer solutions. J. Math. Phts. 4, 686-700. Jackson, E.A., Pasta, J.R. & Waters, J.F. (1968). Thermal conductivity of one-dimensional lattices, J. Comput. Phys. 2, 202-7. Jackson, E.A. (1978). Nonlinearity and irreversibility in lattice dynamics, Rocky Mount. J. Math. 8, 127-46.
Jackson, E. A. & Mistriotis, A. D. (1989). Thermal conductivity of one- and two-dimensional lattices. J. Phys.: Condens. Matter 1, 1223-38. Kruskal, M.D. & Zabusky, N.J. (1964). Stroboscopic perturbation procedure for treating a class of nonlinear wave equations, J. Math. Phys. 5, 231-44. Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani, A. (1983). Relaxation to different stationary states in the Fermi-Pasta-Ulam model, Phys. Rev. A28, 3544-52. Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani A. (1985). Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model, Phys. Rev. A31, 1039-45. MacDonald, R.A. & Tsai, D.H. (1978). Molecular dynamics calculations of energy transport in crystalline solids, Phys. Reports 46, 1-48. Manakov, S.V. (1974). Sot). Phys. JETP 40, 269. Maradudin, A.A., Montroll, E.W., Weiss, G.H. & Ipatova, I.P. (1971). Theory of Lattice Dynamics in the Harmonic Approximation, second eds. Academic Press. Mokross, F. & Buttner H. (1983). Thermal conductivity in the diatomic Toda lattice, J. Phys. C16, 4539-46. Nakazaea, H. (1970). On the Lattice Thermal Conduction, Suppl. Prog. Theor. Phys. 45, 231-62. Northcote, R.S. & Potts, R.B. (1964). Energy sharing and equilibrium for nonlinear systems, J. Math. Phys. 5, 383. Pnevmatikos, S.N. (1985). Solitons in nonlinear atomic chains. In Singularities and Dynamical Systems, ed. S.N. Pnevmatikos, 397-437 North-Holland. Sait6, N., Hirotomi, N. & Ichimura, A. (1975). The induction phenomenon and ergodicity in the anharmonic lattice vibration, J. Phys. Soc. Japan 39, 1431-8. Saito, N., Ooyama, N. & Aizawa, Y. (1970). Computer experiments on ergodicity problems in anharmonic lattice vibrations, Suppl. Prog. Ther. Phys. 45, 209-30. Schrodinger, E. (1914). Zur Dynamik elastisch gekoppelter Punktsysteme, Ann. d. Phys. 44, 916-34. Toda, M. (1975). Studies of a non-linear lattice, Phys. Reports 18C, 1-124. Toda, M. (1967). Vibration of a chain with nonlinear interaction, J. Phys. Soc. Japan 22, 431-36; Wave propagation in anharmonic lattices, 23, 501-6. Toda, M. (1971). Waves in nonlinear lattice, Prog. Theor. Phys. Suppl. No. 45, 174. Toda, M. (1981). Theory of Nonlinear Lattices, Springer-Verlag. Tuck, J.L. & Menzel, M.T. (1972). The superperiod of the nonlinear weighted string (FPU) problem, Adv. Math. 9, 339-407. Visscher, W.M. (1976). Computer studies of Transport properties in Simple models of solids, Meth. Comput. Phys. 15, ed. G. Gilat, pp. 371-408, Academic Press. Wadati, M. (1975). Wave propagation in nonlinear lattice. I, II., J. Phys. Soc. Japan 38, 673-80, 6826.
Walker, G.H. & Ford, J. (1969). Amplitude instability and ergodic behavior for conservative nonlinear oscillator systems, Phys. Rev. 188, 416-32. Zabusky, N.J. (1969). Nonlinear lattice dynamics and energy sharing, J. Phvs. Soc. Japan 26, Suppl.
Lie theory of continuous groups Campbell, J.E. (1903). /ntroductor.t on Lie's Theort g1'finite Continuous Transformations Groups, Oxford, Clarendon Press (reprint: Chelsea, 1966).
606
References by topics
Chevalley, C. (1946). Theory of Lie Groups, Princeton Univ. Press. Coggeshall, S.V. & Axford, R.A. (1986). Lie group invariance properties of radiation hydrodynamic equations and their associated similarity solutions, Phys. Fluids 28, 2398-20. Cohen, A. (1911). Introduction to the Lie Theory of One-Parameter Groups, D.C. Heath & Co. Eisenhart, L.P. (1933). Continuous Groups of Transformations, Princeton Univ. Press. Hill J.M. (1982). Solution of Differential Equations by Means of One-Parameter Groups, Pitman. Ince, E.L. (1956). Ordinary Differential Equations, Dover, Chap. IV. Lie, S. (1893). Vorlesungen ueber Continuierliche Gruppen mit Geometrischen and Anderen Anwendugen, B.C. Teubner, Leipzig. Winternitz, P. (1983). Lie groups and solutions of nonlinear differential equations. In Nonlinear Phenomena Lecture Notes in Physics, 189, ed. K.B. Wolf pp. 265 25 Springer-Verlag.
Lorenz equations Alfsen, K.H. Fr¢yland, J. (1984). Systematics of the Lorenz Model at a = 10, Univ. of Oslo, Institute of Physics, Norway. Birman, J.S. & Williams, R.F. (1983). Knotted periodic orbits in dynamical systems-I: Lorenz's equations, Topology 22, 47-82. Curry, J.H. (1978). A generalized Lorenz system, Comm. Math. Phys. 60, 193-204. Fowler, A.C. & McGuinness, M.J. (1982). A description of the Lorenz attractor at high Prandtl number, Physica 5D, 149-82. Fowler, A.C. & McGuiness, M.J. (1982). Hysteresis in the Lorenz equations, Phys. Letters 92A, 103-6.
Fr¢yland, J. & Alfsen, K.H. (1984). Lyapunov-exponent spectra for the Lorenz model, Phys. Rev. A29, 2928-31. Jackson, E.A. (1985). The Lorenz system: I. The global structure of its stable manifolds, Physica Scripta 32, 469-75; The Lorenz system: II. The homoclinic convolution of the stable manifolds, Physica Scripta 32, 476-81. Kaplan, J.L. Yorke, J.A. (1979). The onset of chaos in a fluid flow model of Lorenz, Ann. NY Acad. Sci. 316, 400-7. Kaplan, J.L. & Yorke, J.A. (1979). Preturbulence: a regime observed in a fluid flow model of Lorenz, Commun. Mathn., Phys. 67, 93-108. Lanford, O.E. (1977). In Turbulence Seminar; Lecture Notes in Mathematics 615, eds. P. Bernard & T. Rativ p. 114, Springer-Verlag. Lorenz, E.N. (1979). On the prevalence of aperiodicity in simple systems. In Global Analysis, Lecture Notes in Mathematics 755, eds. M. Grmela & J.E. Marsden, pp. 53-75, Springer-Verlag. Lorenz, E.N. (1963). Deterministic nonperiodic flow, J. Atoms. Sci. 20, 130-41; The mechanics of vacillation, ibid., 448-64. Mammerville, P. & Pomeau, Y. (1979). Intermittency and the Lorenz model, Phys. Letters 75A, 1-2. Manneville, P. & Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamic sytems, Physica ID, 219-26. McGuiness, M.J. (1985). A computation of the limit capacity of the Lorenz attractor, Physica 16D, 265-75. Richtmyer, R.D. (1986). A study of the Lorenz attractor, science and computers, Adv. Math. Suppl. Studies, 10, 207-19. Robbins, K.A. (1979). Periodic solutions and bifurcation structure at high R in the Lorenz model, J. Appl. Math 36, 457-71. Schmutz, M. & Rueff, M. (1984). Bifurcations schemes of the Lorenz model, Physica 11D, 167-78.
References by topics
607
Sparrow, C. (1980). Bifurcations in the Lorenz Equation, Lecture Notes in Applied Mathematics Springer-Verlag. Uezu, T. & Aizawa, Y. (1982). Topological character of a periodic solution in three-dimensional ordinary differential equation system, Prog. Theor. Phys. 68, 1907-16. Williams, R.F. (1979). The structure of the Lorenz attractor, Publ. Math. I.H.E.S. 50, 73-99. Yorke, J.A. & Yorke, E.D. (1981). Chaotic behavior and fluid dynamics. In Hydrodynamic Instabilities and the Transition to Turbulence, eds. H.L. Swinney & J.P. Gollub, Springer-Verlag, pp. 77-94. Yorke, J.A. & Yorke, E.D. (1979). Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys. 21, 263-77. Lyapunov exponents
Bennetin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: a method for computing all of them, Meccanica 15, 9. Benettin, G., Galgani, L. & Strelcyn, J.M. (1976). Kolmogorov entropy and numerical experiments, Phys. Rev. A14, 2338-45. Feit, S.D. (1978). Characteristic exponents and strange attractors, Commun. Math. Phys 61, 249-60. Oseledec, V.I. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197. Shimada, I. & Nagashima, T. (1979). A numerical approach to ergodic problem of dissipative dynamical systems, Prog. Theor. Phys 61, 1605. Wolf, A., Swift, J.B., Swinney, H.L. & Vastano, J.A. (1985). Determining Lyapunov exponents from a time series, Physica 16D, 285-317.
Mathematical definitions and terminology Choquet-Bruhat, Y., Dewitt-Morette, C. & Dillard-Bleick, M. (1977). Analysis, Manifolds and Physics, North Holland. lyanaga, S. & Kawada, Y. (Eds.) (1980). Encyclopedic Dictionary of Mathematics, Math. Soc. Japan; MIT Press.
Mathematics Ames, W.F. (1965, 1972).Nonlinear Partial Differential Equations in Engineering, Vols I and 11 Academic Press. Appel, K & Haken, W. (1977). Every Planar Map is Four Colorable, 1. Discharging; II. With J. Koch. Reducibility. lu. J. Math 21, 429-90; 491-567. Appel, K. & Haken, W. (1986). The four color proof suffices, Math. Intelligencer 8, 10-20. Arnold, V.I. (1961). The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case, Soviet Math. Dokl. 2, 247-9. Arnold, V.I. (1963). Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math. Surveys 18, No. 6, 85-189. Arnold, V.I. (1964). Dokl. Akad. Nauk. SSSR 156, 9. Arnold, V.I. (1978). Ordinary Differential Equations, M.I.T. Press. Arnold, V.I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations, SpringerVerlag.
Barber, M.N. & Ninham, B.W. (1970). Random and Restricted Walks, Gordon and Breach.
608
References by topics
Billingsley, P. (1965). Ergodic Theory and Information, Wiley. Birkhoff, G.D. (1913). Proof of Poincare's geometric theorem, Trans. A.M.S. 14, 14-22. Birkhoff, G.D. (1922). Surface Transformations and their Dynamical Application, Acta. Math. 43, 1119 (1922).
Bogoliubov, N & Mitropolsky, Y.A. (1961). Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach Science Publishers. Bogoliubov, N.N., Mitropoliskii, Ju.A. & Samoilenko, A.M. (1976). Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag. Borrelli, R.L. & Coleman, C.S. (1987). Differential Equations, A Modeling Approach, Prentice-Hall. Brocker, Th. & Janich, K. (1982). Introduction to Differential Topology, Cambridge Univ. Press. Brown, M. & Neumann, W.D. (1977). Proof of the Poincare-Birkhoff Fixed Point Theorem, Mich. Math. J. 24, 21-31. Buck, R.C. (1981). The Solutions to a Smooth PDE can be Dense in C(1), J. Diff. Eq. 41, 239-244. Cesari, L. (1963). Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Academic Press. Cesari, L. (1976). Functional Analysis, Nonlinear Differential Equations, and the Alternative Method. Lecture Notes in Pure and Applied Mathematics 19, pp. 1-197 Dekker. Chinn, W.G. & Steenrod, N.E. (1966). First Concepts of Topology, MAA. Coddington, E.A. & Levinson, N. (1955). Theory of Ordinary Differential equations, McGraw-Hill Book Co. Cunningham, W.J. (1964). Introduction to Nonlinear Analysis, McGraw-Hill Book Co. De Vogelaere, R. (1958). On the structure of symmetric period solutions of conservative systems, with applications. In Concentrations to the Theory of Nonlinear Oscillations, 4, ed. S. Lefshetz, pp. 53-84. Duffin, R.J. (1981). Rubel's Universal Differential Equation, Proc. Natl. Acad. Sci. USA 78, 4661-2. Eminhizer, C.H., Heileman, R.H.G. & Montroll, E.W. (1976). On a convergent nonlinear perturbation theory without small denominators or secular terms, J. Math. Phys. 17, 121-40. Francis, G.K. (1987). A Topological Picturebook, Springer-Verlag. Furstenberg, H. (1983). Poincare recurrence and number theory. In The Mathematical Heritage of Henri Poincare, Part 2, pp. 193-216, American Math. Soc. Gantmacher, R.R. (1959). The Theory of Matricies, Chelsa Pub. Co. Goursat, E. (1964). A Course in Mathematical Analysis, Dover. Guillemin, V. & Pollack, A. (1974). Differential Topology, Prentice-Hall. Hale, J. (1977). Theory of Functional Differential Equations, Springer-Verlag. Hale, J.K. (1980). Ordinary Differential Equations, R.E. Krieger Pub. Co. Hardy, G.E. (1916). Weierstrass's non-differentiable functions, Tran. Amer. Math. Soc. 17, 30125. Hartman, P. (1964). Ordinary Differential Equations, Wiley Reprinted by Hartman, 1973. Hirsch, M.W. & Smale, S. (1974). Differential Equations, Dynamical Systems and Linear Algebra, Academic Press. Hurewicz, W. & Wallman, H. (1948). Dimension Theory, Princeton Univ. Press. Ince, E.L. (1956). Ordinary Differential Equations, Dover. Lancaster, P. (1985). Theory of Matricies, Academic Press. LaSalle, J.P. & Lefschetz S. (Eds.) (1963). Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press. Lefschetz, S. (1959). Differential Equations; Geometric Theory, Interscience. Levinson, N. & Smith, O.K. (1942). A general equation for relaxation oscillations, Duke Math. J. 9, 287-403.
References by topics
609
MacKay, R.S. & Percival, I.C. (1985). Converse KAM: theory and practice, Commun. Math. Phys. 98, 469-512. Marsden, J.E. & McCracken, M. (1976). The Hopf Bifurcation and its Applications, Applied Math. Sci. 1, Springer-Verlag. Mather, J.N. (1984). Non-existence of Invariant Circles, Ergod. Th & Dynam. Sys. 4, 301-9. McLachlan, N.W. (1964). Theory and Application of Mathieu Functions, Dover. Moser, J. (1962). On Invariant Curves of Area-preserving mappings of an Annulus, Nachr. Akad. Wiss Gottingen Math. Phys. Kl. 2, 1-20. Moser, J. (1967). Convergent Series Expansions for Quasi-periodic Motions, Math. Annalen 169, 136-76.
Morse, P.M. & Feshbach, H. (1953). Methods of Theoretical Physics, McGraw-Hill. Nash, C. & Sen S. (1983). Tolopogy and Geometry for Physicists, Academic Press. Nayfeh, A.H. (1986). Perturbation methods in nonlinear dynamics. In Nonlinear Dynamics Aspects of Accelerators, eds. J.M. Jowett, Month, M. & Turner, S. Lecture Notes in Physics 247, pp. 238-314, Springer-Verlag. Nemytskii, V.V. & Stepanov, V.V. (1960). Qualitative Theory of Differential Equations, Princeton Univ. Press. Nitecki, Z. (1971). Differentiable Dynamics, MIT Press. Olds, C.D. (1963). Continued Fractions, Random House. Rolfsen, D. (1976). Knots and Links, Publish or Perish, Inc., Berkeley, CA. Rohlin, V.A. (1964). Exact endomorphisms of a Lebesque space, Amer. Math. Soc. Trans. 39, 1-36. Rubel, L.A. (1981). A universal differential equation, Bull. (NS) Amer. Math. Soc. 4, 345-9. Sanders, J.A. & Verhulst, F. (1985). Averaging Methods in Nonlinear Dynamical Systems, SpringerVerlag.
Shannon, C.E. (1948). A mathemational theory of communication, Bell Tech. J. 27, 623-56. Smale, S. (1981). The fundamental theorem of algebra and complexity Theory, Bull. Amer. Math. Soc. (N.S.) 4, 1-6. Smart, D.R. (1974). Fixed Point Theorems, Cambridge Univ. Press. Takens, F. (1981). Detecting strange attractors in turbulence. In Dynamical Systems and Turburlence, Warwick 1980. Lecture Notes in Mathematics 898, eds D.A. Rand & L.-S. Yang, Springer-Verlag. Tucker, A.W. & Bailey, H.S. Jr. (1950). Topology, Sci. Amer. 182(1), 18-24. Ulam, S.M. (1960). A Collection of Mathematical Problems Wiley. Ulam, S.M. (1982). Transformations, iterations and mixing flows. In Dynamical Systems II eds. A.R. Bednarek & L. Cesari, Academic Press, pp. 419-26. Weiss, L. (Ed.) (1972). Ordinary Differential Equations, Academic Press. Whitney, H. (1936). Differentiable manifolds, Ann. Math. 37, 645-80. Whitney, H. (1944). The self-intersections of a smooth n-manifold in 2n-space, Ann. Math. (2) 45, 220-46. Whitney, H. (1944). The singularities of a smooth n-manifold in (2n-1)-space, Ann. Math. (2) 45, 247-93. Yakubovich, V.A. & Starzhinskii, V.M. (1985). Linear Differential Equations with Periodic Coefficients, Vols. I and 2, Keter Publishing House, Jerusalem. Yan-Qian, Y. & Cai Sui-lin, et al. (1986). Theory of Limit Cycles, Amer. Math. Soc.; Translation of Math. Monographs 66. Zehnder, E. (1973). Homoclinic points near elliptic fixed points, Common. Pure Appl. Math. 26, 131-82.
610
References by topics
Mechanical and electrical system (also see, classical mechanics) Andronov, A.A. Khaikin, S.E. & Vitt, A.A. (1966). Theory of Oscillators, Pergamon Press. Arnold, V.I. & Avez, A. (1968). Ergodic Problems of Classical Mechanics, W.A. Benjamin, Inc. Hayashi, C. (1985). Nonlinear Oscillations in Physical Systems, Princeton Univ. Press. Kryloff, N. Bogliuboff, N. (1947). Introduction to Non-Linear Mechanics, Annals of Math. Studies, Princeton Univ. Press. Minorsky, N. (1947). Introduction to Nonlinear Mechanics, J.W. Edwards, Ann Arbor, Mich. Minorsky, N. (1962). Non-Linear Oscillations, Van Nostrand Co. Moon, F.C. (1984). Magneto-Solid Mechanics, Wiley. Neyfeh, A.H. & Mook, D.T. (1979). Nonlinear Oscillations, J. Wiley & Sons. Pippard, A.B. (1978). The Physics of Vibration, Cambridge Univ. Press. Schmidt, G. & Tondl, A. (1986). Non-Linear Vibrations, Cambridge Univ. Press. Scott, A.C. (1970). Active and Nonlinear Wave Propagation in Electronics, Willey & Sons, Inc. Stoker, J.J. (1950). Nonlinear Vibrations, Interscience Publishers. Vidyasaga, M. (1978). Nonlinear Systems Analysis, Prentice-Hall.
Meteorology Charney, J.G. & DeVore, J.G. (1979). Multiple flow equilibrium in the atmosphere and blocking, J. Atmos. Sci. 36, 1205-16. Lorenz, E.N. (1980). Attractor sets and quasi-geostrophic equilibrium, J. Atmos. Sci. 37, 1685-1699. Mitchell, K.E. & Dutton J.A. (1981). Bifurcations from stationary to periodic solutions in a low-order mode of forced dissipative barotropic flow, J. Atmos. Sci. 38, 690-716.
Modeling methodology Andrews, J.G. & McLone, R.R. (Eds.) (1976). Mathematical Modeling, Butterworths. Aris, R. (1978). Mathematical Modelling Techniques, Pitman. Aubert, K.E. (1984). Spurious mathematical modelling, Math. Intelligencer 6, 54-60. Dym, C.L. & Ivey, E.S. (1980). Principles of Mathematical Modeling, Academic Press. Farmer, D., Lapedes, A., Packard, N. & Wendroff, B. (Eds.) (1986). Evolution, games and learning; models for adaptation in machines and nature, Physica 22D, Nos. 1-3. Grabert, H. (1982). Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics, 95, Springer-Verlag. Paulre E. (Ed.) (1981). System Dynamics and the Analysis of Change, North-Holland. Saaty, T.L. & Alexander, J.M. (1981). Thinking with Models, Pergamon.
Nerve conduction Adey, W.R. (1975). Evidence for cooperative mechanisms in the susceptibility of cerebral tissue to environmental and intrinsic electric fields. In Functional Linkage in Biomolecular Systems, eds. F.O. Schmit, D.M. Schneider, and D.M. Crothers, Raven Press, NY. Basar, E., Durusan, R., Gorder, A. & Ungan, P. (1969). Combined Dynamics of E.E.G. and evoked potential, Biol. Cybernetics 34, 21. Coley, J.W. & Dodge, F.A. (1966). Digital computer solutions for excitation and propagation of the nerve impulse, Biphys. J. 6, 583-99. Fitzhugh, R. Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1,445-64. Fitzhugh, R. (1969). Mathematical models of excitation and propagation in nerves. In Biological Engineering, ed. H.P. Schwan, McGraw-Hill.
References by topics
611
Harmon, L.D. (1961). Studies with artificial neurons, 1: Properties and functions of an artificial neuron, Kybernetik 1, 89-101. Hodgkin, A.L. & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol. 117, 500-544. Krinskii, V.I., Pertsov, A.M. & Reshetilov, A.N. (1972). Investigation of one mechanism of origin of the etopic focus of excitation in modified Hodgkin-Huxley equations, Biofizika 17, 271-7. Nagumo, J. & Sato, S. (1972). On a response characteristic of a mathematical neuron model, Kybernetik 10, 155-64. Pauwelussen, J.P. (1981). Nerve impulse propagation in a branching nerve system: a simple model, Physica 4D, 67-88. Wiener, N. & Rosenblueth, A. (1946). The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, Specifically in Cardiac Muscle, Arch. Inst. Cardiol. Mexico 16, 205-65. Neural nets
Amari, S. & Arbib, M.A. (eds.) (1982) Competition and Cooperation in Neutral Nets, Lecture Notes in Biomathematics 45, Springer-Verlag. Clark, J.W., Winston, J.V. & Rafelski, J. (1984). Self-organization of neural networks. Phys. Letters 10A, 207-11. Clark, J.W., Rafelski, J. & Winston, J.V. (1985). Brain without mind: computer simulation of neural networks with modifiable neuronal interactions, Phys. Reports 123(4), 215-73. Cooper, L.N. (1973). A Possible Organization of Animal Memory and Learning, in Proceedings of the Nobel Symposium on Collective Properties of Physical Systems, 24, eds. B. Lundquist and L. Lundquist, 252-64. Ermentrout, E.B. & Cowan, J.D. (1979). A mathematical theory of visual hallucination patterns. Biol. Cybernetics 34, 137-50. Ermentrout, E.B. & Cowan, J.D. (1979). Temporal oscillations in neuronal nets, J. MAth. Biology 7, 265-80. Ermentrout, E.B. & Cowan, J.D. (1980). Large scale spatially organized activity in neural nets. SIAM J. Appl. Math 38, 1-21. Hopfield, J.J. (1982). Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79, 2554-8. Hopfield, J.J. (1984). Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA 81, 3088-92. Little, W.A. (1974). The existence of persistent states in the brain, Math. Biosci. 19, 101-20.
Nonlinear superposition (also see Lie-Backlund transformations) Ames, W.F. (1978). Nonlinear superposition for operator equations. In Nonlinear Equations in Abstract spaces, ed. V. Lakshmikantham, pp. 43-66, Academic Press, pp. 43-66. Anderson, R.L., Harnad, J. & Winternitz, P. (1982). Systems of ordinary differential equations with nonlinear superposition principles, Physica 4D, 164-82. Inselberg, A. (1972). Superpositions for nonlinear operators. 1. Strong superpositions and linearizability, J. Math. Anal. Appl. 40, 494-508.
Nonsoliton asymptotic part Ablowitz, M.J. & Segur, H. (1977). Asymptotic solutions of the Korteweg-deVries equation, Studies Appl. Math. 57, 13-44.
612
References by topics
Manakov, S.V. (1974). Nonlinear Fraunhofer diffraction, Soviet Phys. JETP 38, 693-6. Schuur, P.C. (1986). Asymptotic Analysis of Soliton Problems, Springer-Verlag. Zakharov, V.E., Monakov, S.V. (1977). Asymptotic behavior of non-linear wave systems integrated by the inverse scattering method, Soviet Phys. JETP 44, 106-12.
Optical phenomena Abraham, E. & Smith, S.D. (1982). Optical bistability and related devices, Rep. Prog. Phys. 45, 81585.
Eglund, J.C., Snapp, R.R. & Schieve, W.C. (1984). Fluctuations, instabilities and chaos in laser-driven nonlinear ring cavity. In Progress in Optics, 21, ed. E. Wolf, pp. 374-441, North-Holland. Lugiato, L.A. (1984). Theory of optical bistability. In Progress in Optics, 21, ed. E. Wolf, pp. 71-211, North-Holland. Optical bistability, dynamical nonlinearity, and photonic logic, Phil. Trans. Roy. Soc. Lond. 313 1984).
Painleve conjecture Ablowitz, M.J. & Segur, H. (1977). Exact linearization of a Painleve trancendent, Phys. Rev. Letters 38, 1103-6. Ablowitz, M.J., Ramani, A. & Segur, H. (1978). Nonlinear evolution equations and ordinary differential equations, Lett. Nuovo. Cimento 23, 333-8. Ablowitz, M.J., Ramani, A. & Segur, H. (1980). A connection between nonlinear evolution equations and ordinary differential equations of P-type. 1. J. Math. Phys. 21, 715-21. Bountis, T., Segur, H. & Vivaldi, F. (1982). Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A25, 1257-64. Bountis, T.C. (1984). A. Singularity analysis of integrability of chaos in dynamical systems. In Singularities and Dynamical Systems, ed. S. Pnevmatikos, North-Holland. Bountis, T. & Segur, H. (1982). Logarithmic singularities and chaotic behavior in Hamiltonian systems, AIP Conference Proceedings, No. 88 eds. M. Tabor and Y.M. Treve, pp. 279-92, AIP. Chang, Y.F., Tabor, M. & Weiss, J. (1982). Analytic structure of the Henon-Heiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Phys. 23, 531-8. Dorizzi, B. Grammatieos, B. & Ramani A. (1983). A New Class of Integrable Systems, J. Math. Phys. 24, 2282-2288. Dorizzi, B., Grammaticos, B. & Ramani, A. (1984). Explicit integrability for Hamiltonian systems and the Painleve conjecture, J. Math. Phys. 25, 481-5. Ramani, A., Dorizzi, B., Grammaticos, B. & Bountis, T. (1984). Integrabillity and the Painleve property for low-dimensional systems, J. Math. Phys. 25, 878-83. Tabor, M. & Weiss, J. (1981). Analytic structure of the Lorenz system, Phys. Rev. A24, 2157-67.
Patterns Bard, J.B.L. (1981). A model for generating aspects of zebra and other mammalian coat patterns, J. Theor. Biol. 93, 363-85. Boon, J.P. & Noullez, A. (1986). Development, growth, and form in living systems. In On Growth and Form, eds. H.F. Stanley & N. Ostrowsky, Martinus Nijhoff Pub. Gierer, A. & Meinhardt, H. (1972). A theory of biological pattern formation, Kybernetik 12, 30-9. Glansdorff, P. & Prigogine, I. (1971). Thermodynamic Theory of Structure Stability and Fluctuations, Wiley-Interscience.
References by topics
613
Hasegawa, A. (1985). Self-organization processes in continuous media, Adv. in Phys. 34, 1-42. Martinez, H.M. (1972). Morphogenesis and chemical dissipative structures, J. Theor. Biol. 36, 479501.
Meinhardt, H. (1982). Models of Biological Pattern Formation, academic Press. Nicolis, G. & Prigogine, I. (1977). Self-Organization in Nonequilibrium Systems, Wiley-Interscience. Stanley, H.E. & Ostrowsky, N. (Eds.) (1986). On Growth and Form, Martinus Nijhoff Pub. Swindale, N.V. (1980). A model for the formation of ocular dominance stripes, Proc. Roy. Soc. London B208, 243-64. Turing, A.M. (1952). The Chemical Basis of Morphogenesis, Phil. Trans. Roy. Soc. London B327, 37-72.
Pattern dynamics (also see, chemical turbulence, and coupled maps) Bishop, G. (1985). Pattern selection and low-dimensional chaos in systems of coupled nonlinear oscillation. In Dynamical Problems in Soliton Systems, ed. S. Takeno, pp.250-7, Springer-Verlag. Ciliberto, S. & Gollub, J.P. (1984). Pattern competition leads to chaos, Phys. Rev. Letters, 922-5. Gollub, J.P. & Meyer, C.W. (1983). Symmetry-breaking instabilities on a fluid surface, Physica 6D, 337-346.
Kaneko, K. (1983). Transition from torus to chaos accompanied by frequency lockings with symmetry breaking, Prog. Theor. Phys. 69, 1427-42. Kapral, R. (1985). Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators, Phys. Rev. A31, 3868-79. Smale, S. (1974). A Mathematical Model of two cells via Turing's equation, AMS Lecture Notes in App!. Math. 6, 15-26. Wesfreid, J.E. & Zaleski, S. (Eds.) (1984). Cellular Structures in Instabilities, Lecture Notes in Physics 210, Springer-Verlag.
Winfree, AT. (1987). When Time Breaks Down, Princeton Univ. Press. Zhabotinsky, A.M. & Zaikin, A.N. (1973). Autowave processes in distributed chemical systems, J. Theor. Biol. 40, 45-61. Related topics
Anderson, P. W., arrow, K.J. & Pines, D. (Eds.) (1988). The Economy as an Evolving Complex System: Santa Fe Institute Studies in the Sciences of'Complexity, Addison-Wesley. Berry, M.V. (1977). Semi-classical mechanics in phase space: a study of Wigner's function, Phil. Trans. Roy. Soc. London, 287, 237-7 1. Bluman, G.W. & Cole, J.D. (1974). Simulating Methods for Differential Equations, Springer-Verlag. Bohm, D. (1984). Causality and Chance in Modern Physics, Univ. of Pennsylvania Press. Bohm, D. & Peat, F.D. (1987). Science, Order, and Creativity, Bantam. Brillouin, L. (1963). Science and Information Theory, Academic Press. Brillouin, L. (1964). Scientific Uncertainty, and Information, Academic Press. Bruce, J.W., Giblin, P.J. & Gibson, C.G. (1984). Caustics through the looking glass, Math. Intelligence 6, 47-58. Chapman, S. & Cowling, T.R. (1970). The Mathematical Theory of Nonuniform Gases, Cambridge Univ. Press. Davidson, M. (1983). Uncommon Sense, J.P. Tarcher Inc., Los Angeles. Davis, R. (1988). The Cosmic Blueprint, Simon and Schuster. deAlfaro, V. & Rastti, M. (1978). Structural stability theory and phase transition models, Fortschrette der Physik 26, 143-73.
614
References by topics
deGroot, S.R. (1961). Thermodynamics of Irreversible Processes, North-Holland. Gardner, M. (1981). The Laffer curve and other laughs in current economics, Sci. Amer. 245(6), 18. Gardner, M. (1986). Knotted Doughnuts and Other Mathematical Entertainments, W.H. Freeman. Hawking, S.W. (1988). A Brief History of Time, Bantam. Heims, S.J. (1984). John Von Neumann and Norbert Wiener, MIT Press. Jackson, E.A. (1984). Radiation reaction dynamics in an electromagnetic wave and constant magnetic field, J. Math. Phys., 1584-91. Kac, M. (1959). Probability and Related Topics in Physical Sciences, Interscience. Khinchin, A.I. (1949). Mathematical Foundations of Statistical Mechanics, Dover. Laslow, E. (1987). Evolution, The Grand Synthesis, Shambhala Pub., Boston. Leggett, A.J. (1987). The Problems of Physics, Oxford Univ. Press. Leisegang, S. (1953). Zum Astigmatismus von Electronenlinsen, Optik 10, 5-14. Makarov, I.M. (Ed.) (1984). Cybernetics Today, Mir Publishers. Moore-Ede, M.C., Sulzman, F.M. & Fuller, C.A. (1982). The Clocks that Time Us, Harvard Univ. Press.
Mistriotis, M.D. (1986). From Determinism to Stochasticity, Thesis, Dept. Phys., Univ. Illinois, Urbana. Nicolis, G. & Prigogine, I. (1978). Self-Organization in Nonequilibrium Systems, Wiley-Interscience. Pines, D. (Ed.) (1985). Emerging Syntheses in Science, Santa Fe Institute. Pippard, A.B. (1985). Response and Stability, Cambridge Univ. Press. Poincare, J.H. (1929). The Foundations of Science, Science Press. Prigogine, I. (1962). Non-Equilibrium Statistical Mechanics, Interscience. Prigogine, I. & Stengers, I. (1984). Order Out of Chaos, Bantam Books. Prigogine, I. (1980). From Being to Becoming, Freeman. Roederer, J. (1979). Introduction to the Physics and Psychphysics of Music, Springer-Verlag. Risset, J.C. (1982). Stochastic processes in music and art. In Stochastic Processes in Quantum Theory and Statistical Physics, Lecture Notes in Physics, 173, eds. S. Albeverio, Ph. Combe, & M. Sirugue-Collin, Springer-Verlag. Rosen, R. (1985). Anticipatory Systems, Pergamon. Sinclair, R.M., Hosea, J.C. & Sheffield, G.V. (1970). A method for mapping a toroidal magnetic field by storage of phase stabilized electrons, Rev. Sci. Instruments, 41, 1552-9. Stravrondis, O.N. (1972). The Optics of Rays, Wavefronts and Caustics, Academic Press. Ulam, Stanislaw (1909-1984); Los Alamos Science, Number 15, Special Issue (1987). Ulam, S.M. (1965). Introduction to studies of nonlinear problems by E. Fermi, J. Pasta and S.M. Ulam. In Collected Papers of Enrico Fermi, Vol. 2, Univ. Chicago Press. Ulam, S.M. (1983). Adventures of a Mathematician, Scribner. VanVelsen, J.F.C. (1978). On linear response theory and area preserving mappings, Physics Reports 41C, 135-90.
Rossler models Crutchfield, J., Farmer, D., Packard, N., Shaw, R., Jones, G. & Donnelly, J. (1980). Power spectral analysis of a dynamical system, Phys. Letters 76A, 1-4. Farmer, D., Crutchfield, J., Froehling, H., Packard, N. & Shaw, R. (1980). Power spectra and mixing properties of strange attractors, Ann. NYAcad. Sci. 357, 453-72. Fraser, S. & Karpal, R. (1982). Analysis of flow hysteresis by a one-dimensional map, Phys. Rev. A25, 3223-33. Kolata, G. (1984). Order out of chaos in computers, Science 223, 917-9.
References by topics
615
Rossler, O.E. (1983). The chaotic hierarchy, Z. Naturforsch. 38a, 788-801. Rossler, O.E. (1979). Continuous chaos - Four Prototype Equations, Ann. NY Acad. Sci. 316, 376-92. Rossler, O.E. (1977). Continuous chaos. In Synergetics - A Workshop, ed. H. Haken, Springer-Verlag, pp. 184-97.
Similarity analysis Birkhoff, G. (1955). Hydrodynamics, Dover Pub., Inc. Bluman, G.W. & Cole, J.D. (1974). Similarity Methods for Differential Equations, Springer-Verlag. Dickson, L.E. (1924). Differential equations from the Group Standpoint, Annal of Math. 25, 287. Hansen, A.G. (1964) Similarity Analyses of Boundary Value Problems in Engineering, Prentice-Hall, Inc.
Ovsjanikov, L.V. (1958). Group and group invariant solutions of differential equations, Dokladi Akad. Nauk. USSR 118, 439-42. Sedov, L.I. (1959). Similarity and Dimensional Methods in Mechanics, Academic Press. Sine Gordon equation Ablowitz, M.J., Kaup, D.J., Newell, A.C. & Segur, H. (1973). MEthod for solving the SineGordon equations, Phys. Lett. 30, 1262-4. Barone, A., Esposito, F. Magee, C.J. & Scott, A.C. (1971). Theory and applications of the Sine-Gordon equation, Nuovo Cimento, Series 2, 1, 227-67 (Contains extensive references to all applications). Lamb, G.L. Jr. (1971). Analytical descriptions to ultrashort optical pulse propagation in resonant media, Rev. Mod. Phys. 43, 99-124.
Osborne, A. & Stuart, A.E.G. (1978). On the separability of the Sine-Gordon equation and similar quasilinear partial differential equations, J. Math. Phys. 19, 1573-8. Perring, J.K. & Skyrme, T.H.R. (1962). A model unified field equation, Nucl. Phys. 31, 550-5. Rajaraman, A. (1975). Some non-perturbative semi-classical methods in quantum field theory, Physics Reports 21C, 227-313. Rubinstein, J. (1970). Sine-Gordon equation, J. Math. Phys. 11, 258-66.
Singular perturbation, asymptotic expansions Asymptotic Methods and Singular Perturbations, SIAM-AMS Proceedings, 10, AMS, 1976. Cole, J.D. (1968). Perturbation Methods in Applied Mathematics, Blaisdell. Dorodnitsyn, A.A. (1947). Asymptotic solution of the van der Pol equation, Inst. Mech. Acad. Sci., USSR, 11, (Russian). Eckhaus, W. (1973). Matched asymptotic expansions and singular perturbations, North-Holland. Eckhaus, W. (1977). Formal approximations and singular perturbations, SIAM Review 19. Erdelyi, A. (1956). Asymptotic Expansions, Dover. Kruskal, M. (1962). Asymptotic theory of Hamiltonian systems, J. Math. Phys. 3, 806-28. LaSalle, J. (1949). Relaxation oscillations, Quart. J. App. Math. 7, 1. Lebowitz, N.R. & Schaar, R. (1975). Exchange of stabilities in autonomous systems, Studies Appl. Math. 54, 229-60. Meyer, R.E. & Parter, S.V. (Eds.) (1980). Singular Perturbations and Asymptotics, Academic Press. O'Malley, R.E. Jr. (1974). Introduction to Singular Perturbations, Academic Press.
616
References by topics
O'Malley, R.E. Jr. (1977). Singular Perturbation Analysis for Ordinary Differential Equations, Comm. Math. Institute Rijksuniversiteit, Utrecht. Netherlands. O'Malley, R.E. Jr. (1976). Phase plane solutions to some singular perturbation problems, J. Math. Sanders, J.A. & Verhulst, F. (1984). Averaging Methods in Nonlinear Dynamical Systems Appl. Math. Sciences, Springer-Verlag. Verhulst, F. (1983). Asymptotic Analysis of Hamiltonian Systems, Lecture Notes in Mathematics 985, ed. F. Verhulst, Springer-Verlag. Verhulst, F. (Ed.) (1979). Asymptotic Analysis, Lecture Notes in Mathematics, 711, Springer-Verlag. Wasow, W. Asymptotic Expansion for Ordinary Differential Equations, Interscience.
Social phenomena (see references on modeling methodology) Aubin, J.P., Saari, D. & Sigmund, K. (Eds.) (1985). Dynamics of Macrosystems, Lecture Notes in Economic and Mathematical Systems 257, Springer-Verlag. Gazis, D.C. (1967). Mathematical Theory of Automobile Traffic, Science 157, 273-81. Gazis, D.C. (Ed.) (1974). Traffic science, Wiley. Meadows, D.L. & Meadows, D.H. (1974). The Limits to Growth, Universe Books, NY. Meadows, D.L. & Meadows, D.H. (Eds.) (1973). Toward Global Equilibrium, Wright-Allen Press, Cambridge, Mass. Meadows, D.L. et al. (1974). Dynamics of Growth in a Finite World, Wright-Allen Press, Cambridge, Mass. Montroll, E.W. (1978). On some mathematical models of social' phenomena. In Nonlinear Equations in Abstract Spaces, ed. V. Lakshmikantham, pp. 161-216, Academic Press. Musha, T. & Higuchi, H. (1977). In Proceedings of the Symposium on l/f Fluctuations, p. 189, Inst. Elect. Engineers, Tokyo, Japan. Muncaster, R.G. & Zinnes, D.A. (1984). Phase portraits, singularities, and the dynamics of hostility in international relations, SIAM J. Applied Math. Saari, D.G. (1984). The ultimate of chaos resulting from weighted voting systems, Adv. App. Math. 5, 286-308.
Solitons (also see, Korteweg-deVries, Backlund, lattice dynamics and inverse scattering) Ablowitz, M.J. & Segur, H. (1981). Solitons and the Inverse Scattering Transform, SIAM. Bishop, A.R. & Schneider, T. (Eds.) (1978). Solitons and Condensed Matter Physics, Springer-Verlag. Bulllough, R.K. & Dodd, R.K. (1979). Solitons in physics: basic concepts; solitons in mathematics. In Structural Stability in Physics, eds. W. Guttinger and H. Eikemier, pp. 219- 53. Springer-Verlag. Bullough, R.K. & Caudrey, P.J. (Eds.) (1980). Solitons, Topics in Current Physics, 17, SpringerVerlag.
Campbell, D.K., Newell, A.C., Schrieffer, R.J. & Segur, H. (Eds.) (1986). Solitons and Coherent Structures, Physica D18, 1-480. Davydov, A.S. (1985). Solitons in Molecular Systems, D. Reidel. Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. (1982). Solitons and Nonlinear Wave Equations, Academic Press. Eilenberger, G. (1981). Solitons, Springer-Verlag. Flascka, H. & McLaughlin, P.W. (Eds.) (1978). Conference on the theory and application of solitons, Rocky Mountain J. Math. 8, 1-428. Ichikawa, Y.H. & Ino, K.H. (1985). Lax-pair operators for squared-sum and squared-difference eigenfunctions, J. Math. Phys. 26, 1976-8.
References by topics
617
Lamb, G.L. Jr. (1980). Elements of Soliton Theory, Wiley-Interscience. Lonngren, K. & Scott, A. (Eds.) (1978). Solitons in Action, Academic Press. Makhankov, V.G. (1978). Dynamics of classical solitons (in non-integrable systems), Physics Reports 35, 1
Manakov, S.V. & Zakhorov, V.E. (Eds.) (1981). Soliton Theory, Physica 3D, Nos I and 2. Newell, A.C. (1985) Soliton in Mathematics and Physics, SIAM. Newell, A.C. & Redekopp, L.G. (1977). Breakdown of Zakharov-Shabat theory and soliton creation, Phys. Rev. Lett. 38, 377-80. Rajaraman, R. (1982). Solutions and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland. Russell, J.S. (1844). Report on Waves, Report of 14th Meeting, British Assoc. Adv. Sci., pp. 311-90, John Murray, London. Sagdeev, R.Z. (Ed.) (1984). Nonlinear and Turbulent Processes in Physics, Vol. 2, Harwood Acad. Pub. Scott, A.C., Chu, F.Y.F. & McLaughlin, D.W. (1973). The soliton: a new concept in applied science, Proc. IEEE 61, 1443-83. Segur, H. (1986). Some open problems, Physica 18D, 1-12. Takeno, S. (Ed.) (1985). Dynamical Problems in Soliton Systems Springer-Verlag. Wilhelmsson, H. (Ed.) (1979). Solitons in physics. Physica Scripta 20 (3/4). Zakharov, V.E. & Shabat, A.B. (1972). Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media, Soviet Phys. JETP 34, 62-69.
Solitons, direct method (Hirota) Ablowitz, M.J. & Satsuma, J. (1978). Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys. 19, 2180-6. Hirota, R. (1985). Fundamental properties of the binary operators in soliton theory and their generalization. In Dynamic Problems in Soliton Systems, ed. S. Takeno, pp. 42-9, Springer-Verlag. Hirota, R. (1971). Exact solution of the Korteweg-deVries equation for multiple collisions of solitons, Phys. Rev. Lett. 27, 1192-4. Hirota, R. (1976). Direct methods of finding exact solutions of nonlinear evolution equations. In Backlund Transformations, ed. R.M. Miura Lecture Notes in Mathematics 515, Springer-Verlag. Hirota, (1985). Direct methods in soliton theory. In Solitons: Topics of Modern Physics, eds. R.K. Bullough & P.J. Candrey. Springer-Verlag. Nakamura, A. (1979). A direct method of calculating periodic wave solutions to nonlinear evolution equations. 1. Exact two-periodic wave solution, J. Phys. Soc. Japan, 47, 170. Solitons, higher dimensional Ablowitz, MJ. (1981). Remarks on nonlinear evolution equations and inverse scattering transform. In Nonlinear Phenomena in Physics and Biology, eds. R.H. Enns et at. Ablowitz, M.J. & Nachman, A.I. (1986). Multidimensional nonlinear evolution equations and inverse scattering, Physica 18D, 223-41. Anderson, D.L.T. (1971). Stability of time-dependent particle-like solutions in nonlinear field theories II, J. Math. Phys. 12, 945-52. Davydov, A.S. (1985). Solitons in Molecular Systems, D. Reidel. Fokas, A.S., & Ablowitz, M.J. (1983). The inverse scattering transform for multi-dimensional (2 + 1) problems. In Nonlinear Phenomena, Lecture Notes in Physics 189, ed. K.B. Wolf, pp. 139-54. Springer-Verlag. Gibbon, J.D., Freeman, N.C. & Johnson, R.S. (1978). Correspondence between the classical gd4.
618
References by topics
Double and single Sine-Gordan equations for three-dimensional solitons, Physics Letters 65A, 3800-4. Kadomtsev, B.B. & Petviashvili, V.I. (1970). Sov. Phys. Doklady, 15, 539-41. Kako, F. & Yajima, N. (1980). Interaction of ion-acoustic solitons in two-dimensional space, J. Phys. Soc. Japan. 49, 2063-71. Newton, R.G. (1978). Three-dimensional solitons, J. Math. Phys. 19, 1068-73. Rebbi, C. (1979). Solitons, Sci. Amer. 240(2), 92-116. Strauss, W.A. (1977). Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55, 14962.
Yajima, N., Oikawa, M. & Satsuma, J. (1978). Interaction of ion-acoustic solitons in three-dimensional space, J. Phys. Soc. Japan 44, 1711-14. Yajima, N. (1985). Soliton resonance in plasmas. In Dynamical Problems in Soliton Systems, ed. S. Takeno, pp. 144-52, Springer-Verlag.
Standard and area-preserving maps: microtron instability Chirikov, B.V. (1979). A universal instability of many-dimensional oscillator systems Physics Reports 52, 265-376. Collet, P., Eckmann, J.-P. & Koch, H. (1981). On universality for area-preserving maps of the plane, Physica 3D, 457-67. Greene, J.M. (1986). How a swing behaves, Physica 18D, 427-7. Greene, J.M. (1979). A method of determining a stochastic transition, J. Math. Phys. 20, 1183-201. Greene, J.M., MacKay, R.S., Vivaldi, F. & Feigenbaum, M.J. (1981). Universal behavior in families of area-preserving maps, Physica 3D, 468-86. Ichikawa, Y.H., Kamimura, T. & Hatori, T. (1987). Stochastic diffusion in the standard map. Physica 29D. 247-55. Jowett, J.M., Month, M. & Turner, S. (Eds.) (1985). Nonlinear Dynamics Aspects of Particle Accelerators, (Lecture Notes in Physics, 247, Springer-Verlag. MacKay, R.S. (1986). Transitions to chaos in area-preserving maps. Nonlinear Dynamics Aspects of Particle Accelerators, eds. J.M. Jowett, M. Month, and S. Turner, pp. 390-454, SpringerVerlag.
Melekhin, V.N. (1976). Phase dynamics of particles in a microtron and the problem of stochastic instability of nonlinear systems, Sov. Phys. JETP 41, 803-8. Sinclair, R.M., Hoser, J.C. & Sheffield, G.V. (1970). A method for mapping a torodial magnetic field by storage of phase stabilized electrons, Rev. Sci. Instruments 141, 1552-9.
Surveys, general
Abraham, R. & Shaw, C. Dynamics: The Geometry of Behavior; Part one: Periodic Behavior (1982); Part two: Chaotic Behavior (1983); Part three: Global Behavior (1983); Part four: Bifurcation Behavior (1984), Aerial Press, Santa Cruz, CA. Davis, H.T. (1962). Introduction to Nonlinear Differential and Integral Equations, Dover. Diner, S., Fargue, D. & Lochak, G. (Eds.) (1986). Dynamical Systems: A Renewal of Mechanism, World Scientific. Hirsch, M.W. (1984). The dynamical systems approach to diferential equations, Bull. Amer. Math. Soc. 11, 1-64. Von Karmen, T. (1940). The engineer grapples with nonlinear problems (Fifteenth Josiah Willard Gibbs Lecture) Bull. Amer. Math. Soc. 46, 615-83.
References by topics
619
Synergetics Haken, H. (1977). Synergetics, An Introduction, Springer-Verlag; Vol. 2: A Workshop (ibid). Haken, H. Advanced Synergetics, Springer-Verlag. Kozak, J.J. (1979). Nonlinear Problems in the Theory of Phase Transitions, Adv. Chem. Phys. 40, pp. 229-368, Interscience Pub. Pacault, A. & Vidal, C. (Eds.) (1979). Synergetics: Far from Equilibrium, Springer-Verlag. Zabusky, N.J. (1981). Computational synergetics and mathematical innovation, J. Comput. Phys. 43, 195-249 (1981).
Turbulent-like dynamics (also see, chemical systems, and explosive dynamics) Barenblatt, G.I., Iooss, G. & Joseph, D.D. (Eds.) (1983). Nonlinear Dynamics and Turbulence, Pitman. Berry, M.V. (1978). Regular & Irregular motion. In Topics in Nonlinear Dynamics, ed. S. Jorna, pp. 16-120, AIP. Conf. Proc. No. 46, AIP. Frisch, U. (1986). Fully developed turbulence: where do we stand? In Dynamical Systems, A Renewal of Mechanism, eds. S. Diner, D. Fargue & G. Lochak, pp. 13-28, World Scientific. Hopf, E. (1948). A mathematical example displaying features of turbulence, Comm. Pure Appl. Math. 1, 303-22. Lorenz, E.N. (1964). The problem of deducing the climate from the governing equations, Tellus 16, 1-11.
Martin, P.C. (1975). The onset of turbulence: a review of recent developments in theory and experiment, in Proceedings of the International Conference on Statistical Physics, Budapest 1975, eds. L. Pal & P. Szepfalusy, 69-96, North-Holland. McLaughlin, J. (1976). Successive bifurcations leading to stochastic behavior, J. Stat. Phys. 15, 30726.
McLaughlin, J.B. & Martin P.C. (1975). Transition to turbulence in a statistically stressed fluid system, Phys. Rev. A12, 186-203. Normand, C. & Pomeau, Y. (1977), convective instability: a physicist's approach, Rev. Mod. Phys. 49, 581-624. Orszag, S.A. (1977). Lectures on the statistical theory of turbulence. In Fluid Dynamics, eds. R. Balian J.-L. Peube, Gordon and Breach, pp.237-368. Ruelle, D. & Takens, F. (1971). On the nature of turbulence, Comm. Math. Phys. 20, 167-92, 23, 3434.
Ruelle, D. (1978). Dynamical systems with turbulent behavior. In Mathematical Problems in Theoretical Physics, eds. G. Del'Atonio, S. Doplicher & G. Jona-Lasinio. Lecture Notes in Physics, 80, Springer-Verlag. Smale, S. (1977). Dynamical systems and turbulence. In Turbulence Seminar, Lecture Notes in Mathematics, 615, Springer-Verlag, 48-70. Stuart, J.T. (1971). Nonlinear stability theory, Ann. Rev. Fluid Mech. 3, 347-70. Swinney, H.W. & Gollub, J.P. (Eds.) (1985). Hydrodynamic Instabilities and the Transition to Turbulence, Springer-Verlag. Treve, Y.M. (1975). Theory of chaotic motion with application to controlled fusion research. In Topics in Nonlinear Dynamics, ed. S. Jorna, pp. 147-220, North-Holland.
Turbulence, onset
A. Experimental Ahlers, G. & Behringer, R. (1978). Evolution of turbulence from the Rayleigh-Benard instability, Phys. Rev. Lett. 40, 712-16.
620
References by topics
Ahlers, G., Behringer, R.P. (1978). The Rayleigh-Benard instability and the evolution of turbulence, Prog. Theor. Phys. Suppl. 64, 186-201. Behringerm, R.P. & Ahlers, G. (1977). Heat transport and critical slowing down near the RayleighBenard instability in cylindrical containers, Phys. Letters 62A, 329-31. Benjamin, T.B. (1978). Bifurcation phenomena in steady flows of a viscous fluid I. theory, II. Experiment, Proc. Roy. Soc. A359, 1-26, 27-43. Berge, P. (1979). Experiments on Hydrodynamic Instabilities and the Transition to Turbulence, Lecture Notes in Physics, 104, pp. 289-308, Springer-Verlag. Busse, F.H. (1980). Convections in a rotating layer: a simple case of turbulence, Science 208, 173-5. Coles, D. (1965). Transition in circular couette flow, J. Fluid Mech. 21, 385-425. Fenstermacher, P.R., Swinney, H.L. & Gollub, J.P. (1979). Dynamic instabilities and the transition to chaotic Taylor Vortex flow, J. Fluid Mech. 94, 103-28. Fenstermacher, P.R., Swinney, H.L., Benson, S.V. & Gollub, J.P. (1979). Bifurcations to periodic, quasiperiodic and chaotic regimes in rotating and convecting fluids, Ann. NY Acad. Sci. 316, 652-6. Gollub, J.P. & Freilich, M.H. (1974). Optical heterodyne study of the Taylor instability, in a Rotating fluid, Phys. Rev. Lett. 33, 1465-8. Gollub, J.P. & Swinney, H.L. (1975). Onset of turbulence in a rotating fluid, Phys. Rev. Lett. 35, 927-30. Gollub, J.P. & Benson, S.V. (1978). Chaotic response to periodic perturbation of convecting fluid, Phys. Rev. Lett. 41, 948. Gollub, J.P. & Meyer, C.W. (1983). Symmetry-breaking instabilities on a fluid surface, Physica 6D, 337-46. Koschmieder, E. L. (1981). Experimental aspects of hydrodynamic instabilities. In Order and Fluctuations in Equilibrium and Nonequilibrium Statistical Mechanics, eds. G. Nicolis, G. Dewel & J.W. Turner, pp. 159-88, Wiley. Meyer, R.E. (Ed.) (1981). Transition and Turbulence, Academic Press. Swinney, H.L. & Gollub, J.P. (Eds.) (1985). Hydrodynamic Instabilities and the Transition to Turbulence, Topics in Applied Physics, 45, Springer-Verlag. Swinney, H.L. & Gollub, J.P. (1978). The transition to turbulence, Phys. Today 31(8) 41.
B. Theoretical (also see, Lorenz equations) Drazin, P.G. & Reid, W.H. (1981). Hydrodynamic Stability, Cambridge Univ. Press. Eckmann, J.-P. (1981). Roads to turbulence in dissipative dynamical systems, Rev. Mod. Phys. 53, 643.
Feigenbaum, M.J. (1979). The onset spectrum of turbulence, Phys. Letters 74A, 375-8. Newhouse, S., Ruelle, D. & Takens, F. (1978). Occurrence of strange axiom A attractors near quasiperiodic flows on T, m _> 3. Commun. Math. Phys. 64, 35-40. Orszag, S.A. & Kells, L.C. (1980). Transition to turbulence in plane poiseccille and plane couette flow, J. Fluid Mech. 96, 159-205. Orszag, S.A. & Patera, A.T. (1981). Subcritical transition to turbulence in plane shear flows. In Transition and Turbulence, ed. R.E. Meyer, pp. 127-46, Academic Press. Packard, N.H., Crutchfield, J.P., Farmer, J.D., & Shaw, R.S. (1980). Geometry from a time series, Phys. Rev. Lett. 45, 712-16. Ruelle, D. (1978). Dynamical Systems with Turbulent Behavior. In Mathematical Problems in Theoretical Physics, eds. G. Dell'Antonio, S. Doplicher & G. Jona-Lasino, Lecture Notes in Physics, 80, pp. 341-60, Springer-Verlag.
References added at 1991 reprinting
621
Sagdeev, R.Z. (Ed.) (1984). Nonlinear and Turbulent Processes in Physics, Vols. 1, 2, and 3, Harwood Acad. Pub. Takens, F. (1981). Detecting Strange Attractors in Turbulence in Dynamical Systems and Turbulence. Warwick 1980, Lecture Notes in Mathematics 898, eds. D.A. Rand & L.-S. Young, SpringerVerlag.
Yahata, H. Temporal Development of the Taylor vortices in a Rotating Fluid, Prog. Theor. Phys. Suppl. 64, 165-85.
References added at 1991 reprinting Abraham, N.B. (Ed.) (1989). Quantitative Measures of Complex Dynamical Systems. Plenum Press. Amit, D.J. (1989). Modeling Brain Function. Cambridge Univ. Press. Barnsley, M. (1988). Fractals Everywhere. Academic Press. Barrow, J.D. (1988). The World within the World. Clarendon Press, Oxford. Barrow, J.D. & Tipler, F.J. (1988). The Anthropic Cosmological Principle. Oxford Univ. Press. Bedford, T. & Swift, J. (Eds.) (1988). New Directions in Dynamical Systems. Cambridge Univ. Press. Bloxham, J. & Gubbins, D. (1989). The evolution of the Earth's magnetic field. Scientific American
261, *6, 68-75. Cairns-Smith, A.G. (1982). Genetic Takeover, and the Mineral Origins of Life. Cambridge Univ. Press.
Chang, S.J. & Wright, J. (1981). Transitions and distribution functions for chaotic systems. Phys. Rev. A22, 1419-33.
Clark, J.W., Rafelski, J. & Winston, W. (1985). Brain without mind: computer simulation of neural networks with modifiable neuronal interactions. Phys. Reports 123, 215-73. Croom, F.H. (1989). Principles of Topology, Sanders. Deutsch, D. (1985). Quantum theory, the Church-Turing principle and universal quantum computer. Proc. R. Soc. Lond. A400, 97-117. Duncan, R. & Weston-Smith, M. (Eds.) (1978). The Encyclopaedia of Ignorance, Pergamon Press. Edelman, G.M. (1987). Neural Darwinism. Basic Books. Escande, D.F. (1985). Stochasticity in classical hamiltonian systems: universal aspects. Phys. Reports 121 (3 & 4), 165-261. Fox, R.F. (1988). Energy and the Evolution of Life. W.H. Freeman. Fox, S. (1988). The Emergence of Life. Basic Books. Freeman, W.J. (1991). The physiology of perception. Scientific American 264 (2), 78-85. Frisch, U. & Orszag, S.A. (1990). Turbulence: challenges for theory and experiment. Physics Today 43, # 1, 24-32. Froyland, J. (1983). Lyapunov exponents for multidimensional orbits. Phys. Letters 97A, 8-10. Gelperin, A. & Tank, D.W. (1990). Odour-modulated collective network oscillations of olfactory interneurons in a terrestrial mollusc. Nature 345, 437-40. Goel, N.S. & Thompson, R.L. (1988). Computer Simulations of Self-organization in Biological Systems. Macmillan.
622
References added at 1991 reprinting
Goldberger, A.L., Rigney, D.R. & West, B.J. (1990). Chaos and fractals in human physiology. Scientific American 262, #2, 40-9. Harrison, L.G. (1987). What is the status of reaction-diffusion theory thirty-four years after Turing? J. Theor. Biol. 125, 369-84. Huberman, B.A. & Crutchfield, J.R. (1979). Chaotic states of anharmonic systems in periodic fields. Phys. Rev. Letters 43, 1743-7. Jackson, E.A. & Mistriotis, A.D. (1989). Thermal conductivity of one- and two-dimensional lattices. J. Phys.: Condens. Matter 1, 1223-38. Jensen, M.H., Kadanoff, L.P., Libchaber, A., Procaccia, I. & Stavans, J. (1985). Global universality at the onset of chaos: results of a forced Rayleigh-Benard experiment. Phys. Rev. Letters 55, 2798-801. Landford, O.E. (1987). Circle mappings. pp. 1-17. In Recent Developments in Mathematical Physics, ed. H. Mitter & L. Pitner. Springer-Verlag. Langton, C.G. (Ed.) (1989). Artificial Life (Vol. VI, Santa Fe Institute), Addison-Wesley. Marchall, C. (1990). The Three-body Problem. Elsevier. Moon, F.C. (1987). Chaotic Vibrations, Wiley-Interscience. Newhouse, S. (1979). The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. IHES 50, 101-51. Paladin, G. & Vulpiani, A. (1987). Anomalous scaling laws in multifractal objects. Phys. Reports 156, 147-225. Palmore, J.I. & McCauley, J.L. (1987). Shadowing by computable chaotic orbits. Phys. Letters A122, 399-402. Peliti, L. & Vulpiani, A. (Eds.) (1988). Measures of Complexity, Springer-Verlag. Popper, K. (1968). The Logic of Scientific Discovery, Harper and Row. Sagdeev, R.Z., Usikov, D.A. & Zaslavsky, G.M. (1988). Nonlinear Physics. Harwood Academic. Schaffer, W.M. (1988). Perceiving order in the chaos of nature. In Evolution of Life Histories of Mammals, ed. M.S. Boyce, pp. 313-50. Schroeder, M. (1990). Fractals, Chaos, Power Laws. W.H. Freeman. Skarda, C.A. & Freeman, W.J. (1987). How brains make chaos in order to make sense of the world, Behavioral and Brain Sciences 10, 161-95. Svirezhev. Yu. M. & Passekov, V.P. (1990). Fundamentals of Mathematical Evolutionary Genetics. Kluwer. Thompson, J.M.T. & Stewart, H.B. (1986). Nonlinear Dynamics and Chaos. Wiley. van Bendegem, J.P. (1987). Finite Empirical Mathematics; Outline of a Model. Rijsuniversitent Gent. Zeeman, E.C. (1988). Stability of Dynamical Systems, Nonlinearity 1, 115-55.
Index Numbers in bold refer to citations in volume 2; others to volume 1.
Abel equation 74 Ablowitz-Kaup-Newell-Segur formulation 406 Ablowitz-Ramani-Segur conditions 297 abstract dynamics 50 accelerator 34 accelerator mode 35 action-angle equations 35 action-angle variables 267, 10, 47 activators-inhibitors 310, 467, 522 adaptive systems 344ff Adelphic integrals, Whittaker's 39, 257, 547 Airy function 363ff algorithmic complexity 514ff almost periodic function 238, 543 alpha-limit set 347, 410, 44 analysis-synthesis 430 analytic methods 14 Andronov and Pontriagin 2, 102 Andronov-Hopf bifurcation 274ff, 278, 279, 301, 303, 177, 189, 243, 312; Ex 7.10
aperiodic solutions 164 area-preserving maps 3,37,42,43, 50, 80ff Arnold diffusion 257ff, 261, 263 `superconductivity' channels 258 Arnold tongues 205 Arnold's cat map 87ff, 96, 108 astronomy 74 asymptotic expansions 268,298; Appendix I, 416 asymptotic solution (see omega-limit set); patterns 453 asymptotic stability 41 attractors 95, 104, 175, 217, 234, 283, 340, 342, 346ff, 4, 24, 147ff multiple 27, 118, 177, 329 autocatalytic oscillators 272, 283ff, 288ff autonomous equations 19, 36, 251, 253ff, 267 averaging method (see KBM method)
Backlund-Lie transformations 404, 410ff invariant type 413ff banana regions 39, 62, 66 basins of attractions 217, 347, 20, 27, 130, 168, 329
Belousov-Zhabotinskii chemical oscillations 159, 223ff, 323ff Bendixson's criterion 245, 522 Benjamin's machine 118 Bernoulli equation 74, 76, 335 Bernoulli sequence 175, 337, 16, 19, 53, 539 semi-infinite, 172 Bernoulli shift 213 Bessel functions 353 beta shadowing 215 bifurcation 82, 381 bifurcation phenomena 73 bifurcation (see catastrophes) 19, 33, 381 Andronov-Hopf 274ff, 304, 177, 189, 243, 312; Ex. 7.10
chemical turbulence 445ff Curie point 98 cusp-butterfly 125 double point 89 exchange of stabilities 90 global 300ff hard 276
homoclinic and heteroclinic 162ff imperfect 93 lines 231 period doubling 154, 162, 32, 184, 206 period tripling 360 pitch-fork 154 reverse 165 saddle-node 88, 300 singular points 88, 89 soft 276
Index
624 bifurcation (see catastrophes) (Cont.) supercritical, subcritical 97, 276 symmetry-breaking 345, 3496, 354, 186 tangent 169ff transcritical 97 Turing 244 binary operator 398ff
biological dynamics 283ff, 323, 1, 240, 330ff, 509ff, 518ff Birkhoff xiii, 2, 410, 6, 251 Birkhoff's limit sets 281, 346, 410, 530, 538 bistable subharmonic 317 bistability 130ff, 296, 311, 317, 335, 189, 197, 211, 454 screw and spiral 215 blue-sky catastrophe 296 Bogoliubov-Krylov-Mitropolsky (see KBM method) Bohm's `order' 510ff Bolzano-Weierstrass theorem 179, 412 Boltzmann equation 12, 18
bouncing ball on a vibrating plate 34 boundaries, fractal 173 bounded stability 42, 265 Boussinesq equation 356 breather solitons 398 Brouwer's fixed point theorem 150, 181, 246, 247, 4, 224
Bruns' theorem xii, 1 Brusselator equation 329, 342, 420, 23, 305 Burgers' equation 361, 362, 365, 411 butterfly catastrophe 115, 116, 123ff CA, see cellular automata canonical transformations 253, 269, Appendix L cantilever spring 92, 316 Cantor-Lebesgue function 63, 204, 334 Cantor correspondence 384 Cantor sets 154, 185, 187, 19, 24, 204, 257, 540 'fat' 65, 168
self-similar 189 `thin' 65
Cantor sponge 257 capacity (dimension) 59ff, 187ff cardioId 32
Cartesian product space 26, 30 Cartwright and Littlewood xiv, 2, 299, 322, 16ff, Appendix J, 529 cat map 87ff, 96, 109 catastrophe set, elementary 107 catastrophes 19, 73 blue-sky 296 `butterfly' 115, 116, 123ff
cusp 111, 116, 119ff, 133, 308
error 336 fold 94, 110, 116
generalized 154, 163
machines 318ff
sets 98, 101, 107, 381 systems 73 theory 107ff
Cauchy data (initial conditions) 32 `causality' 524, 526ff caustics 79ff cellular automata 5, 13, 429, 454ff association 459 code number 465 configuration 455, 456 dynamic rule 456, 463 entropies 473ff; shortcomings 478 Garden-of-Eden 484, 493, 495 Hamming distance 461 invertible 493ff legal 457, 459 'life' 487ff, 495 molluscs 465ff nature 494 neighborhoods 456, 462, 482, 487, 492 quiescent condition 457 oscillating `molecules' 481 parallel map 457 particle-like dynamics 478ff range 456 rule `decomposition' 468ff rule number 463 self-reproduction 455, 482ff, 484 state values 455 totalistic 464 transition function 457 two-dimensional 482 von Neumann 483, 499 von Neumann and Ulam 454 Wolfram's dynamic classes 464ff cellular differential equations 428 center 229 center cycle 133
center manifold 276, 284, 131 center of mass, solitons 281, 385 chaos 166, 177, 22ff, 46ff, see strange attractor and homoclinic/heteroclinic points 40ff and order 27, 65, 11811, 172, 176 CA 467ff, 473ff chemical 323ff conservative 77, 92, 96 dynamos 161 entrainment 445 forced oscillators 329ff, 356ff, 90ff gliders 491
Hinon and Heiles 77 intermittent 188ff KO set 18; see curious set Li and Yorke 178
Index Lorenz attractor 145ff; system 189 Lyapunov exponents 182 multiple 27, 118ff noise 330ff, 323 screw-type 208 spiral-type 208 transients 29, 160, 172ff
turbulence 327ff various types 173, 193, 26, 198ff, 206 chaotic dynamics 166, 182, 210, 330, 331, 337, 340, 345, 350, 352, 354, 355, 358ff, 18ff, 2211, 71, 78, 14611, 160ff, 204, 289ff
chaotic parameter regions 332, 352 chaotic region 118 characteristic exponents 228, 275, 236, 243 multipliers 44 chemical oscillations 285, 343, 222ff, 323 turbulence 446ff chirality 183 Chirikov 34 resonance-overlap criterion 67ff, 71 circadian rhythms 323 circle map 197ff classical dynamic system 51 cnoidal wave 227 coarse system 103 codimension 54, 109 coexistent dynamics 13, 217; see bistability coexisting chaos and periodicity 27 collective variables 8, also see normal modes, solitons complex systems 331, 464ff, 505ff
complexity
algorithmic 514ff logical depth 516ff noncomputability 517 stability 521 computability 103, 211ff, 246 computations 210, 93, 495, also see lattice maps computer integration program Appendix H, 414 configuration space 26 configurations, CA 455, 456 conservation laws 371, 494 infinite number 371, 418ff conservative system 143, 231, 355, 3; see measure-preserving constants of the motion 36, 39, 250, 256, 296, 353
additive 39, 285 Adelphic integrals 257 algebraic 13, 284, 285 cellular automata 494 time independent 37 continued fraction 62 continuity equation 371, 418 control parameters 19, 82
625
map space 154ff, 167, 218ff phase space 20, 83ff, 112ff, 276, 279, 280 control space 21, 111, 114, 230, 231, 236, 277, 301, 244 convoluted manifold 170ff Conway's `Life' CA 487ff, 493, Ex. 10.31 Couette flow 234ff coupled clocks 203 coupled maps 431ff inhomogeneous 444 lattice maps 445 coupled oscillators 197, 203, 322, 287, 304 coupled piano strings 203 cover, 103 minimal 109 crisis 172, 29 critical energy 282, 290 critical points 36; see fixed points, singular points, Painleve property critical singular point 295 cross-ratio theorem 75, 415 curious set, Ko 337, 340, 16ff, 531, 537, 542 cusp catastrophe 111, 116, 119ff, 133, 308, 313 dual 111 cusped-diamond 121 cyclic variables 247 cylinder 26, 39, 69
`Dali' limit cycle 214 damped motion 238; also see gradient systems `Daruma' flow 248 degree of ODE 18 Degrees of freedom 12 `Degrees' of order (Bohm) 510ff delay situation 98 delayed logistic equation I dendrite structure 32 determinism, mathematical and physical 11, 145, 173, 210ff, 523ff
devil's staircase 63, 204, 334, 324 diatomic Toda lattice 281ff diffeomorphism 22, 36, 51, 377 difference equations (DE) 5, 10, 142; see maps diffusion, Arnold 2576, 261, 263 diffusive coupling (also see Turing instability) 241, 304ff, 432 digital simulation world 210 digraph method 168, Appendix E, 400 dimensions of sets 58ff, 187ff, 379, 380, 387ff, 329ff direct method (Hirota) 398 directed orbit 53 discharge tube 97, 291 discrete space-time 497ff; see lattice maps dispersion 352, 362, 421 dissipative structure 308ff, 362
626
Index
dissipative systems 143, 343, 362; see gradient systems double point 89 dual cusp 111, 116 Duffing equation 163, 268, 343 cusp catastrophe 313 forced 309, 320 inverted 349, 360 nonharmonic 343; Ueda's study 344 Dulac's criterion 248 Dym's equation 410 dynamic dimensions 12 dynamic entropies, see entropies dynamic phase transitions 19, 82, 231, 302, 144, 163ff, 169, 176, 179, 182ff, 209, 328, 447ff dynamic relations 524, 527 dynamic rule numbers, CA 456, 463 dynamic symmetry breaking 181ff dynamic systems 13, 50ff dynamo systems 155ff
Earth's magnetic field history 156ff `Eaters', CA 490
Euclidean space 19, 378 Euler characteristic 249 Euler strut 92 'Eulerian' models 428 Euler's equations 44, 296 Euler's theorem 242 eventually periodic 151, 217, 97ff, 473ff, 391 evolution 331, 335, 339 exchange of stability 90, 92, 98, 100, 154 excitable medium 449ff, 491 existence of solutions 30, 34 for all t 34; also see flows exogenous variables 19, 323 experiments 14, 64, 93, 97, 118ff, 128, 130, 159, 164, 203, 251, 252, 285, 290, 291, 316, 328, 331, 348, 349, 356ff, 66, 155, 323ff explosive instabilities 35, 44, 46 exponent
characteristic 228, 275 Lyapunov 182; see Lyapunov exponential separation 182; also see linear separation extended phase space 317, 336, 340, 342, 2, 20ff
echo waves 227
ecology 432, 521; also see biology, logistic, Lotka-Volterra economics 146; see references; Related Topics effective frequency 309
catastrophe 336 computations 93, 495ff; see FINITE, lattice
family of oriented phase curves global 226, 129ff, 163ff, 199ff local 229ff, 129ff portrait 21, 229ff, 302 fast switches 296, 212 fast variables, see slowly varying variables feedback, see autocatalytic Feigenbaum's universal functions 160ff Fermi-Pasta-Ulam 256, 259ff, 286, 348, 350ff Fermi's conjecture 261 Fermi's little discovery 261 Fermi's theorem 256ff Feynman quotations 15, 496 Fibonacci numbers 64, 97, 477 Fick's Law 265,266 Field-Noyes equations 224ff FINITE, THE 210, 478, 496, 523ff first order differential equations, Chapter 2, 18ff Fitzhugh-Nagumo model 292 fixed point theorem, see Brouwer's fixed points 36, 149 (map); 227; see Brouwer's 5, 235ff of flows 36, 227ff Floquet theorem 237ff flows 35, 51 in phase space 35ff measure-preserving 46 fluctuation 98 fluid systems 361 fluids, bifurcations, see Rayleigh-Benard 163,
maps experimental 11, 210ff
focus 229
Einstein quotation 15 elliptic functions 255, 322, 406, 297, 354, 392 elliptic point 229, 44, 54ff embedding 27, 30, 388ff, 317ff, 326 empirical sciences 210, 93, 114ff, 478, 496, 507ff, 524ff endogenous variables 19, 323 entrainment 203, 322ff, 330ff, 352 of chaos Ex. 10.8 Entropies, dynamic Kolmogorov-Sinai 101, 112, 117 of CA 473ff
production 519 specific spatial 474 specific temporal 475 topological 101, 105, 117, 474 equations of variation, see Poincare equilibrium, see fixed points equipartitioning of energy 259ff, 264, 286 equivalence class of phase portraits 21ff ergodic 181, 193, 381, 106, 249, 256, 259, 264, 285ff errors bounds on averaging method 271
151ff, 328
627
Index fold catastrophe 94, 110, 116 forced oscillators 308, 329 entrainment 203, 327, 333 hysteresis effect 311, 331, 335, 350, 364 subharmonic 315ff, 333ff Fourier transform 380 fractal boundaries 20, 28, 31ff, 190, 273, 470ff fractal dimensions 60, 61, 387, 217ff fractal `Tori' 340, 342, 22 fractals 59ff, 135, 347, 387, 21, 23ff, 93, 217ff Fredkin's invertible CA 499 Fredkin's postulate 210, 497 Fredkin's 'self-reproduction' 482 Frenkel-Kontorva model 36, 390 frequency shift 256, 257, 265, 288, 309 fusion 37, 39, 65ff Garden-of-Eden configurations 488ff, 493, 495
Gardner-Greene-Kruskal-Miura solution 373ff Gel'fand-Levitan-Marchenko theorem 379 generator (fractals) 64 generic properties 11, 100, 238, 380, see structural stability ghosts 100 gliders, CA 491 global analysis 232ff global bifurcations 300ff, 144, 165ff, 179 global characteristics 20, 232 global constants of the motion 39, 284
Golden mean 63, 64,89 gradient systems 20, 73, 236ff, 339 Hamiltonian systems 20, 43, 45, 46, 55, 236, 238, 394ff, 6, 246, 256, 269, 302; see KAM theorem normal form 247 steepness 258 Hamming distance 461 handedness 183 hard excitation 314 hard nonlinearity 258, 310, 316, 271 hares eat lynx? 285 harmonic oscillators 42, 55, 47, 353; relativistic 299 harmonic-plus-hard-core 268, 286 Hausdorff dimension 66, 379 heartbeat 323, 492 heat conduction, see lattice thermal conductivity Hbnon-Heiles Hamiltonian 6, 74ff, 272, 302 Henon's area preserving map 80ff
Hbnon's `strange attractor' map 22ff dimension 222 Hessian 254, 255, 547, 551 heteroclinic orbit 234, 261, 163, 177 heteroclinic point 48ff, 90, 177 heterodyning 327
hexagonal cells 492 Hirota's direct method 39811 historical outline xii Hofbauer's theorem 341ff homeomorphism 22, 36, 377 homoclinic explosion 177 homoclinic orbit 103, 235, 261, 301, 306, 164 homoclinic point 48, 89 homogeneous functions 242 homotopy 391 Hopf bifurcation; see Andronov-Hopf Hopf-Cole transformation 365ff, 372, 411 Hopf theorem 278 horseshoe map (Smale) Appendix K, 538 Hugo, Victor 15 Hurwitz (Routh-Hurwitz) Criterion 236 hyperbolic singular point 230, 44, 46ff hypercycles 332ff elementary 337 hyperelliptic functions 297 hysteresis effects 96, 110ff, 118ff, 134, 289, 292, 311ff, 331, 335, 364, 20, 189, 197
imbedding - see embedding immersion 7, 321ff `nearly' Ex. 2.26, 7, 74ff imperfect bifurcation 93 implicit function theorem 38, 84, 254 index (see Poincarb index) INFINITE, THE 210ff, 93, 430, 497, 517 information 60 conservation 494 production 113ff, 118 information dimension 61 instabilities (see stabilities) explosive 35, 44, 46
integrable system 246ff, 284, 296 Painleve properly 298 separations linear in time 251ff, 283, 288 integrability condition 411 conjecture 298 integral invariants 44, Appendix C, 393 manifolds 25, 37 integrality (Denbigh) 519 integrals of the motion (see constants of the motion) integrating factors 77, 78, 240 intermittency 169, 119, 188, 444 invariant curve of maps 44 invariant measure 191 inverse map 190 inverse scattering transform (IST) 337, 338, 379, 406
history 367ff
628
Index
inverse scattering transform (IST) (Cont.) vis-d-vis Fourier 380 inverted Dulling equation 349, 360 invertible CA 493ff involution 250, 284 and independence 250 involutive transformation 85 irreversibility 259, 265, 287, 294, 344, 458, 495, 525
islands, coherent 77, 84, 87, 119, 289ff isochronous solutions 41, 437, 519 isoenergetic nondegeneracy 551 isolating integrals of the motion 40, 284ff
Jacobi elliptic functions 255, 322, 406 Jacobian: matrix and determinant 38, 44ff, 106ff, 230
Julia sets 20, 29ff jump phenomena 120, 288ff, 311ff, 338, 348, 212ff
Kadomtsev-Petviashvili equation 421, 422 KAM-curve breakup 57ff, 76ff KAM surfaces 55, 60, 62, 66, 79 KAM theorem 239, 253ff, 256, 259, Appendix L, 543 KAM tori 355ff kanonische Normalsysteme (Fermi) 256 Kepler problem 250 Klein bottle 206, 389, 319 kneading action 180, 339, 342, 21ff, 205, 215ff, 327
knots 391, 12; see torus knots knotted limit cycles 13, 181ff Koch triadic curve 64 Kolmogorov 59, 211, 259, 547 Kolmogorov-Arnold-Moser (see KAM) Kolmogorov-Sinai dynamic entropy 101, 112, 117
'Kolmogorov's' (Albrecht, Gatzke, Haddad, & Wax) Theorem 287ff Korteweg and deVries 277, 349 Korteweg-deVries equation 360, 362, 373, 378, 389,404 modified 372, 405, 410, 412 Kruskal and Zabusky 350ff Krylov-Bogoliubov-Mitropolsky (KBM) method 264ff, 69 autonomous 267ff errors 271 nonautonomous 319
K-sets 107, 111, 112, 114, 115, 120ff, 125, 313
laboratory studies 14 Laffer curve 146ff Lagrange stability 42, 340
Lagrangian models 427ff Landau's equation 74, 99 Landau-Lifshitz subset 385 laser model 101 lattice dynamics 36, 259, 269, 281ff, 350 continuum limit 351ff, 359 harmonic 352 Toda 274 lattice thermal conductivity 265ff, 292ff lattice maps 216ff coupled 445 Lax formulation 387ff Lax's equation 405 legal CA 457, 459 Leggett, quotation 512 Lennard-Jones potential 262, 274 Levinson's analysis xiv, 2, 299, 332, 11f, Appendix J, 529, 539 Levi's analysis 335, 339, 11ff Li and Yorke chaos 178 Lie groups, see references Lienard's equation 5 Lienard's phase plane 295, 335 'Life' 312, 509ff CA 487ff, 495 lift-reinjection 198, 201 light caustics 79ff limit cycle 234, 273 (see van der Pol), 304ff, 176, 343
'Dali' 214 knotted 186 linked 182ff limit point 378, 383, 410 limit surface 176 linear analysis 227, 126ff, 232ff, 241ff, 352, 379
linear operator, superposition 3 linear PDE 7, 77 linear and nonlinear interrelations 3, 6ff; also see Riccati equation; 223ff Hopf-Cole transformation; inverse scattering method linear separation with time 251ff, 283, 288 linked limit cycles 182ff Liouville's theorem, generalized 45ff, 231, Appendix C, 393, 3 on submanifold 47 Liouville equation 16, 394, 412 Lipschitz condition 31, 78, 191, 355 living systems 330ff; also see Lottka-Volterra, logistic systems logical depth (Bennett) 516ff logistic equation 75, 100 delayed 1 logistic map 148ff, 181, 212, 432, 515 complex 30 discrete 216 Lorenz attractor 145ff
629
Index dimensions 219, 222 Lorenz bifurcations 144, 177 global 162ff, 179 Lorenz convolution 170ff Lorenz equations 142, 154, 162 Lorenz manifolds 165ff Lorenz match ('map') 149ff, 202ff Lorenz model 138ff bistability 197 chaos and order 177 dynamo 155ff fluid ring 151ff linked, stable limit cycles 181 linked and knotted limit cycles 182ff Lyapunov exponents 1% overview 187, 189 power spectrum 211 Lotka-Volterra equations 47, 283ff, 339ff Lyapunov exponents 182ff, 347, 26ff, 112, 117, 190ff, 207, 237, 326, 461
dimension of strange attractor 217ff Lorenz 1% Rossler 207 Lyapunov function 43, 46, 105, 340 Lyapunov stability 41 Lyapunov's first theorem 42 Lyapunov's reducibility theorem 238 Lyapunov's theorem 232 magnetic field reversals 155ff magnetic fields 35, 65, 67 magnetic surfaces 37, 65 magneto-elastic dynamics 348, 362 Mandelbrot xv, 59, 61, 64, 135, 29 set 29ff, 32 manifolds 24, 51, 380, 390 convoluted 170ff integral 37 slow and fast, see jump phenomena; integration 415 stable, unstable 46ff, 89, 130, 137ff, 165ff maps: Chapters 2, 4, 6 cat 87ff, % conservative 3, 33ff, 38, Ex. 2.27 coupled 431ff fixed point 149, 247 fold-over 180, 217ff Henon's 22ff, 80ff homotopic 392 involutive 85 logistic, see logistic map nonconservative 3, 22 on lattices 93ff parallel 457 period-n point 149, 52 random, ensemble 94
relations to flows 56; see Poincare map S-maps 149 suspensions 58, 206
tent, see tent map topologically conjugate 181 mathematical abstractions 11, 50ff mathematical terms, Appendix A, 376 mathematics ris-d-vis empirics 11, 169, 210ff,
4%,523 Mathieu equation 237 Maxwell situation 97, 112 Maxwell's rule 114 measure-preserving flows 46, 51, 191, 232 measure-preserving maps Ex. 2.27, 3, 33ff measure of sets 51, 65, Appendix B, 378 Menger and Uryshon, dimension 59, 387 metamodels 507, 513 metastable 112, 122 metric 20, 66, 378 Metropolis-Stein-Stein (MSS) 158ff, 353, 360
microtron 34 middle third set (Cantor) 63 minimal limit sets 410 mirror-image dynamics 82 Miura's transformation 372, 412 Mitropolsky, see KBM mixing 192ff, 381, 90, 201, 210, 264 Mobius strip 30, 206, 222, 136, 185 mode coupling 47, 260ff, 338ff modeling 10ff, 505ff, 523ff; see sensitivity to initial conditions predictable, unpredictable 11, 212ff molecular models 268ff molluscs 465ff
Moore neighborhoods 487 'morphogens' 241, 309, 428 morphogenesis 136, 240ff, 304, 311ff Morse functions 106, 116 Moser (also Kolmogorov-Arnold-Moser) integral example 251 Moser Calso Kolmogorov-Arnold-Mose) twist theorem 57 movable critical point 294 multistability 330, 344, 184, 189; also see basins of attraction, and attractors myths 8ff natural selection 339 near-integrable 41ff, 253ff, 281ff, 543ff neighborhoods, see cellular automata neurons 292, 323, 491 Newton map 30 Newton quotation 505 Newtonian system 235 node 229, 130 nonautonomous systems 19, 319, 344
Index
630
nonequilibrium statistical mechanics 265 nonlinear oscillators, Chapter 5 epilogue 365 preview 251 nonlinear phenomena 3, 6 nonlinear superposition 75ff, 414, 415, 416 nonuniqueness 32ff normal modes 243, 260ff, 286, 434 normal phase functions (Khinchin) 287 nowhere dense set 65, 379 N-sphere 30 N-torus 30
nullcline 225 omega-limit set 281, 304ff, 340, 342, 410, 25, 44, 148; also see attractors open systems 222ff, 265, 304ff, 330, 343ff, 505,
518ff
optical bistability 130ff, 293 orbit 21, 51 orbital stability 41 order and organization 509ff, 516ff ordinary differential equations (ODE) 5, 12, 18, 33, 226
oregonator 223ff organization 518ff organization, constant 333 organization of ideas 12 organizing center 109 orientation preserving 22, 206, 392, 49 oscillations biological 283ff, 323 bouncing ball 91 chemical 159, 283, 342, 2221f, 323 entrainment 203, 322ff, 330ff intermittency 169ff, 119, 187ff, 444 `kicked' 68 limit cycles, see limit cycles overviews of types 251, 329f, 365 relaxation 388ff semiperiodic 165 overdamped dynamics 73, 86; see gradient dynamics overlap of resonances 67ff
pacemaker 322 Painleve's property 294ff defined 297 integrability conjecture 298 Painlev6 transcendents 295 Painleve's theorem xiii, 1 parabolic point 45 parallel map 457 partial differential equations (PDE) 3, 7, 13, 297, 35211, 361, 365, 398, 410, 428
partition 103ff
refinement 105 passive oscillators autonomous 251, 253ff nonautonomous 251, 308ff, 348ff patterns: see dissipative structure, Turing's instabilities cellular automata 461, 466ff, 470ff coupled lattice maps 448ff
coupled maps 437ff excitable medium 491ff interdynamic 120 molluscs 465ff multiple-periodic 451 traveling 454 Peano's curve 59, 384 pendulum 251, 253ff, 362ff hard-spheres 73 upside-down 349ff, 361 period eventual 151, 217, 97ff, 473ff pendulum 256 relaxation oscillator 298, 420, Appendix 1, 416 period-doubling bifurcations 152, 350, 353, 357ff, 185ff
period-n point 149, 152 period-three implies chaos 178
not inRI 28,31
periodic patterns, multiple 451 phase coherence 208 phase-control space 20, 83ff, 112ff, 276, 279, 280 phase portrait 21, 53 phase space 20, 283, 288 embedding 317ff extended 317, 336, 342, 2 morphogen 242, 309, 311 toroidal, see extended piano strings 203, 328 piecewise-linear functions xiii, 299, 260 dynamics 91 Pippard, quotation 15 pitchfork bifurcation 154 plasma oscillations 74, 241, 361 Poincare 1, 15, 44, 51, 59, 82, 382, 386, 393, 100, 285
quotations 15, 55 Poincare first return map 55, 57, 135, 535; also see Poincare map Poincare index of a curve 243ff, 286
of singular point 245, 249 Poincare integral invariants 44, Appendix C, 393
Poincare-Bendixson theorem 280ff, 327, 409, 14 Poincare map, numerical 56, 72ff Poincare maps 53ff, 169, 181, 198, 317, 2, 9, 11, 17, 52, 761f, 135, 184, 201, 202, 326ff
Index Poincare recurrence 52ff, 98 Poincare rotation number 200, 333, 11 Poincare stability 41 Poincare surface of section 55ff, 318, 6, 8, 282, 326, 317
computation 56, 6, 201 Poincare's last geometric theorem (PoincareBirkhoff) 42, 64 Poincarb's theorem, see Historical Survey 256ff Poincare's variational equation 32ff, 252 Poisson bracket 250 Pol, van der, see van der Pol polar representation 39 polynomial nonlinearities 74ff polynomial oscillators 253, 262, 329 Poston's machine 118 predator-prey equations, see Lotka-Volterra predictability 11, 116 probability distribution functions 45, 183, 192, 394 projections 145
to `physical' variables 4, 6 proteins and fractals 64 pseudo-orbit 215 P-type equations 297 examples 299ff
quadratures 74 qualitative aspects 14 qualitative universality 158ff quantitative universality 147, 160ff quantum uncertainty 479 phenomena 498 quarter period 407 quasi-linear system 7 quasi-periodic functions 238ff, 254ff, 543 quasi-species 333 quiescent condition 457 quiescent-excited-refractory 492
random-map ensemble 94 Rannou's maps 93ff rationally independent frequencies 238, 249 Rayleigh equation 272 Rayleigh-Benard convection 163, 141 reaction-diffusion system 306, 345, 467 recurrence
Fermi-Pasta-Ulam 260, 348 theorems 52ff recurrent point 381 reductionism 430, 505ff refining sequence 105 refractionless potential 381 relaxation oscillations 288ff; Appendix 1, 416, 5, 212ff, 226
relativistic dynamics 242, 299
631
repellors 100 replicator equations 338ff, 341 resonance, nonlinear 310, 421 resonance-overlap 67ff, 72 resonant coupling 319 'Reversible' CA, see invertible CA Riccati equation 74, 75, 101, 299, 372, 415, 416 Riemann invariants 358ff RLC circuit 356 Rossler's models 32, 198ff Lyapunov exponents 207 rotation number (Poincare) 200, 333, 11, 41, 62, 249, 537
round-off errors 212, 495 Rubel's theorem (universal differential equation) 33 Ruelle-Takens strange attractor 25 rule number, CA 463 Runge-Kutta integration method, Appendix H, 414 Russell's 'soliton' observation 348 saddle connection 301, 303 saddle point 229, 233, 235, 261 saddle-cycle 133, 134, 183 saddle-focus 132 saddle-node 88, 229, 244, 300, 130, 137, 138, 199
Sawada-Kotera equation 405 scaling 60ff, 160ff, 187ff
Schrodinger equation 8, 373, 374, 388 (inverse scattering method) derivative nonlinear 404 nonlinear 390, 403, 410 pairs 406 Schrodinger's harmonic lattice solution 353 Schwarzian derivative 148; Appendix D, 396 screw-type chaos 208 secular terms 265 self-avoiding (random) walk 64 self-deterministic 145 self-exciting systems, see autocatalytic self-reproduction 455, 462, 482ff, 509 self-similarity 61ff, 361, 31
semi-periodic sequence 165, 119, 440 semi-stable points 201 sensitivity to initial conditions 182, 185, 19, 151, 190, 343, 461, 468, 470 separatrices 229, 234, 235, 259ff, 350ff, 44 sets (various types) Appendix A, 376 shadowing orbit 215 Sharkovsky's theorem 168, 178 Shaw's variant of forced van der Pol oscillator 340, 21, 531 shift dynamics 472, 542 shockwave, displacement 355, 363, 366 simplex space 337
632
Index
sine-Gordon equation 370, 390ff, 405, 408, 415, 418, 423 Lamb's solution 392 Singer's theorem 149, 398 singular perturbations 74, 288, 294, 295, 299; Appendix 1, 416 singular points, see fixed points, bifurcations, Poincare-Bendixson theorem singular solutions 40, 78ff, 294
Sinh-Gordon equation 408 situation, delay 98 slowly varying variables 267, 320, 69, 212, 226 Smale's horseshoe map, Appendix K, 539 Smale's theorem 342 Smale-Turing morphogenesis 311ff, 342 small divisors 63, 239, 544ff soft excitations 309, 314 soft nonlinearities 253, 256, 258, 310, 349ff solitaires 348ff, 357 solitons 267, 268, 280, 368ff
breather mode 398 cellular automata 480 center of mass 385 coupled lattice maps 450, 453 Hirota's solutions 398ff instantons 423 kink, anti-kink 393, 417 Landau-Lifshitz subset 385 one soliton 382, 393 pure 381ff resonance interactions 421ff Toda 281ff topological 393 two soliton 383, 394ff virtual 422 (0,4)npulses 418 solvable by quadradures 74 spheres 25, 27, 30, 248 spiral (focus) 229 stabilities: Lyapunov, Poincare, Lagrange 41, 42 asymptotic 41 exchange at bifurcations 90ff fixed points of a map in R1, 150, 153 in R2 227ff global (structural) 102ff, 300 multi 330, 344 orbital 41 Thom 103 uniform 41 stable manifold 46ff, 130ff, 165ff standard map 33ff, 68, 86, 93 statistical mechanics 8, 16, 394, 341; see lattice
thermal conductivity; Fermi-Pasta-Ulam stellarator 36 'sticky-island' effect 237 Stocker and Haag model 299; Appendix I, 416
strange attractor 186, 342, 347, 22ff, 29, 93, 151, 202, 204, 342, 531; also see curious set Lyapunov exponents, dimension 217ff strange basins 347 stretching and folding 180, 339, 342, 201, 205, 215ff, 327
stroboscopic record 318, 356ff structural stability 19, 102ff, 147, 380 and computability 103, 246 Sturm-Liouville equation 74, 294 strut dynamics 316 subcritical bifurcation 95, 97, 118, 132, 276, 304, 178
subharmonic bifurcations, see period doubling subharmonic resonances 315ff, 345, 346, 348ff, 357ff
supercritical bifurcation 97 (see pitchfork bifurcation) superstable 2"-cycle 156 surface of section, see Poincare suspension of maps 58, 206 symbolic dynamics 173; see Bernoulli sequences symmetry-breaking 344, 345, 349ff, 354, 181ff synergetics, examples 158ff, 187ff, 212, 217ff, 302, 303ff, 333, 342, 345, 350ff, 382, 99, 138ff, 162ff, 259ff, 281ff, 292ff, 437ff, 464ff, 487ff, 349
synthesis-analysis 43ff swallowtail catastrophe 115, 116 taffey machine 215ff
tangent bifurcation 169ff; see semistable points Taylor vortices 208, 327ff tent map 148, 181, 184, 207, 107, 117 suspension 206ff Thom, Rene, xv, 14, 134ff Thom's perspectives 134 Thom's theorem 116 Toda equation 275, 284
Flaschka-Henon-Manakov constants 284 Hamiltonian 74 lattices 274, 281ff, 304 potential 79, 273 solitary waves 274ff topological concepts 14, Appendix B, 382 topological entropy 101, 105, 117 topological equivalence 22, 58, 384 topological invariant 23, 384 topological orbital equivalence 18, 21, 24, 53, 231, 392, 342 topological solitons 393 topologically conjugate 58, 181 topology Appendix B, 382 tori 25, 28, 30, 55, 104, 249, 283, 355, 10ff, 20, 134, 248ff
Index toroidal space (solid torus) 318, 2, 39 torus knots 30, 318, 355, 186, 321 totalistic CA 464 transcritical bifurcation 97 transient chaos, see chaos transition function 457 transverse manifold 54 'Turbulence' 329, 440, 444, 451ff, 478 chemical 449ff Turing 308 Turing diffusive coupling 227, 241 Turing instability 240ff, 304ff Turing machine 483, 488 Turing-Smale analysis 311ff twist map 41, 65, 67 Ulam xiv, 2, 181, 182; see Fermi-Ulam ultra(sub)harmonics 315, 318 umbilic catastrophes 116 unbounded solutions 34, 42, 46; see explosive uncertainty exponent, fraction 175 uncertainty, macroscopic 104 quantum-like 479 uniqueness of inversions 38; see implicit function theorem uniqueness of solutions 30, 33, 38, 78 universal circuit (Khaikin) 214 universal differential equations 33 universal scaling 160ff universal sequences 147, 158ff, 360 unpredictability It, 210f; see sensitivity to initial conditions unstable manifold, see stable manifolds upside-down pendulum 349f1', 361 U-sequences 158, 360
'Value' of dynamics 519 van der Pol xiii, 2 van der Pol equation 272, 323ff, 315
633 Cartwright-Littlewood, Levinson, Levi studies 332, 16ff, Appendix J, 529 Duff ing 282, 303; generalized 307
forced 332, 16; Shaw's variant 340, 21 van der Pol and van der Mark experiment 330, 19, 529
van der Pol variables 324 van der Waal equation 112 variational equations (Poincare) 191, 232, 252 velocity vector field 53 violin string 290 virial theorem 340 viscous dynamics (see gradient systems) von Neumann xiv, 2, 515
von Neumann neighborhood 482 von Neumann's questions 453ff, 499 vortex (center) 229 vortices, fluid 327ff
walking stick region 205 wandering point 381 Weierstrass fractal curves 29 wigwam 116 wild, geometrically 26 winding number 200, 11 windows 166, 173, 353, 359, 361, 32, 187ff, 196, 207
Wintner's condition 34 Wolfram's CA classifications 464, 467, 468 Wronskian, see Jacobian
Yajima-Oikawa-Satsuma equation 421 Yajima-Oikawa-Satsuma resonance 421ff Zabusky and Kruskal 348, 3518, 367ff Zakharov and Shabat's solution 390, 422 Zeeman's catastrophe machine 119, 123 Zeitgeber 323
