Applied Mathematical Sciences Volume 38

Editors F. John J. E. Marsden L. Sirovich

Advisors H. Cabannes M. Ghil J. K. Hale J. Keller J. P. LaSalle G. B. Whitham

Applied Mathematical Sciences

1. John: Partial Differential Equations, 4th ed. (cloth) 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. (cloth) 4. Percus: Combinatorial Methods. 5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-Linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space.

10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost

Periodic Solutions. 15. Braun: Differential Equations and Their Applications, 2nd ed. (cloth) 16. Lefschetz: Applications of Algebraic Topology. 17. CollatzlWetterling: Optimization Problems. 18. Grenander: Pattern SyntheSis: Lectures in Pattern Theory, Vol. I. 19. Marsden/McCracken: The Hopf Bifurcation and Its Applications. 20. Driver: Ordinary and Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. (cloth) 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and

Unconstrained Systems. 27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29. de Veubeke: A Course in ElastiCity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturm ian Theory for Ordinary Differential Equations. 32. Meis/Marcowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Pertubation Methods in Applied Mathematics. (cloth) 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Kallen: Dynamic Meteorology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Stochastic Motion. (cloth)

A. J. Lichtenberg M. A. Lieberman

Regular and Stochastic Motion

With 140 Figures

Springer Science+Business Media, LLC

A. J. Lichtenberg M. A. Lieberman Department of Electrical Engineering and Computer Sciences

University of California Berkeley, CA 94720 USA

Editors

F. John Courant Institute of Mathematical Sciences

New York University New York, NY 10012 USA

J. E. Marsden Department of

Mathematics University of California Berkeley, CA 94720 USA

L. Sirovich Division of Applied Mathematics

Brown University Providence, RI 02912 USA

AMS Subject Classifications: 70Kxx, 60Hxx, 34Cxx, 35Bxx

Library of Congress Cataloging in Publication Data Lichtenberg, Allan J.

Regular and stochastic motion. (Applied mathematical sciences; v. 38) Bibliography: p. Includes index. 1. Nonlinear oscillations. 2. Stochastic processes.

3. Hamiltonian systems. I. Lieberman, M. A (Michael A) II. Title. III. Series: Applied mathematical sciences (Springer-Verlag New York Inc.); v. 38. QA1.A647 vol. 38 [QA867.5] 510s [531'.322] 82-19471

©1983 by Springer Science+Business Media New York

Originally published by Springer-Verlag N ew York Heidelberg Berlin in 1983 Softcover reprint of the hardcover 1st edition 1983

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media. LLC.

Typeset by Computype, Inc., St. Paul, MN.

9 8 7 6 5 432 1

978-1-4757-4259-6 ISBN ISBN 978-1-4757-4257-2 (eBook) DOI 10.1007/978-1-4757-4257-2

To Elizabeth and Marlene

Preface

This book treats stochastic motion in nonlinear oscillator systems. It describes a rapidly growing field of nonlinear mechanics with applications to a number of areas in science and engineering, including astronomy, plasma physics, statistical mechanics and hydrodynamics. The main em­phasis is on intrinsic stochasticity in Hamiltonian systems, where the stochastic motion is generated by the dynamics itself and not by external noise. However, the effects of noise in modifying the intrinsic motion are also considered. A thorough introduction to chaotic motion in dissipative systems is given in the final chapter.

Although the roots of the field are old, dating back to the last century when Poincare and others attempted to formulate a theory for nonlinear perturbations of planetary orbits, it was new mathematical results obtained in the 1960's, together with computational results obtained using high speed computers, that facilitated our new treatment of the subject. Since the new methods partly originated in mathematical advances, there have been two or three mathematical monographs exposing these developments. However, these monographs employ methods and language that are not readily accessible to scientists and engineers, and also do not give explicit tech­niques for making practical calculations.

In our treatment of the material, we emphasize physical insight rather than mathematical rigor. We present practical methods for describing the motion, for determining the transition from regular to stochastic behavior, and for characterizing the stochasticity. We rely heavily on numerical computations to illustrate the methods and to validate them.

The book is intended to be a self contained text for physical scientists and engineers who wish to enter the field, and a reference for those

V111 Preface

researchers already familiar with the methods. It may also be used as an advanced graduate textbook in mechanics. We assume that the reader has the usual undergraduate mathematics and physics background, including a mechanics course at the junior or senior level in which the basic elements of Hamiltonian theory have been covered. Some familiarity with Hamiltonian mechanics at the graduate level is desirable, but not necessary. An exten­sive review of the required background material is given in Sections 1.2 and 1.3.

The core ideas of the book, concerning intrinsic stochasticity in Hamilto­nian systems, are introduced in Section 1.4. Our subsequent exposition in Chapters 2-6 proceeds from the regular to the stochastic. To guide the reader here, we have "starred" (*) the sections in which the basic material appears. These "starred" sections form the core of our treatment. Section 2.4a on secular perturbation theory is of central importance. The core material has been successfully presented as a 30 lecture-hour graduate course at Berkeley.

In addition to the core material, other major topics are treated. The effects of external noise in modifying the intrinsic motion are presented in Section 5.5, (using the results of Section 5.4d) for two degrees of freedom, in Section 6.3 for more than two degrees of freedom, and an application is given in Section 6.4. Our description of dissipative systems in Chapter 7 can be read more-or-less independently of our treatment of Hamiltonian sys­tems. For studying the material of Chapter 7, the introduction in Section 1.5 and the material on surfaces of section in Section 1.2b and on Liapunov exponents in Sections 5.2b and 5.3 should be consulted. The topic of period doubling bifurcations is presented in Section 7.2b, 7.3a and Appendix B (see also Section 3.4d). Other specialized topics such as Lie perturbation methods (Section 2.5), superconvergent perturbation methods (Section 2.6), aspects of renormalization theory (Sections 4.3, 4.5), non-canonical meth­ods (Section 2.3d), global removal of resonances (Section 2.4d and part of 2.5c), variational methods (Sections 2.6b and 4.6), and modulational diffu­sion (Section 6.2d) can generally be deferred until after the reader has obtained some familiarity with the core material.

This book has been three and a half years in the writing. We have received encouragement from many friends and colleagues. We wish to acknowledge here those who reviewed major sections of the manuscript. The final draft has been greatly improved by their comments. Thanks go to H. D. I. Abarbanel, J. R. Cary, B. V. Chirikov, R. H. Cohen, D. F. Escande, J. Ford, J. Greene, R. H. G. Helleman, P. J. Holmes, J. E. Howard, O. E. Lanford, D. B. Lichtenberg, R. Littlejohn, B. McNamara, H. Motz, C. Sparrow, J. L. Tennyson and A. Weinstein. Useful comments have also been received from G. Casati, A. N. Kaufman, I. C. Percival and G. R. Smith. We are also pleased to acknowledge the considerable influ­ence of the many published works in the field by B. V. Chirikov. Many of

Preface ix

the ideas expressed herein were developed by the authors while working on grants and contracts supported by the National Science Foundation, the Department of Energy, and the Office of Naval Research. One of the authors (A.J.L.) acknowledges the hospitality of St. Catherine's College, Oxford, and one of the authors (M.A.L.) acknowledges the hospitality of Imperial College, London, where much of the manuscript was developed.

Contents

List of Symbols

Chapter 1

Overview and Basic Concepts

1.1. An Introductory Note * 1.2. Transformation Theory of Mechanics

* 1.2a. Canonical Transformations * 1.2b. Motion in Phase Space *1.2c. Action-Angle Variables

1.3. Integrable Systems *1.3a. One Degree of Freedom

1.3b. Linear Differential Equations 1.3c. More than One Degree of Freedom

*1.4. Near-Integrable Systems *1.4a. Two Degrees of Freedom *l.4b. More than Two Degrees of Freedom

1.5. Dissipative Systems l.5a. Strange Attractors 1.5b. The Lorenz System

Chapter 2

Canonical Perturbation Theory

2.1. Introduction 2.1 a. Power Series

* Starred sections indicate core material.

xvii

1 7 7

12 20 23 23 28 31 41 42 54 56 57 59

63

63 66

xii

2.1 b. Asymptotic Series and Small Denominators 2.1 c. The Effect of Resonances

*2.2. Classical Perturbation Theory *2.2a. One Degree of Freedom *2.2b. Two or More Degrees of Freedom

2.3. Adiabatic Invariance *2.3a. Introduction and Basic Concepts *2.3b. Canonical Adiabatic Theory *2.3c. Slowly Varying Harmonic Oscillator 2.3d. Noncanonical Methods

2.4. Secular Perturbation Theory *2.4a. Removal of Resonances *2.4b. Higher-Order Resonances *2.4c. Resonant Wave-Particle Interaction 2.4d. Global Removal of Resonances

2.5. Lie Transformation Methods 2.5a. General Theory 2.5b. Deprit Perturbation Series 2.5c. Adiabatic Invariants

2.6. Superconvergent Methods 2.6a. Kolmogorov's Technique 2.6b. Singly Periodic Orbits

Chapter 3

Mappings and Linear Stability

*3.1. Hamiltonian Systems as Canonical Mappings *3.la. Integrable Systems *3.lb. Near-Integrable Systems *3.lc. Hamiltonian Forms and Mappings

*3.2. Generic Behavior of Canonical Mappings *3.2a. Irrational Winding Numbers and KAM Stability *3.2b. Rational Winding Numbers and Their Structure *3.2c. Complete Description of a Nonlinear Mapping *3.2d. A Numerical Example

3.3. Linearized Motion 3.3a. Eigenvalues and Eigenvectors of a Matrix

*3.3b. Two-Dimensional Mappings *3.3c. Linear Stability and Invariant Curves

*3.4. Fermi Acceleration *3.4a. Physical Problems and Models *3.4b. Numerical Results *3.4c. Fixed Points and Linear Stability *3.4d. Bifurcation Phenomena *3.4e. Hamiltonian Formulation

*3.5. The Separatrix Motion *3.5a. Driven One-Dimensional Pendulum *3.5b. The Separatrix Mapping

Contents

68 70 71 71 76 85 85 88 92 94

100 101 107 112 119 123 125 126 130 138 141 143

150

151 151 155 157 158 159 168 172 176 178 179 183 186 190 191 194 198 201 205 206 208 211

Contents

Chapter 4

Transition to Global Stochasticity

*4.1. Introduction *4.1a. Qualitative Description of Criteria *4.lb. The Standard Mapping

*4.2. Resonance Overlap *4.2a. Rationale for Criteria *4.2b. Calculation of Overlap Criteria

4.3. Growth of Second-Order Islands 4.3a. Elliptic Fixed Points 4.3b. The Separatrix

*4.4. Stability of High-Order Fixed Points *4.4a. Basic Elements of Greene's Method *4.4b. Numerical Evaluation

4.5. Renormalization for Two Resonances 4.6. Variational Principle for KAM Tori

*4.7. Qualitative Summary and Conclusions

Chapter 5

Stochastic Motion and Diffusion

5.1. Introduction 5.2. Definitions and Basic Concepts

*5.2a. Ergodicity *5.2b. Liapunov Characteristic Exponents 5.2c. Concepts of Stochasticity 5.2d. Randomness and Numerical Errors

5.3. Determination of Liapunov Exponents and KS Entropy 5.3a. Analytical Estimates 5.3b. Numerical Methods

5.4. Diffusion in Action Space *5.4a. The Fokker-Planck Equation *5.4b. Transport Coefficients *5Ac. Steady-State and Transient Solutions

5Ad. Higher-Order Transport Corrections 5.5. The Effect of Extrinsic Stochastic Forces

5.5a. Introduction 5.5b. Diffusion in the Presence of Resonances

Chapter 6

Three or More Degrees of Freedom

*6.1. Resonance in Multidimensional Oscillations *6.la. Geometflc Relations *6.1 b. Examples of Arnold Diffusion

xiii

214

214 215 218 226 226 227 232 232 236 239 239 243 249 256 257

259

259 260 260 262 268 273 277 279 280 285 286 289 291 293 300 300 303

309

309 310 316

xiv

6.2. Diffusion Rates along Resonances *6.2a. Stochastic Pump Diffusion Calculation 6.2b. Coupling Resonance Diffusion 6.2c. Many Resonance Diffusion 6.2d. Modulational Diffusion

6.3. Extrinsic Diffusion 6.3a. Resonance Streaming 6.3b. Diffusion of a Parameter

6.4. Diffusion in Toroidal Magnetic Fields 6.4a. Magnetic Islands 6.4b. Drift Surfaces and Diffusion in Static Fields 6.4c. Time-Varying Fields 6.4d. The Self-Consistent Problem

6.5. Many Degrees of Freedom

Chapter 7

Dissipative Systems

7.1. Simple and Strange Attractors 7.1 a. Basic Properties 7.1 b. Examples of Strange Attractors 7.lc. Geometric Properties of Strange Attractors

7.2. One-Dimensional Noninvertible Maps 7.2a. Basic Properties 7.2b. Periodic Behavior 7.2c. Chaotic Motion

7.3. Two-Dimensional Maps and Related Flows 7.3a. Period-Doubling Bifurcations 7.3b. Motion near a Separatrix 7.3c. Calculation of Invariant Distributions

7.4. The Fluid Limit 7.4a. Fourier Mode Expansions 7.4b. The Transition to Turbulence

Appendix A

Applications

A.I. Planetary Motion A.2. Accelerators and Beams A.3. Charged Particle Confinement A.4. Charged Particle Heating A.5. Chemical Dynamics A.6. Quantum Systems

Contents

322 322 330 333 335 344 345 351 356 357 363 366 372 375

380

380 381 386 392 396 396 399 411 422 422 426 434 442 443 447

453

453 455 457 459 461 462

Contents xv

Appendix B

Hamiltonian Bifurcation Theory 464

Bibliography 471

Author Index 483

Subject Index 489

List of Symbols

The major uses of symbols throughout the book are given first. Impor­tant special uses within a section are noted by section number. Minor uses do not appear. Scalars appear in italic type; vectors appear in boldface italic type; and tensors and matrices appear in boldface sans serif type.

a

A A A ~

b

B

B c c C d

D

e

coefficient; parameter; an' Fourier coefficient; ak' Fourier coefficient; ai' a2' continued fraction expansion amplitude; An' Fourier coefficient (2.6) vector potential linear transformation matrix part of generating function (3.1); ~m' Melnikov-Arnold inte­gral coefficient; constant; parameter; bn , Fourier coefficient; bij' coefficient matrix (3.3) magnetic field; friction coefficient; value of Jacobian (7.3); Bn , Fourier coefficient (2.6) coefficient matrix (3.3) velocity of light; constant; coefficient; parameter diagonalized coefficient (2.6) constant; parameter of one-dimensional map derivative; differential; parameter; fractal dimension (7.1); distance between neighboring trajectories (5.3) diffusion coefficient; total derivative (2.5); Melnikov distance (7.3); D*, normalized diffusion coefficient; DQL> quasilinear diffusion coefficient electric charge; base of natural logarithm; parameter; en' error (2.6); e, basis vector

xviii List of Symbols

E energy o complete elliptic integral of the second kind j function, forcing term; mean residue (4.4); invariant function

(5.2); j(sub), fast f function; mapping function F force coefficient in Hamiltonian; F\, F2 , generating functions §" elliptic integral of the first kind g function; gravity g function G nonlinearity coefficient in Hamiltonian § part of generating function (3.1) h function; hk' KS entropy H Hamiltonian; H o, value of Hamiltonian (energy); HO,H\,

zero order, first order parts of H, etc.; H, transformed H; Ii, transformed to rotating coordinates; H r , entropy (5.2)

% transformed H (l.3b) summation index, R; i(sub), ith coordinate

I invariant; transformed action; normalized momentum in standard map; 1\ '/2' involutions

I identity matrix g identity function on phase space j summation index J action; Jo,J\, etc., zero, first-order parts; J\,J2' components;

], transformed J; J, transformed to rotating coordinates :f :fn' Bessel function of the first kind k integer; summation index; (integer) period of fixed point

(3.4); amplitude or (subscripted) component of wave vector; perturbation amplitude (6.2); kT, thermal energy (6.4)

k wave vector K stochasticity parameter; transformed Hamiltonian (6.2); K2 ,

stochasticity parameter for separatrix map (4.3) % complete elliptic integral of the first kind I integer; summation index; angular momentum (1.3) L Lagrangian; Lie operator; Ls ' shear length (6.4) m integer; summation index; m\, m2 , etc., Fourier path integers

(5.4d) m integer vector M phase advance parameter in Fermi map; mass; number of

continuous derivatives (3.2); normalized potential amplitude (4.5); M(sub) magnitude

M Jacobian matrix '!)]L mapping n normalized time units; integer; summation index; n(sub),

step number n integer vector (3.1, 2.4)

List of Symbols XlX

N degrees of freedom; N(sub), Nth coordinate 0L noise power density o O(sub); unperturbed part; O(sub), linearized value (C) order of p momentum; integer; summation index (2.4); bifurcation fam­

ily (4.1) p vector momentum P probability distribution; invariant distribution; transformed

momentum; normalized perturbation amplitude (4.5) q coordinate; integer; Fourier transform variable (5.4d) Q Qo, ratio of driving frequency to linearized oscillator fre­

quency; inverse rotation number; Qjm, Fourier amplitude (2.6)

r integer; harmonic number; radius; parameter (1.5, 7.4) r vector position R axial ratio of phase space ellipse (F / G)I/2; residue; radius;

R(sub), resonance value; Ra , Rayleigh number (7.4) s integer; harmonic number; s(sub), slow; s(sub), secondary;

s(sub), stable; sx(sub), on separatrix S generating function F2 ; generalized stochasticity parameter

(4.5); scaling parameter (6.3b); shear parameter (6.4); S(sub), stochastic part (6.1)

t time T period; mapping; Lie evolution operator (2.5) ~ renormalization operator u normalized velocity; function of coordinates (1.2); periodic

function (3.2) U potential v velocity; function of coordinates (1.2); periodic function (3.2) V Hamiltonian perturbation amplitude; VI' V2 , potential am­

plitudes (4.5); Vk , Fourier harmonics of perturbation ampli­tude; Vp ' p dimensional volume (5.2)

w Lie generating function; deviation from separatrix energy HI tangent vector ~x (5.2, 5.3) W ~, transition probability; W( I), transition probability x coordinate; mapping variable; x·, value of x at extremum

(7.2) x vector mapping variable; position X phase space variable in differential equations; guiding center

coordinates (2.2); normalized amplitude (4.5); attractor (7.1); XI' Fourier component (7.2)

X eigenvector matrix y coordinate; mapping variable; nice variable (2.3) Y phase space variable in differential equations; guiding center

coordinates (2.2); normalized amplitude (4.5)

xx

z Z lx

p y r r 6

t (J

8 e

IC

h A A

'IT

II P a

"

List of Symbols

coordinate; nice coordinate (2.3) phase space variable in differential equations rotation number (frequency ratio); lXi' invariant (1.2); lx/,

golden mean (4.4); momentum variable (6.1); rescaling pa­rameter (7.2, 7.3) parameter; momentum variable (6.1) rescaling parameter (7.2) Fourier coefficient (2.4); flow (S.Sb) antisymmetric matrix (3.3) variation; 6w (or 6.1), frequency (or action) distance between resonances; rescaling parameter (7.1, 7.2); dissipation factor (7.3); 61( ), periodic delta function; 6( ), Dirac delta func­tion; 6ij> Kronecker delta function variation; fl.w (or fl.J), small increment in frequency (or action) about a zero-order value; fl.wmax (or fl.J max>, separatrix half-width in frequency (or action) perturbation strength; perturbation ordering parameter; small quantity tn' random variable (6.3b) angle; angle conjugate to an action vector angle 2'IT or I (3.4); temperature variable (7.4) rotational transform 2'ITlX

elliptic integral argument, thermal conductivity (7.4) eigenvalue; unit step (4.S); modulation amplitude (6.2d) matrix of eigenvalues Fourier harmonic; perturbation amplitude; volume contrac­tion rate magnetic moment; parameter; phase space measure (S.2); coupling coefficient (6.2); extrinsically diffusing parameter (6.3b) collision frequency; viscosity (7.4) elliptic integral argument; angle variable (3.2); random vari­able (6.3b) pi product gyroradius (PL in 6.4) Liapunov exponent; phase of eigenvalue; parameter; stan­dard deviation (S.5b) summation; surface of section normalized time; time interval; phase space density (1.2); £.t,

slow time; (2.2, 2.3); "D' diffusion time (6.3); "e' collision time (6.4) pendulum angle; angle conjugate to an action; phase angle; toroidal angle (6.4)

List of Symbols

x

n A A A a (') v [, ] C)

( )' const

< > { } (1 (") ( ) A ( ) .1

II ( ) I I II II sgn det Tr In exp div

xxi

potential; potential amplitude; transformed angle (1.3); <Po, potential amplitude constant angle; argument of Bessel function (4.3); angle in action space (6.3) phase angle; fixed angle parameter (3.1, 3.2); angle in action space (6.3); toroidal angle (6.4); stream function (7.4) radian frequency; wo, linear frequency; slowly varying fre­quency (2.5) driving frequency; cyclotron frequency scalar vector matrix or tensor partial derivative total derivative with respect to time gradient Poisson bracket transformed to new coordinates; step of a mapping; renor­malized quantity (4.5) derivative with respect to argument; renormalized quantity constant average part over angles oscillating part integral of oscillating part (2.3d) transformed to rotating coordinates; unit vector wedge operator (volume of parallelpiped) perpendicular (sub) parallel (sub) matrix magnitude norm sign of determinant of matrix trace of matrix natural logarithm exponential divergence