The physics of chaos in Hamiltonian systems

George M. Zaslavsky1860947956, 9781860947957, 9781860948619

This book aims to familiarize the reader with the essential properties of the chaotic dynamics of Hamiltonian systems by avoiding specialized mathematical tools, thus making it easily accessible to a broader audience of researchers and students. Unique material on the most intriguing and fascinating topics of unsolved and current problems in contemporary chaos theory is presented. The coverage includes: separatrix chaos; properties and a description of systems with non-ergodic dynamics; the distribution of Poincaré recurrences and their role in transport theory; dynamical models of the Maxwell s Demon, the occurrence of persistent fluctuations, and a detailed discussion of their role in the problem underlying the foundation of statistical physics; the emergence of stochastic webs in phase space and their link to space tiling with periodic (crystal type) and aperiodic (quasi-crystal type) symmetries. This second edition expands on pseudochaotic dynamics with weak mixing and the new phenomenon of fractional kinetics, which is crucial to the transport properties of chaotic motion. The book is ideally suited to all those who are actively working on the problems of dynamical chaos as well as to those looking for new inspiration in this area. It introduces the physicist to the world of Hamiltonian chaos and the mathematician to actual physical problems.The material can also be used by graduate students.

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Table of contents :
CONTENTS……Page 10
PREFACE TO THE FIRST EDITION……Page 6
PREFACE TO THE SECOND EDITION……Page 9
1.1 Coexistence of the Dynamical Order and Chaos……Page 14
1.2 The Standard Map (Kicked Rotator)……Page 16
1.3 The Web-Map (Kicked Oscillator)……Page 21
1.4 Perturbed Pendulum……Page 27
1.5 Perturbed Oscillator……Page 30
1.6 Billiards……Page 32
Conclusions……Page 34
2.1 Nonlinear Resonance and Chain of Islands……Page 36
2.2 Overlapping of Resonances……Page 41
2.3 The Separatrix Map……Page 43
2.4 Stochastic Layer……Page 48
2.5 Hidden Renormalisation Group Near the Separatrix……Page 54
2.6 Renormalisation of Resonances……Page 63
2.7 Stochastic Layer of the Standard Map……Page 64
Conclusions……Page 67
3.1 Non-universality of the Scenario……Page 69
3.2 Collapsing Islands……Page 76
3.3 Blinking Islands……Page 82
3.4 Boundary Islands……Page 87
3.5 Self-Similar Set of Islands……Page 89
3.6 Ballistic Mode Islands……Page 99
3.7 General Comments About the Islands……Page 101
Conclusions……Page 102
4.1 Beyond the KAM-Theory……Page 104
4.2 Web-Tori……Page 106
4.3 Width of the Stochastic Web……Page 114
4.4 Transition from the KAM-Tori to Web-Tori……Page 116
Conclusions……Page 119
5.1 Fractal Dynamics……Page 121
5.2 Generalised Fractal Dimension……Page 124
5.3 Renormalisation Group and Generalised Fractal Dimension……Page 126
5.4 Multi-Fractal Spectra……Page 127
Conclusions……Page 132
6.1 Poincare Recurrences……Page 133
6.2 Poissonian Distribution of Recurrences……Page 135
6.3 Non-Ergodicity, Stickiness and Quasi-Traps……Page 138
6.4 Renormalisation Formulas for the Exit Time Distribution……Page 144
6.5 Fractal Time……Page 148
6.6 Fractal and Multi-Fractal Recurrences……Page 150
6.7 Multi-Fractal Space-Time and Its Dimension Spectrum……Page 154
6.8 Critical Exponent for the Poincare Recurrences……Page 157
6.9 Rhombic Billiard……Page 158
Conclusions……Page 161
7.1 The Dynamical Foundation of Statistical Physics……Page 163
7.2 Fractal Traps and Maxwell’s Demon……Page 165
7.3 Coupled Billiards……Page 171
7.4 Contacted Cassini-Sinai Billiards……Page 178
7.5 Weak Mixing and Stickiness……Page 185
7.6 Persistent Fluctuations……Page 186
Conclusions……Page 189
8.1 Stochastic Webs……Page 191
8.2 Stochastic Web with Quasi-Crystalline Symmetry……Page 194
8.3 Stochastic Web Skeleton……Page 198
8.4 Symmetries and Their Dynamical Generation……Page 208
8.5 The Width of the Stochastic Web……Page 212
Conclusions……Page 217
COLOUR PLATES (C.1 C.8)……Page 218
9.1 General Remarks……Page 227
9.2 Four-Dimensional Map for the Motion in Magnetic Field……Page 228
9.3 Multi-Web Structures in the Phase Space……Page 232
9.4 Equilibrium of the Atomic Chains……Page 237
9.5 Discretisation……Page 241
Conclusions……Page 244
10.1 Fokker-Planck-Kolmogorov (FPK) Equation……Page 246
10.2 Transport for the Standard Map and Web-Map……Page 251
10.3 Dynamics in the Potential with q-Fold Symmetry……Page 255
10.4 More Examples of the Anomalous Transport……Page 258
10.5 Levy Processes……Page 261
10.6 The Weierstrass Random Walk……Page 264
Conclusions……Page 266
11.1 Fractional Generalisation of the Fokker-Planck-Kolmogorov Equation (FFPK)……Page 267
11.2 Evolution of Moments……Page 272
11.3 Method of the Renormalisation Group for Kinetics (RGK)……Page 274
11.4 Complex Exponents and Log-Periodicity……Page 279
Conclusions……Page 282
12.1 Definitions……Page 284
12.2 Billiards with Pseudochaotic Dynamics……Page 286
12.3 Filamented Surfaces……Page 293
12.4 Bar-in-Square Billiard……Page 295
12.5 Renormalisation Group Equation for Recurrences……Page 298
12.6 Recurrences in the Multi-Bar-Billiard……Page 301
Conclusions……Page 303
Appendix 1 THE NONLINEAR PENDULUM……Page 305
Appendix 2 SOLUTION TO THE RENORMALISATION TRANSFORM EQUATION……Page 309
Appendix 3 FRACTIONAL INTEGRO-DIFFERENTIATION……Page 311
Appendix 4 FORMULAS OF FRACTIONAL CALCULUS……Page 317
REFERENCES……Page 322
INDEX……Page 334