Hamiltonian Chaos Beyond the KAM Theory : Dedicated to George M. Zaslavsky (1935-2008)

by Luo, Albert C. J.; Afraimovich, Valentin

• ISBN13: 9783642127175
• ISBN10: 3642127177
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-03-07
• Publisher: Springer Verlag

Summary

'œHamiltonian Chaos Beyond the KAM Theory: Dedicated to George M. Zaslavsky (1935'”2008)' covers the recent developments and advances in the theory and application of Hamiltonian chaos in nonlinear Hamiltonian systems. The book is dedicated to Dr. George Zaslavsky, who was one of three founders of the theory of Hamiltonian chaos. Each chapter in this book was written by well-established scientists in the field of nonlinear Hamiltonian systems. The development presented in this book goes beyond the KAM theory, and the onset and disappearance of chaos in the stochastic and resonant layers of nonlinear Hamiltonian systems are predicted analytically, instead of qualitatively. The book is intended for researchers in the field of nonlinear dynamics in mathematics, physics and engineering. Dr. Albert C.J. Luo is a Professor at Southern Illinois University Edwardsville, USA. Dr. Valentin Afraimovich is a Professor at San Luis Potosi University, Mexico.

Author Biography

Dr. Albert C.J. Luo is a Professor at Southern Illinois University Edwardsville, USA. Dr. Valentin Afraimovich is a Professor at San Luis Potosi University, Mexico.

Table of Contents

Stochastic and Resonant Layers in Nonlinear Hamiltonian Systems	p. 1
Introduction	p. 1
Stochastic layers	p. 4
Geometrical description	p. 5
Approximate criterions	p. 9
Resonant layers	p. 17
Layer dynamics	p. 19
Approximate criterions	p. 23
A periodically forced Duffing oscillator	p. 27
Approximate predictions	p. 28
Numerical illustrations	p. 34
Discussions	p. 47
References	p. 48
A New Approach to the Treatment of Separatrix Chaos and Its Applications	p. 51
Introduction	p. 52
Heuristic results	p. 52
Mathematical and accurate physical results	p. 53
Numerical evidence for high peaks in ¿E(¿f) and their rough estimations	p. 54
Accurate description of the peaks and of the related phenomena	p. 54
Basic ideas of the approach	p. 55
Single-separatrix chaotic layer	p. 60
Rough estimates. Classification of systems	p. 61
Asymptotic theory for systems of type I	p. 62
Asymptotic theory for systems of type II	p. 71
Estimate of the next-order corrections	p. 79
Discussion	p. 83
Double-separatrix chaos	p. 85
Asymptotic theory for the minima of the spikes	p. 89
Theory of the spikes' wings	p. 108
Generalizations and applications	p. 114
Enlargement of a low-dimensional stochastic web	p. 117
Slow modulation of the wave angle	p. 119
Application to semiconductor superlattices	p. 120
Discussion	p. 121
Conclusions	p. 121
Appendix	p. 122
Lower chaotic layer	p. 122
Upper chaotic layer	p. 137
References	p. 138
Hamiltonian Chaos and Anomalous Transport in Two Dimensional Flows	p. 143
Introduction	p. 143
Point vortices and passive tracers advection	p. 145
Definitions	p. 145
Chaotic advection	p. 146
A system of point vortices	p. 148
Definitions	p. 148
Dynamics of systems with two or three point vortices	p. 150
Dynamics of two vortices	p. 150
Dynamics of three vortices	p. 151
Vortex collapse and near collapse dynamics of point vortices	p. 152
Vortex collapse	p. 153
Vortex dynamics in the vicinity of the singularity	p. 153
Chaotic advection and anomalous transport	p. 155
A brief history	p. 156
Definitions	p. 157
Anomalous transport in incompressible flows	p. 160
Tracers (passive particles) dynamics	p. 162
Transport properties	p. 169
Origin of anomalous transport	p. 174
General remarks	p. 180
Beyond characterizing transport	p. 180
Chaos of field lines	p. 180
Local Hamiltonian dynamics	p. 180
An ABC type flow	p. 182
Targeted mixing in an array of alternating vortices	p. 185
Conclusion	p. 189
References	p. 189
Hamiltonian Chaos with a Cold Atom in an Optical Lattice	p. 193
Short historical background	p. 194
Introduction	p. 195
Semiclassical dynamics	p. 196
Hamilton-Schrödinger equations of motion	p. 196
Regimes of motion	p. 198
Stochastic map for chaotic atomic transport	p. 200
Statistical properties of chaotic transport	p. 202
Dynamical fractals	p. 204
Quantum dynamics	p. 208
Dressed states picture and nonadiabatic transitions	p. 209
Wave packet motion in the momentum space	p. 211
Quantum-classical correspondence and manifestations of dynamical chaos in wave-packet atomic motion	p. 219
References	p. 221
Using Stochastic Webs to Control the Quantum Transport of Electrons in Semiconductor Superlattices	p. 225
Introduction	p. 226
Superlattice structures	p. 228
Semiclassical electron dynamics	p. 231
Electron drift velocity	p. 234
Current-voltage characteristics: theory and experiment	p. 236
Electrostatics and charge domain structure	p. 238
Tailoring the SL structure to increase the number of conductance resonances	p. 240
Energy eigenstates and Wigner functions	p. 243
Summary and outlook	p. 247
References	p. 249
Chaos in Ocean Acoustic Waveguide	p. 255
Introduction	p. 255
Basic equations	p. 258
Parabolic equation approximation	p. 259
Geometrical optics. Hamiltonian formalism	p. 260
Modal representation of the wave field	p. 262
Ray-based description of normal modes	p. 263
Ray chaos	p. 264
Statistical description of chaotic rays	p. 264
Environmental model	p. 266
Wiener process approximation	p. 267
Distribution of ray parameters	p. 269
Smoothed intensity of the wave field	p. 270
Ray travel times	p. 272
Timefront	p. 272
Statistics of ray travel times	p. 274
Modal structure of the wave field under conditions of ray chaos	p. 278
Coarse-grained energy distribution between normal modes	p. 278
Transient wave field	p. 280
Conclusion	p. 287
References	p. 289
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