Hamiltonian chaos and fractional dynamics

Cover......Page 1
Hamiltonian Chaos and Fractional Dynamics......Page 4
PREFACE......Page 6
Contents......Page 10
Part 1 Chaotic Dynamics......Page 16
1.1 Hamiltonian equations......Page 18
1.2 Phase space dynamics......Page 20
1.3 Action–angle variable (one degree of freedom)......Page 23
Problems......Page 26
2.1 Pendulum......Page 28
2.2 Oscillations in the infinite potential well......Page 31
2.3 Magnetic moments......Page 32
2.4 Field line behaviour......Page 33
2.5 Hamiltonian equations for the ABC-flow......Page 35
Problems......Page 37
3.1 The Liouville–Arnold theorem on integrability......Page 38
3.2 Consequences of the integrability......Page 40
3.3 Non-integrability and the Kozlov condition......Page 41
3.4 Resonances......Page 43
3.5 Non-linear resonance and chain of islands......Page 44
3.6 Kolmogorov–Arnold–Moser (KAM) theory......Page 47
Problems......Page 49
4.1 Natural measure......Page 52
4.2 Ergodicity, mixing, and weak mixing......Page 54
4.3 Local instability and Lyapunov exponents......Page 57
4.4 Hyperbolic systems......Page 61
4.5 Entropy of dynamical systems......Page 63
4.6 Definition of chaotic dynamics......Page 68
4.7 Chirikov resonance overlapping criteria......Page 69
Notes......Page 70
Problems......Page 71
5.1 Mapping the dynamics......Page 72
5.2 Universal and standard map......Page 75
5.3 Web map (kicked oscillator)......Page 79
5.4 Kepler map......Page 83
Notes......Page 85
Problems......Page 86
6.1 Description of models......Page 88
6.2 Separatrix map......Page 91
6.3 The stochastic layer......Page 93
6.4 The stochastic layer of the standard map......Page 96
6.5 Hidden renormalization group near the separatrix......Page 98
6.6 Renormalization of resonances......Page 104
6.7 Hidden renormalization for coupled oscillators......Page 106
Notes......Page 109
Problems......Page 110
7.1 Stochastic webs......Page 112
7.2 Stochastic webs with quasi-crystalline symmetry......Page 114
7.3 Stochastic web skeleton......Page 117
7.4 Symmetries and their dynamical generation......Page 125
7.5 Width of the stochastic web......Page 129
7.6 Symmetry in art and nature......Page 132
Notes......Page 137
Problems......Page 138
8.1 Small non-linearity......Page 140
8.2 Web-Tori......Page 142
8.3 Width of the stochastic web......Page 149
8.4 Transition from KAM-Tori to Web-Tori......Page 150
Problems......Page 153
9.1 Topological non-universality of chaos......Page 154
9.2 Examples with billiards......Page 157
9.3 Accelerator mode islands......Page 158
9.4 Ballistic mode islands......Page 166
9.5 Cantori......Page 167
9.6 Sticky domains and escapes......Page 169
Notes......Page 170
Problems......Page 171
Part 2 Fractality of Chaos......Page 172
10.1 Fractal dynamics......Page 174
10.2 Generalized fractal dimension......Page 176
10.3 Renormalization group and generalized fractal dimension......Page 177
10.4 Multifractal spectra......Page 179
10.5 Thermodynamic interpretation......Page 182
10.6 Complex dimension and log-periodicity......Page 184
Notes......Page 185
Problems......Page 186
11.1 Poincaré theorem on recurrences......Page 188
11.2 Recurrence time distributions and Kac lemma......Page 189
11.3 Distribution of recurrences in uniform mixing......Page 192
11.4 More asymptotics on recurrences......Page 195
Problems......Page 201
12.1 Definition of the dynamical trap......Page 202
12.2 Hierarchical-islands trap (HIT)......Page 204
12.3 Renormalization for the exit time distribution......Page 208
12.4 Stochastic layer trap......Page 211
Notes......Page 213
13.1 Fractal time......Page 216
13.2 Fractal and multifractal recurrences......Page 219
13.3 Multifractal space-time and its dimension spectrum......Page 222
13.4 Critical exponent for the Poincaré recurrences......Page 224
Problems......Page 226
Part 3 Chaotic kinetics......Page 228
14.1 Time scales......Page 230
14.2 Fokker–Planck–Kolmogorov (FPK) equation......Page 232
14.3 Detailed balance principle......Page 235
14.4 Solutions and normal transport......Page 236
14.5 Growth of entropy......Page 237
14.6 Kolmogorov conditions and conflict with dynamics......Page 238
14.7 Truncated distributions......Page 240
Notes......Page 241
Problems......Page 242
15 Lévy process, Lévy flights, and Weierstrass random walk......Page 244
15.1 Lévy distribution......Page 245
15.2 Lévy process......Page 246
15.3 Poincaré recurrences and Feller’s theorems......Page 249
15.4 Lévy flights and conflict with dynamics......Page 250
15.5 Weirstrass random walks (WRW)......Page 255
Notes......Page 258
Problems......Page 259
16.1 Derivation of FKE......Page 260
16.2 Conditions for the FKE......Page 264
16.3 Evolution of moments (transport)......Page 265
16.4 Conflict with dynamics......Page 267
16.5 Dynamical origin of critical exponents......Page 268
16.6 Principles of simulations......Page 272
Notes......Page 273
Problems......Page 274
17.1 Space-time scalings......Page 276
17.2 Log-periodicity......Page 278
17.3 Duality of the dynamics and the origin of multi-fractality......Page 280
17.4 Multifractional kinetics......Page 282
Notes......Page 287
18.1 Solutions to FKE (series)......Page 288
18.2 Solutions to FKE (separation of variables)......Page 290
18.3 Continuous time random walk (CTRW)......Page 291
18.4 Lévy walks and other generalizations of CTRW......Page 294
18.6 Subdiffusion and superdiffusion......Page 296
Notes......Page 299
Problems......Page 300
19.1 Billiards in polygons......Page 302
19.2 Continued fractions and scalings of trajectories......Page 306
19.3 Fractional kinetics of irrational trajectories......Page 311
19.4 More examples of pseudochaos......Page 318
Notes......Page 324
Problems......Page 325
Part 4 Applications......Page 328
20 Complexity and entropy of dynamics......Page 330
20.1 Complexity in phase space......Page 331
20.2 Symbolic and topological complexities......Page 332
20.3 Topological and metric entropies......Page 335
20.4 Conflict with dynamics......Page 338
Problems......Page 339
21.1 Definitions of complexity function......Page 340
21.2 Probability of  ∈-divergence......Page 343
21.3 Calculation of local complexity function......Page 344
21.4 Flight complexity function......Page 346
21.5 Entropy function......Page 348
21.6 Polynomial and mixed complexities and anomalous transport......Page 350
21.7 Travelling waves and Riemann invariants of entropy and complexity......Page 352
Notes......Page 354
Problems......Page 355
22.1 Zermelo’s and Loschmidt’s paradoxes......Page 356
22.3 Anomalous properties of the Sinai and Bunimovich billiards......Page 359
22.4 Maxwell’s Demon and Chaos......Page 361
22.5 Maxwell’s Demon as a dynamical model......Page 363
22.7 Comments on dynamical cooling and chaos erasing......Page 367
Notes......Page 370
23.1 Beltrami flows with q-symmetry......Page 372
23.2 Compressible helical flows......Page 374
23.3 Compressible flow with quasi-symmetry......Page 382
Notes......Page 385
Problems......Page 386
24.1 Basic equations for point vortices and for advection......Page 388
24.2 Advection in three vortices......Page 391
24.3 Transport of advected particles (vortices)......Page 398
Notes......Page 404
Problems......Page 405
A: Elliptic integrals and elliptic functions......Page 408
B: Spectrum of the Kepler problem......Page 409
C: Fractional integro-diferentiation......Page 411
D: Formulas of fractional calculus......Page 414
References......Page 418
D......Page 432
I......Page 433
R......Page 434
W......Page 435