Chaos: From Simple Models to Complex Systems (Series on Advances in Statistical Mechanics)

Contents......Page 16
Preface......Page 6
Historical note......Page 8
Overview of the book......Page 12
Hints on how to use/read this book......Page 14
Introduction to Dynamical Systems and Chaos......Page 23
1.2 The nonlinear pendulum......Page 25
1.3 The damped nonlinear pendulum......Page 27
1.4 The vertically driven and damped nonlinear pendulum......Page 28
1.5 What about the predictability of pendulum evolution?......Page 30
1.6 Epilogue......Page 32
2.1 Ordinary Differential Equations (ODE)......Page 33
2.1.1 Conservative and dissipative dynamical systems......Page 35
A: Symplectic structure and Canonical Transformations......Page 37
B: Integrable systems and Action-Angle variables......Page 39
2.1.2 Poincar´eMap......Page 41
2.2 Discrete time dynamical systems: maps......Page 42
2.2.1.1 The H´enon Map......Page 43
2.2.1.2 Two-dimensional symplectic maps......Page 45
2.3 The role of dimension......Page 47
2.4 Stability theory......Page 48
2.4.1 Classification of fixed points and linear stability analysis......Page 49
BoxB.2 A remark on the linear stability of symplectic maps......Page 51
2.4.2 Nonlinear stability......Page 52
2.4.2.1 Limit cycles......Page 53
2.5 Exercises......Page 55
3.1 The logisticmap......Page 59
BoxB.3 Topological conjugacy......Page 67
3.2 The Lorenzmodel......Page 68
BoxB.4 Derivation of the Lorenz model......Page 73
3.3 The H´enon-Heiles system......Page 75
3.4 What did we learn and what will we learn?......Page 80
BoxB.5 Correlation functions......Page 83
3.6 Exercises......Page 84
4.1 An informal probabilistic approach......Page 87
4.2 Time evolution of the probability density......Page 90
A: Finite states Markov Chains......Page 94
B: Continuous Markov processes......Page 97
4.3 Ergodicity......Page 99
4.3.1 An historical interlude on ergodic theory......Page 100
BoxB.7 Poincar´e recurrence theorem......Page 101
4.3.2 Abstract formulation of the Ergodic theory......Page 103
4.4 Mixing......Page 106
4.5 Markov chains and chaoticmaps......Page 108
4.6 Natural measure......Page 111
4.7 Exercises......Page 113
5.1 Strange attractors......Page 115
5.2 Fractals and multifractals......Page 117
5.2.1 Box counting dimension......Page 120
5.2.2 The stretching and folding mechanism......Page 122
5.2.3 Multifractals......Page 125
BoxB.8 Brief excursion on Large Deviation Theory......Page 130
5.2.4 Grassberger-Procaccia algorithm......Page 131
5.3 Characteristic Lyapunov exponents......Page 133
BoxB.9 Algorithm for computing Lyapunov Spectrum......Page 137
5.3.1 Oseledec theorem and the law of large numbers......Page 138
5.3.2.1 Lyapunov exponents are topological invariant......Page 140
5.3.2.2 Relationship between Lyapunov exponents of flows and Poincar´e maps......Page 141
5.3.3 Fluctuation statistics of finite time Lyapunov exponents......Page 142
5.3.4 Lyapunov dimension......Page 145
BoxB.10 Mathematical chaos......Page 146
B: SRB measure......Page 147
C: The Arnold cat map......Page 148
5.4 Exercises......Page 149
6.1 The scenarios for the transition to turbulence......Page 153
6.1.1 Landau-Hopf......Page 154
BoxB.11 Hopf bifurcation......Page 156
BoxB.12 The Van der Pol oscillator and the averaging technique......Page 157
6.1.2 Ruelle-Takens......Page 159
6.2 The period doubling transition......Page 161
6.2.1 Feigenbaum renormalization group......Page 164
6.3 Transition to chaos through intermittency: Pomeau-Manneville scenario......Page 167
6.4 Amathematical remark......Page 169
6.5 Transition to turbulence in real systems......Page 170
6.5.1 A visit to laboratory......Page 171
6.6 Exercises......Page 173
7.1 The integrability problem......Page 175
7.1.1 Poincar´e and the non-existence of integrals of motion......Page 176
7.2 Kolmogorov-Arnold-Moser theorem and the survival of tori......Page 177
BoxB.13 Arnold diffusion......Page 182
7.3 Poincar´e-Birkho. theorem and the fate of resonant tori......Page 183
7.4 Chaos around separatrices......Page 186
BoxB.14 The resonance-overlap criterion......Page 190
7.5 Melnikov’s theory......Page 193
7.5.1 An application to the Duffing’s equation......Page 196
7.6 Exercises......Page 197
Advanced Topics and Applications: From Information Theory to Turbulence......Page 199
8.1 Chaos, randomness and information......Page 201
8.2 Information theory, coding and compression......Page 205
8.2.1 Information sources......Page 206
8.2.2 Properties and uniqueness of entropy......Page 207
8.2.3 Shannon entropy rate and its meaning......Page 209
BoxB.15 Transient behavior of block-entropies......Page 212
8.2.4 Coding and compression......Page 214
8.3 Algorithmic complexity......Page 216
BoxB.16 Ziv-Lempel compression algorithm......Page 218
8.4.1 Partitions and symbolic dynamics......Page 219
8.4.2 Kolmogorov-Sinai entropy......Page 222
8.4.3 Chaos, unpredictability and uncompressibility......Page 225
8.5 Concluding remarks......Page 227
8.6 Exercises......Page 228
9.1 Finite-resolution versus in.nite-resolution descriptions......Page 231
9.2.1 Channel capacity......Page 235
9.2.2 Rate distortion theory......Page 237
BoxB.18 ε-entropy for the Bernoulli and Gaussian source......Page 240
9.3 ε-entropy in dynamical systems and stochastic processes......Page 241
9.3.1 Systems classification according to e-entropy behavior......Page 244
BoxB.19 ε-entropy from exit-times statistics......Page 246
9.4 The finite size lyapunov exponent (FSLE)......Page 250
9.4.1 Linear vs nonlinear instabilities......Page 255
9.4.2 Predictability in systems with different characteristic times......Page 256
9.5 Exercises......Page 259
10.1 Chaos in silico......Page 261
BoxB.20 Round-off errors and floating-point representation......Page 263
10.1.1 Shadowing lemma......Page 264
10.1.2 The effects of state discretization......Page 266
10.2 Chaos detection in experiments......Page 269
10.2.1.1 Choice of delay time......Page 273
10.2.1.2 Choice of the embedding dimension......Page 274
10.2.1.3 The necessary amount of data......Page 276
10.3 Can chaos be distinguished from noise?......Page 277
BoxB.22 Lyapunov exponents from experimental data......Page 272
10.3.2 Scale-dependent signal classiffication......Page 278
10.3.3 Chaos or noise? A puzzling dilemma......Page 280
10.3.3.1 Indeterminacy due to finite resolution......Page 281
10.3.3.2 Indeterminacy due to fnite block length effects......Page 282
10.4.1 Data prediction......Page 285
10.4.2 Data modeling......Page 286
11.1 Celestial mechanics......Page 289
11.1.1 The restricted three-body problem......Page 291
11.1.2.1 The chaotic motion of Hyperion......Page 295
11.1.2.2 Asteroids......Page 296
BoxB.23 A symplectic map for Halley comet......Page 298
11.1.2.3 Long time behavior of the Solar system......Page 299
11.2 Chaos and transport phenomena in fluids......Page 301
BoxB.24 Chaos and passive scalar transport......Page 302
11.2.1.1 Eulerian vs Lagrangian chaos......Page 305
11.2.1.2 Lagrangian chaos in point-vortex systems......Page 308
BoxB.25 Point vortices and the two-dimensional Euler equation......Page 310
11.2.1.3 Lagrangian Chaos in the ABC flow......Page 311
11.2.2.1 Transport in a model of the Gulf Stream......Page 312
11.2.2.2 Standard and Anomalous diffusion in a chaotic model of transport......Page 314
BoxB.26 Relative dispersion in turbulence......Page 317
11.2.3 Advection of inertial particles......Page 318
11.3 Chaos in population biology and chemistry......Page 321
11.3.1 Population biology: Lotka-Volterra systems......Page 322
11.3.2 Chaos in generalized Lotka-Volterra systems......Page 326
11.3.3 Kinetics of chemical reactions: Belousov-Zhabotinsky......Page 329
BoxB.27 Michaelis-Menten law of simple enzymatic reaction......Page 333
11.3.4 Chemical clocks......Page 334
BoxB.28 A model for biochemical oscillations......Page 336
11.4 Synchronization of chaotic systems......Page 338
11.4.1 Synchronization of regular oscillators......Page 339
11.4.2 Phase synchronization of chaotic oscillators......Page 341
11.4.3 Complete synchronization of chaotic systems......Page 345
12.1 Systems and models for spatiotemporal chaos......Page 351
12.1.1.1 Reaction diffusion systems......Page 352
12.1.1.2 Rayleigh B´enard convection......Page 354
12.1.1.3 Complex Ginzburg-Landau and Kuramoto-Sivanshisky equations......Page 355
12.1.1.4 Coupled map lattices......Page 357
12.1.1.7 Delayed ordinary differential equations......Page 358
12.1.2 Networks of chaotic systems......Page 359
12.2 The thermodynamic limit......Page 360
12.3.1 An overview......Page 362
12.3.2 “Spatial” and “Temporal” Lyapunov exponents......Page 363
12.3.3 The comoving Lyapunov exponent......Page 365
12.3.4 Propagation of perturbations......Page 366
BoxB.29 Stable chaos and supertransients......Page 370
12.3.5 Convective chaos and sensitivity to boundary conditions......Page 372
12.4 Non-equilibrium phenomena and spatiotemporal chaos......Page 374
Directed Percolation......Page 375
Multiplicative Noise......Page 376
Kardar-Parisi-Zhang equation and surface roughening......Page 377
12.4.1 Spatiotemporal perturbations and interfaces roughening......Page 378
12.4.2 Synchronization of extended chaotic systems......Page 380
12.4.3 Spatiotemporal intermittency......Page 383
12.5.1 Scale-dependent description of high-dimensional systems......Page 385
12.5.2 Macroscopic chaos: low dimensional dynamics embedded in high dimensional chaos......Page 387
13.1 Fluids as dynamical systems......Page 391
13.2.1 Three dimensional ideal fluids......Page 395
13.2.2 Two dimensional ideal fluids......Page 396
13.2.3 Phenomenology of three dimensional turbulence......Page 397
BoxB.31 Intermittency in three-dimensional turbulence: the multifractal model......Page 401
13.2.4 Phenomenology of two dimensional turbulence......Page 404
13.3.1 On the number of degrees of freedom of turbulence......Page 407
13.3.2 The Galerkin method......Page 409
13.3.3 Point vortices method......Page 410
13.3.4 Proper orthonormal decomposition......Page 412
13.3.5 Shell models......Page 413
13.4 Predictability in turbulent systems......Page 416
13.4.1 Small scales predictability......Page 417
13.4.2 Large scales predictability......Page 419
13.4.3 Predictability in the presence of coherent structures......Page 423
14.1 An influential unpublished paper......Page 427
14.1.1 Toward an explanation: Solitons or KAM?......Page 431
14.2.1 Beyond metrical transitivity: a physical point of view......Page 433
14.2.2 Physical questions and numerical results......Page 434
14.2.3 Is chaos necessary or sufficient for the validity of statistical mechanical laws?......Page 437
14.3 Final remarks......Page 439
BoxB.32 Pseudochaos and diffusion......Page 440
Epilogue......Page 443
Bibliography......Page 449
Index......Page 477