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[资源] Chaos - From Simple Models to Complex Systems

Preface v
Introduction vii
Introduction to Dynamical Systems and Chaos
1. First Encounter with Chaos 3
1.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The nonlinear pendulum . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The damped nonlinear pendulum . . . . . . . . . . . . . . . . . . . 5
1.4 The vertically driven and damped nonlinear pendulum . . . . . . . 6
1.5 What about the predictability of pendulum evolution? . . . . . . . 8
1.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2. The Language of Dynamical Systems 11
2.1 OrdinaryDifferential Equations (ODE) . . . . . . . . . . . . . . . 11
2.1.1 Conservative and dissipative dynamical systems . . . . . . 13
BoxB.1 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Poincar´eMap . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Discrete time dynamical systems: maps . . . . . . . . . . . . . . . 20
2.2.1 Two dimensionalmaps . . . . . . . . . . . . . . . . . . . . 21
2.3 The role of dimension . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Stability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Classification of fixed points and linear stability analysis . 27
BoxB.2 A remark on the linear stability of symplectic maps . . . . 29
2.4.2 Nonlinear stability . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3. Examples of Chaotic Behaviors 37
3.1 The logisticmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
xv
xvi Chaos: From Simple Models to Complex Systems
BoxB.3 Topological conjugacy . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Lorenzmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
BoxB.4 Derivation of the Lorenz model . . . . . . . . . . . . . . . 51
3.3 The H´enon-Heiles system . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 What did we learn and what will we learn? . . . . . . . . . . . . . 58
BoxB.5 Correlation functions . . . . . . . . . . . . . . . . . . . . . 61
3.5 Closing remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4. Probabilistic Approach to Chaos 65
4.1 An informal probabilistic approach . . . . . . . . . . . . . . . . . . 65
4.2 Time evolution of the probability density . . . . . . . . . . . . . . 68
BoxB.6 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 An historical interlude on ergodic theory . . . . . . . . . . 78
BoxB.7 Poincar´e recurrence theorem . . . . . . . . . . . . . . . . . 79
4.3.2 Abstract formulation of the Ergodic theory . . . . . . . . . 81
4.4 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5 Markov chains and chaoticmaps . . . . . . . . . . . . . . . . . . . 86
4.6 Naturalmeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5. Characterization of Chaotic Dynamical Systems 93
5.1 Strange attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Fractals and multifractals . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 Box counting dimension . . . . . . . . . . . . . . . . . . . . 98
5.2.2 The stretching and folding mechanism . . . . . . . . . . . . 100
5.2.3 Multifractals . . . . . . . . . . . . . . . . . . . . . . . . . . 103
BoxB.8 Brief excursion on Large Deviation Theory . . . . . . . . 108
5.2.4 Grassberger-Procaccia algorithm . . . . . . . . . . . . . . . 109
5.3 Characteristic Lyapunov exponents . . . . . . . . . . . . . . . . . . 111
BoxB.9 Algorithm for computing Lyapunov Spectrum . . . . . . . 115
5.3.1 Oseledec theorem and the law of large numbers . . . . . . 116
5.3.2 Remarks on the Lyapunov exponents . . . . . . . . . . . . 118
5.3.3 Fluctuation statistics of finite time Lyapunov exponents . 120
5.3.4 Lyapunov dimension . . . . . . . . . . . . . . . . . . . . . 123
BoxB.10 Mathematical chaos . . . . . . . . . . . . . . . . . . . . . 124
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6. From Order to Chaos in Dissipative Systems 131
6.1 The scenarios for the transition to turbulence . . . . . . . . . . . . 131
6.1.1 Landau-Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . 132
BoxB.11 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . 134
Contents xvii
BoxB.12 The Van der Pol oscillator and the averaging technique . 135
6.1.2 Ruelle-Takens . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 The period doubling transition . . . . . . . . . . . . . . . . . . . . 139
6.2.1 Feigenbaum renormalization group . . . . . . . . . . . . . . 142
6.3 Transition to chaos through intermittency: Pomeau-Manneville
scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Amathematical remark . . . . . . . . . . . . . . . . . . . . . . . . 147
6.5 Transition to turbulence in real systems . . . . . . . . . . . . . . . 148
6.5.1 A visit to laboratory . . . . . . . . . . . . . . . . . . . . . 149
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7. Chaos in Hamiltonian Systems 153
7.1 The integrability problem . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.1 Poincar´e and the non-existence of integrals of motion . . . 154
7.2 Kolmogorov-Arnold-Moser theorem and the survival of tori . . . . 155
BoxB.13 Arnold diffusion . . . . . . . . . . . . . . . . . . . . . . . 160
7.3 Poincar´e-Birkhoff theorem and the fate of resonant tori . . . . . . 161
7.4 Chaos around separatrices . . . . . . . . . . . . . . . . . . . . . . 164
BoxB.14 The resonance-overlap criterion . . . . . . . . . . . . . . 168
7.5 Melnikov’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.5.1 An application to the Duffing’s equation . . . . . . . . . . 174
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Advanced Topics and Applications: From Information
Theory to Turbulence
8. Chaos and Information Theory 179
8.1 Chaos, randomness and information . . . . . . . . . . . . . . . . . 179
8.2 Information theory, coding and compression . . . . . . . . . . . . . 183
8.2.1 Information sources . . . . . . . . . . . . . . . . . . . . . . 184
8.2.2 Properties and uniqueness of entropy . . . . . . . . . . . . 185
8.2.3 Shannon entropy rate and its meaning . . . . . . . . . . . 187
BoxB.15 Transient behavior of block-entropies . . . . . . . . . . . 190
8.2.4 Coding and compression . . . . . . . . . . . . . . . . . . . 192
8.3 Algorithmic complexity . . . . . . . . . . . . . . . . . . . . . . . . 194
BoxB.16 Ziv-Lempel compression algorithm . . . . . . . . . . . . . 196
8.4 Entropy and complexity in chaotic systems . . . . . . . . . . . . . 197
8.4.1 Partitions and symbolic dynamics . . . . . . . . . . . . . . 197
8.4.2 Kolmogorov-Sinai entropy . . . . . . . . . . . . . . . . . . 200
BoxB.17 R´enyi entropies . . . . . . . . . . . . . . . . . . . . . . . 203
8.4.3 Chaos, unpredictability and uncompressibility . . . . . . . 203
xviii Chaos: From Simple Models to Complex Systems
8.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9. Coarse-Grained Information and Large Scale Predictability 209
9.1 Finite-resolution versus infinite-resolution descriptions . . . . . . . 209
9.2 ε-entropy in information theory: lossless versus lossy coding . . . . 213
9.2.1 Channel capacity . . . . . . . . . . . . . . . . . . . . . . . 213
9.2.2 Rate distortion theory . . . . . . . . . . . . . . . . . . . . . 215
BoxB.18 ε-entropy for the Bernoulli and Gaussian source . . . . . 218
9.3 ε-entropy in dynamical systems and stochastic processes . . . . . 219
9.3.1 Systems classification according to ε-entropy behavior . . . 222
BoxB.19 ε-entropy from exit-times statistics . . . . . . . . . . . . 224
9.4 The finite size lyapunov exponent (FSLE) . . . . . . . . . . . . . . 228
9.4.1 Linear vs nonlinear instabilities . . . . . . . . . . . . . . . 233
9.4.2 Predictability in systems with different characteristic times 234
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10. Chaos in Numerical and Laboratory Experiments 239
10.1 Chaos in silico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
BoxB.20 Round-off errors and floating-point representation . . . . 241
10.1.1 Shadowing lemma . . . . . . . . . . . . . . . . . . . . . . . 242
10.1.2 The effects of state discretization . . . . . . . . . . . . . . 244
BoxB.21 Effect of discretization: a probabilistic argument . . . . . 247
10.2 Chaos detection in experiments . . . . . . . . . . . . . . . . . . . . 247
BoxB.22 Lyapunov exponents from experimental data . . . . . . . 250
10.2.1 Practical difficulties . . . . . . . . . . . . . . . . . . . . . . 251
10.3 Can chaos be distinguished from noise? . . . . . . . . . . . . . . . 255
10.3.1 The finite resolution analysis . . . . . . . . . . . . . . . . . 256
10.3.2 Scale-dependent signal classification . . . . . . . . . . . . . 256
10.3.3 Chaos or noise? A puzzling dilemma . . . . . . . . . . . . 258
10.4 Prediction andmodeling from data . . . . . . . . . . . . . . . . . . 263
10.4.1 Data prediction . . . . . . . . . . . . . . . . . . . . . . . . 263
10.4.2 Data modeling . . . . . . . . . . . . . . . . . . . . . . . . . 264
11. Chaos in Low Dimensional Systems 267
11.1 Celestialmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
11.1.1 The restricted three-body problem . . . . . . . . . . . . . . 269
11.1.2 Chaos in the Solar system . . . . . . . . . . . . . . . . . . 273
BoxB.23 A symplectic map for Halley comet . . . . . . . . . . . . 276
11.2 Chaos and transport phenomena in fluids . . . . . . . . . . . . . . 279
BoxB.24 Chaos and passive scalar transport . . . . . . . . . . . . 280
11.2.1 Lagrangian chaos . . . . . . . . . . . . . . . . . . . . . . . 283
Contents xix
BoxB.25 Point vortices and the two-dimensional Euler equation . 288
11.2.2 Chaos and diffusion in laminar flows . . . . . . . . . . . . . 290
BoxB.26 Relative dispersion in turbulence . . . . . . . . . . . . . . 295
11.2.3 Advection of inertial particles . . . . . . . . . . . . . . . . 296
11.3 Chaos in population biology and chemistry . . . . . . . . . . . . . 299
11.3.1 Population biology: Lotka-Volterra systems . . . . . . . . . 300
11.3.2 Chaos in generalized Lotka-Volterra systems . . . . . . . . 304
11.3.3 Kinetics of chemical reactions: Belousov-Zhabotinsky . . . 307
BoxB.27 Michaelis-Menten law of simple enzymatic reaction . . . 311
11.3.4 Chemical clocks . . . . . . . . . . . . . . . . . . . . . . . . 312
BoxB.28 A model for biochemical oscillations . . . . . . . . . . . . 314
11.4 Synchronization of chaotic systems . . . . . . . . . . . . . . . . . . 316
11.4.1 Synchronization of regular oscillators . . . . . . . . . . . . 317
11.4.2 Phase synchronization of chaotic oscillators . . . . . . . . . 319
11.4.3 Complete synchronization of chaotic systems . . . . . . . . 323
12. Spatiotemporal Chaos 329
12.1 Systems and models for spatiotemporal chaos . . . . . . . . . . . . 329
12.1.1 Overview of spatiotemporal chaotic systems . . . . . . . . 330
12.1.2 Networks of chaotic systems . . . . . . . . . . . . . . . . . 337
12.2 The thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . 338
12.3 Growth and propagation of space-time perturbations . . . . . . . . 340
12.3.1 An overview . . . . . . . . . . . . . . . . . . . . . . . . . . 340
12.3.2 “Spatial” and “Temporal” Lyapunov exponents . . . . . . 341
12.3.3 The comoving Lyapunov exponent . . . . . . . . . . . . . . 343
12.3.4 Propagation of perturbations . . . . . . . . . . . . . . . . . 344
BoxB.29 Stable chaos and supertransients . . . . . . . . . . . . . . 348
12.3.5 Convective chaos and sensitivity to boundary conditions . 350
12.4 Non-equilibrium phenomena and spatiotemporal chaos . . . . . . . 352
BoxB.30 Non-equilibrium phase transitions . . . . . . . . . . . . . 353
12.4.1 Spatiotemporal perturbations and interfaces roughening . 356
12.4.2 Synchronization of extended chaotic systems . . . . . . . . 358
12.4.3 Spatiotemporal intermittency . . . . . . . . . . . . . . . . 361
12.5 Coarse-grained description of high dimensional chaos . . . . . . . . 363
12.5.1 Scale-dependent description of high-dimensional systems . 363
12.5.2 Macroscopic chaos: low dimensional dynamics embedded
in high dimensional chaos . . . . . . . . . . . . . . . . . . 365
13. Turbulence as a Dynamical System Problem 369
13.1 Fluids as dynamical systems . . . . . . . . . . . . . . . . . . . . . . 369
13.2 Statistical mechanics of ideal fluids and turbulence phenomenology 373
13.2.1 Three dimensional ideal fluids . . . . . . . . . . . . . . . . 373
xx Chaos: From Simple Models to Complex Systems
13.2.2 Two dimensional ideal fluids . . . . . . . . . . . . . . . . . 374
13.2.3 Phenomenology of three dimensional turbulence . . . . . . 375
BoxB.31 Intermittency in three-dimensional turbulence:
the multifractal model . . . . . . . . . . . . . . . . . . . . . 379
13.2.4 Phenomenology of two dimensional turbulence . . . . . . . 382
13.3 From partial differential equations to ordinary differential
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
13.3.1 On the number of degrees of freedom of turbulence . . . . 385
13.3.2 The Galerkinmethod . . . . . . . . . . . . . . . . . . . . . 387
13.3.3 Point vorticesmethod . . . . . . . . . . . . . . . . . . . . . 388
13.3.4 Proper orthonormal decomposition . . . . . . . . . . . . . 390
13.3.5 Shell models . . . . . . . . . . . . . . . . . . . . . . . . . . 391
13.4 Predictability in turbulent systems . . . . . . . . . . . . . . . . . . 394
13.4.1 Small scales predictability . . . . . . . . . . . . . . . . . . 395
13.4.2 Large scales predictability . . . . . . . . . . . . . . . . . . 397
13.4.3 Predictability in the presence of coherent structures . . . 401
14. Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study 405
14.1 An influential unpublished paper . . . . . . . . . . . . . . . . . . . 405
14.1.1 Toward an explanation: Solitons or KAM? . . . . . . . . . 409
14.2 A random walk on the role of ergodicity and chaos for equilibrium
statisticalmechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 411
14.2.1 Beyond metrical transitivity: a physical point of view . . . 411
14.2.2 Physical questions and numerical results . . . . . . . . . . 412
14.2.3 Is chaos necessary or sufficient for the validity of statistical
mechanical laws? . . . . . . . . . . . . . . . . . . . . . . . 415
14.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
BoxB.32 Pseudochaos and diffusion . . . . . . . . . . . . . . . . . 418
Epilogue 421
Bibliography 427
Index 455

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