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Chaos - From Simple Models to Complex Systems
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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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