Contents -- Preface -- Introduction -- Historical note -- Overview of the book -- Hints on how to use/read this book -- Introduction to Dynamical Systems and Chaos -- 1. First Encounter with Chaos -- 1.1 Prologue -- 1.2 The nonlinear pendulum -- 1.3 The damped nonlinear pendulum -- 1.4 The vertically driven and damped nonlinear pendulum -- 1.5 What about the predictability of pendulum evolution? -- 1.6 Epilogue -- 2. The Language of Dynamical Systems -- 2.1 Ordinary Differential Equations (ODE) -- 2.1.1 Conservative and dissipative dynamical systems -- BoxB.1 Hamiltonian dynamics -- A: Symplectic structure and Canonical Transformations -- B: Integrable systems and Action-Angle variables -- 2.1.2 PoincaréMap -- 2.2 Discrete time dynamical systems: maps -- 2.2.1 Two dimensional maps -- 2.2.1.1 The Hénon Map -- 2.2.1.2 Two-dimensional symplectic maps -- 2.3 The role of dimension -- 2.4 Stability theory -- 2.4.1 Classification of fixed points and linear stability analysis -- BoxB.2 A remark on the linear stability of symplectic maps -- 2.4.2 Nonlinear stability -- 2.4.2.1 Limit cycles -- 2.4.2.2 Lyapunov Theorem -- 2.5 Exercises -- 3. Examples of Chaotic Behaviors -- 3.1 The logisticmap -- BoxB.3 Topological conjugacy -- 3.2 The Lorenzmodel -- BoxB.4 Derivation of the Lorenz model -- 3.3 The Hénon-Heiles system -- 3.4 What did we learn and what will we learn? -- BoxB.5 Correlation functions -- 3.5 Closing remark -- 3.6 Exercises -- 4. Probabilistic Approach to Chaos -- 4.1 An informal probabilistic approach -- 4.2 Time evolution of the probability density -- BoxB.6 Markov Processes -- A: Finite states Markov Chains -- B: Continuous Markov processes -- C: Dynamical systems with additive noise -- 4.3 Ergodicity -- 4.3.1 An historical interlude on ergodic theory -- BoxB.7 Poincaré recurrence theorem.

4.3.2 Abstract formulation of the Ergodic theory -- 4.4 Mixing -- 4.5 Markov chains and chaoticmaps -- 4.6 Natural measure -- 4.7 Exercises -- 5. Characterization of Chaotic Dynamical Systems -- 5.1 Strange attractors -- 5.2 Fractals and multifractals -- 5.2.1 Box counting dimension -- 5.2.2 The stretching and folding mechanism -- 5.2.3 Multifractals -- BoxB.8 Brief excursion on Large Deviation Theory -- 5.2.4 Grassberger-Procaccia algorithm -- 5.3 Characteristic Lyapunov exponents -- BoxB.9 Algorithm for computing Lyapunov Spectrum -- 5.3.1 Oseledec theorem and the law of large numbers -- 5.3.2 Remarks on the Lyapunov exponents -- 5.3.2.1 Lyapunov exponents are topological invariant -- 5.3.2.2 Relationship between Lyapunov exponents of flows and Poincaré maps -- 5.3.3 Fluctuation statistics of finite time Lyapunov exponents -- 5.3.4 Lyapunov dimension -- BoxB.10 Mathematical chaos -- A: Hyperbolic sets and Anosov systems -- B: SRB measure -- C: The Arnold cat map -- 5.4 Exercises -- 6. From Order to Chaos in Dissipative Systems -- 6.1 The scenarios for the transition to turbulence -- 6.1.1 Landau-Hopf -- BoxB.11 Hopf bifurcation -- BoxB.12 The Van der Pol oscillator and the averaging technique -- 6.1.2 Ruelle-Takens -- 6.2 The period doubling transition -- 6.2.1 Feigenbaum renormalization group -- 6.3 Transition to chaos through intermittency: Pomeau-Manneville scenario -- 6.4 Amathematical remark -- 6.5 Transition to turbulence in real systems -- 6.5.1 A visit to laboratory -- 6.6 Exercises -- 7. Chaos in Hamiltonian Systems -- 7.1 The integrability problem -- 7.1.1 Poincaré and the non-existence of integrals of motion -- 7.2 Kolmogorov-Arnold-Moser theorem and the survival of tori -- BoxB.13 Arnold diffusion -- 7.3 Poincaré-Birkho. theorem and the fate of resonant tori -- 7.4 Chaos around separatrices -- BoxB.14 The resonance-overlap criterion.

7.5 Melnikov's theory -- 7.5.1 An application to the Duffing's equation -- 7.6 Exercises -- Advanced Topics and Applications: From Information Theory to Turbulence -- 8. Chaos and Information Theory -- 8.1 Chaos, randomness and information -- 8.2 Information theory, coding and compression -- 8.2.1 Information sources -- 8.2.2 Properties and uniqueness of entropy -- 8.2.3 Shannon entropy rate and its meaning -- BoxB.15 Transient behavior of block-entropies -- 8.2.4 Coding and compression -- 8.3 Algorithmic complexity -- BoxB.16 Ziv-Lempel compression algorithm -- 8.4 Entropy and complexity in chaotic systems -- 8.4.1 Partitions and symbolic dynamics -- 8.4.2 Kolmogorov-Sinai entropy -- BoxB.17 Rényi entropies -- 8.4.3 Chaos, unpredictability and uncompressibility -- 8.5 Concluding remarks -- 8.6 Exercises -- 9. Coarse-Grained Information and Large Scale Predictability -- 9.1 Finite-resolution versus in.nite-resolution descriptions -- 9.2 ε-entropy in information theory: lossless versus lossy coding -- 9.2.1 Channel capacity -- 9.2.2 Rate distortion theory -- BoxB.18 ε-entropy for the Bernoulli and Gaussian source -- 9.3 ε-entropy in dynamical systems and stochastic processes -- 9.3.1 Systems classification according to e-entropy behavior -- BoxB.19 ε-entropy from exit-times statistics -- 9.4 The finite size lyapunov exponent (FSLE) -- 9.4.1 Linear vs nonlinear instabilities -- 9.4.2 Predictability in systems with different characteristic times -- 9.5 Exercises -- 10. Chaos in Numerical and Laboratory Experiments -- 10.1 Chaos in silico -- BoxB.20 Round-off errors and floating-point representation -- 10.1.1 Shadowing lemma -- 10.1.2 The effects of state discretization -- BoxB.21 Effect of discretization: a probabilistic argument -- 10.2 Chaos detection in experiments -- 10.2.1 Practical difficulties -- 10.2.1.1 Choice of delay time.

10.2.1.2 Choice of the embedding dimension -- 10.2.1.3 The necessary amount of data -- 10.2.1.4 Role of noise -- BoxB.22 Lyapunov exponents from experimental data -- 10.3 Can chaos be distinguished from noise? -- 10.3.1 The finite resolution analysis -- 10.3.2 Scale-dependent signal classiffication -- 10.3.3 Chaos or noise? A puzzling dilemma -- 10.3.3.1 Indeterminacy due to finite resolution -- 10.3.3.2 Indeterminacy due to fnite block length effects -- 10.4 Prediction andmodeling from data -- 10.4.1 Data prediction -- 10.4.2 Data modeling -- 11. Chaos in Low Dimensional Systems -- 11.1 Celestial mechanics -- 11.1.1 The restricted three-body problem -- 11.1.2 Chaos in the Solar system -- 11.1.2.1 The chaotic motion of Hyperion -- 11.1.2.2 Asteroids -- BoxB.23 A symplectic map for Halley comet -- 11.1.2.3 Long time behavior of the Solar system -- 11.2 Chaos and transport phenomena in fluids -- BoxB.24 Chaos and passive scalar transport -- 11.2.1 Lagrangian chaos -- 11.2.1.1 Eulerian vs Lagrangian chaos -- 11.2.1.2 Lagrangian chaos in point-vortex systems -- BoxB.25 Point vortices and the two-dimensional Euler equation -- 11.2.1.3 Lagrangian Chaos in the ABC flow -- 11.2.2 Chaos and diffusion in laminar flows -- 11.2.2.1 Transport in a model of the Gulf Stream -- 11.2.2.2 Standard and Anomalous diffusion in a chaotic model of transport -- BoxB.26 Relative dispersion in turbulence -- 11.2.3 Advection of inertial particles -- 11.3 Chaos in population biology and chemistry -- 11.3.1 Population biology: Lotka-Volterra systems -- 11.3.2 Chaos in generalized Lotka-Volterra systems -- 11.3.3 Kinetics of chemical reactions: Belousov-Zhabotinsky -- BoxB.27 Michaelis-Menten law of simple enzymatic reaction -- 11.3.4 Chemical clocks -- BoxB.28 A model for biochemical oscillations -- 11.4 Synchronization of chaotic systems.

11.4.1 Synchronization of regular oscillators -- 11.4.2 Phase synchronization of chaotic oscillators -- 11.4.3 Complete synchronization of chaotic systems -- 12. Spatiotemporal Chaos -- 12.1 Systems and models for spatiotemporal chaos -- 12.1.1 Overview of spatiotemporal chaotic systems -- 12.1.1.1 Reaction diffusion systems -- 12.1.1.2 Rayleigh Bénard convection -- 12.1.1.3 Complex Ginzburg-Landau and Kuramoto-Sivanshisky equations -- 12.1.1.4 Coupled map lattices -- 12.1.1.5 Nonlinear lattices: the Fermi-Pasta-Ulam model -- 12.1.1.6 Fully developed turbulence -- 12.1.1.7 Delayed ordinary differential equations -- 12.1.2 Networks of chaotic systems -- 12.2 The thermodynamic limit -- 12.3 Growth and propagation of space-time perturbations -- 12.3.1 An overview -- 12.3.2 "Spatial" and "Temporal" Lyapunov exponents -- 12.3.3 The comoving Lyapunov exponent -- 12.3.4 Propagation of perturbations -- BoxB.29 Stable chaos and supertransients -- 12.3.5 Convective chaos and sensitivity to boundary conditions -- 12.4 Non-equilibrium phenomena and spatiotemporal chaos -- BoxB.30 Non-equilibrium phase transitions -- Directed Percolation -- Multiplicative Noise -- Kardar-Parisi-Zhang equation and surface roughening -- 12.4.1 Spatiotemporal perturbations and interfaces roughening -- 12.4.2 Synchronization of extended chaotic systems -- 12.4.3 Spatiotemporal intermittency -- 12.5 Coarse-grained description of high dimensional chaos -- 12.5.1 Scale-dependent description of high-dimensional systems -- 12.5.2 Macroscopic chaos: low dimensional dynamics embedded in high dimensional chaos -- 13. Turbulence as a Dynamical System Problem -- 13.1 Fluids as dynamical systems -- 13.2 Statistical mechanics of ideal fluids and turbulence phenomenology -- 13.2.1 Three dimensional ideal fluids -- 13.2.2 Two dimensional ideal fluids.

13.2.3 Phenomenology of three dimensional turbulence.