| |
| |
| |
| Chaotic Dynamics | |
| |
| |
| |
| Hamiltonian dynamics | |
| |
| |
| |
| Hamiltonian equations | |
| |
| |
| |
| Phase space dynamics | |
| |
| |
| |
| Action-angle variable (one degree of freedom) | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Examples of Hamiltonian dynamics | |
| |
| |
| |
| Pendulum | |
| |
| |
| |
| Oscillations in the infinite potential well | |
| |
| |
| |
| Magnetic moments | |
| |
| |
| |
| Field line behaviour | |
| |
| |
| |
| Hamiltonian equations for the ABC-flow | |
| |
| |
| Problems | |
| |
| |
| |
| Perturbed dynamics | |
| |
| |
| |
| The Liouville-Arnold theorem on integrability | |
| |
| |
| |
| Consequences of the integrability | |
| |
| |
| |
| Non-integrability and the Kozlov condition | |
| |
| |
| |
| Resonances | |
| |
| |
| |
| Non-linear resonance and chain of islands | |
| |
| |
| |
| Kolmogorov-Arnold-Moser (KAM) theory | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Chaotic dynamics | |
| |
| |
| |
| Natural measure | |
| |
| |
| |
| Ergodicity, mixing, and weak mixing | |
| |
| |
| |
| Local instability and Lyapunov exponents | |
| |
| |
| |
| Hyperbolic systems | |
| |
| |
| |
| Entropy of dynamical systems | |
| |
| |
| |
| Partitioning and coarse-graining | |
| |
| |
| |
| Kolmogorov-Sinai (KS) entropy | |
| |
| |
| |
| Topological entropy | |
| |
| |
| |
| Physical interpretation | |
| |
| |
| |
| Entropy and Lyapunov exponents | |
| |
| |
| |
| Definition of chaotic dynamics | |
| |
| |
| |
| Chirikov resonance overlapping criteria | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Physical models of chaos | |
| |
| |
| |
| Mapping the dynamics | |
| |
| |
| |
| Universal and standard map | |
| |
| |
| |
| Web map (kicked oscillator) | |
| |
| |
| |
| Kepler map | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Separatrix chaos | |
| |
| |
| |
| Description of models | |
| |
| |
| |
| Separatrix map | |
| |
| |
| |
| The stochastic layer | |
| |
| |
| |
| The stochastic layer of the standard map | |
| |
| |
| |
| Hidden renormalization group near the separatrix | |
| |
| |
| |
| Renormalization of resonances | |
| |
| |
| |
| Hidden renormalization for coupled oscillators | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Weak chaos and symmetry | |
| |
| |
| |
| Stochastic webs | |
| |
| |
| |
| Stochastic webs with quasi-crystalline symmetry | |
| |
| |
| |
| Stochastic web skeleton | |
| |
| |
| |
| Symmetries and their dynamical generation | |
| |
| |
| |
| Width of the stochastic web | |
| |
| |
| |
| Symmetry in art and nature | |
| |
| |
| |
| Symmetry and chaos | |
| |
| |
| |
| Ornamental patterns | |
| |
| |
| |
| Patterns in nature | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Beyond the KAM-theory | |
| |
| |
| |
| Small non-linearity | |
| |
| |
| |
| Web-Tori | |
| |
| |
| |
| Width of the stochastic web | |
| |
| |
| |
| Transition from KAM-Tori to Web-Tori | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Phase space of chaos | |
| |
| |
| |
| Topological non-universality of chaos | |
| |
| |
| |
| Examples with billiards | |
| |
| |
| |
| Accelerator mode islands | |
| |
| |
| |
| Ballistic mode islands | |
| |
| |
| |
| Cantori | |
| |
| |
| |
| Sticky domains and escapes | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Fractality of Chaos | |
| |
| |
| |
| Fractals and chaos | |
| |
| |
| |
| Fractal dynamics | |
| |
| |
| |
| Generalized fractal dimension | |
| |
| |
| |
| Renormalization group and generalized fractal dimension | |
| |
| |
| |
| Multifractal spectra | |
| |
| |
| |
| Thermodynamic interpretation | |
| |
| |
| |
| Complex dimension and log-periodicity | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Poincare recurrences | |
| |
| |
| |
| Poincare theorem on recurrences | |
| |
| |
| |
| Recurrence time distributions and Kac lemma | |
| |
| |
| |
| Distribution of recurrences in uniform mixing | |
| |
| |
| |
| More asymptotics on recurrences | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Dynamical traps | |
| |
| |
| |
| Definition of the dynamical trap | |
| |
| |
| |
| Hierarchical-islands trap (HIT) | |
| |
| |
| |
| Renormalization for the exit time distribution | |
| |
| |
| |
| Stochastic layer trap | |
| |
| |
| Notes | |
| |
| |
| |
| Fractal time | |
| |
| |
| |
| Fractal time | |
| |
| |
| |
| Fractal and multifractal recurrences | |
| |
| |
| |
| Multifractal space-time and its dimension spectrum | |
| |
| |
| |
| Critical exponent for the Poincare recurrences | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Chaotic kinetics | |
| |
| |
| |
| General principles of kinetics | |
| |
| |
| |
| Time scales | |
| |
| |
| |
| Fokker-Planck-Kolmogorov (FPK) equation | |
| |
| |
| |
| Detailed balance principle | |
| |
| |
| |
| Solutions and normal transport | |
| |
| |
| |
| Growth of entropy | |
| |
| |
| |
| Kolmogorov conditions and conflict with dynamics | |
| |
| |
| |
| Truncated distributions | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Levy process, Levy flights, and Weierstrass random walk | |
| |
| |
| |
| Levy distribution | |
| |
| |
| |
| Levy process | |
| |
| |
| |
| Poincare recurrences and Feller's theorems | |
| |
| |
| |
| Levy flights and conflict with dynamics | |
| |
| |
| |
| Weirstrass random walks (WRW) | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Fractional kinetic equation (FKE) | |
| |
| |
| |
| Derivation of FKE | |
| |
| |
| |
| Conditions for the FKE | |
| |
| |
| |
| Evolution of moments (transport) | |
| |
| |
| |
| Conflict with dynamics | |
| |
| |
| |
| Dynamical origin of critical exponents | |
| |
| |
| |
| Principles of simulations | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Renormalization group of kinetics (RGK) | |
| |
| |
| |
| Space-time scalings | |
| |
| |
| |
| Log-periodicity | |
| |
| |
| |
| Duality of the dynamics and the origin of multi-fractality | |
| |
| |
| |
| Multifractional kinetics | |
| |
| |
| Notes | |
| |
| |
| |
| Fractional kinetic equation: solutions and modifications | |
| |
| |
| |
| Solutions to FKE (series) | |
| |
| |
| |
| Solutions to FKE (separation of variables) | |
| |
| |
| |
| Continuous time random walk (CTRW) | |
| |
| |
| |
| Levy walks and other generalizations of CTRW | |
| |
| |
| |
| Conflict with dynamics | |
| |
| |
| |
| Subdiffusion and superdiffusion | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Pseudochaos | |
| |
| |
| |
| Billiards in polygons | |
| |
| |
| |
| Continued fractions and scalings of trajectories | |
| |
| |
| |
| Fractional kinetics of irrational trajectories | |
| |
| |
| |
| More examples of pseudochaos | |
| |
| |
| |
| Rhombic billiard | |
| |
| |
| |
| More billiards | |
| |
| |
| |
| Saw-tooth web map | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Applications | |
| |
| |
| |
| Complexity and entropy of dynamics | |
| |
| |
| |
| Complexity in phase space | |
| |
| |
| |
| Symbolic and topological complexities | |
| |
| |
| |
| Topological and metric entropies | |
| |
| |
| |
| Conflict with dynamics | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Complexity and entropy functions | |
| |
| |
| |
| Definitions of complexity function | |
| |
| |
| |
| Probability of [epsilon]-divergence | |
| |
| |
| |
| Calculation of local complexity function | |
| |
| |
| |
| Flight complexity function | |
| |
| |
| |
| Entropy function | |
| |
| |
| |
| Polynomial and mixed complexities and anomalous transport | |
| |
| |
| |
| Travelling waves and Riemann invariants of entropy and complexity | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Chaos and foundation of statistical mechanics | |
| |
| |
| |
| Zermelo's and Loschmidt's paradoxes | |
| |
| |
| |
| Historical comments | |
| |
| |
| |
| Paradox of recurrence | |
| |
| |
| |
| Paradox of reversibility | |
| |
| |
| |
| Boltzmann's comments | |
| |
| |
| |
| Chaos and the paradoxes | |
| |
| |
| |
| Anomalous properties of the Sinai and Bunimovich billiards | |
| |
| |
| |
| Maxwell's Demon and Chaos | |
| |
| |
| |
| Maxwell's Demon as a dynamical model | |
| |
| |
| |
| Comments on the application of Ergodic Theory | |
| |
| |
| |
| Comments on dynamical cooling and chaos erasing | |
| |
| |
| Notes | |
| |
| |
| |
| Chaotic advection (dynamics of tracers) | |
| |
| |
| |
| Beltrami flows with q-symmetry | |
| |
| |
| |
| Compressible helical flows | |
| |
| |
| |
| Compressible flow with quasi-symmetry | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| |
| Advection by point vortices | |
| |
| |
| |
| Basic equations for point vortices and for advection | |
| |
| |
| |
| Advection in three vortices | |
| |
| |
| |
| Transport of advected particles (vortices) | |
| |
| |
| Notes | |
| |
| |
| Problems | |
| |
| |
| Appendices | |
| |
| |
| |
| Elliptic integrals and elliptic functions | |
| |
| |
| |
| Spectrum of the Kepler problem | |
| |
| |
| |
| Fractional integro-diferentiation | |
| |
| |
| |
| Formulas of fractional calculus | |
| |
| |
| References | |
| |
| |
| Index | |