CONTENTS -- PREFACE TO THE FIRST EDITION -- PREFACE TO THE SECOND EDITION -- 1 DISCRETE AND CONTINUOUS MODELS -- 1.1 Coexistence of the Dynamical Order and Chaos -- 1.2 The Standard Map (Kicked Rotator) -- 1.3 The Web-Map (Kicked Oscillator) -- 1.4 Perturbed Pendulum -- 1.5 Perturbed Oscillator -- 1.6 Billiards -- Conclusions -- 2 SEPARATRIX CHAOS -- 2.1 Nonlinear Resonance and Chain of Islands -- 2.2 Overlapping of Resonances -- 2.3 The Separatrix Map -- 2.4 Stochastic Layer -- 2.5 Hidden Renormalisation Group Near the Separatrix -- 2.6 Renormalisation of Resonances -- 2.7 Stochastic Layer of the Standard Map -- Conclusions -- 3 THE PHASE SPACE OF CHAOS -- 3.1 Non-universality of the Scenario -- 3.2 Collapsing Islands -- 3.3 Blinking Islands -- 3.4 Boundary Islands -- 3.5 Self-Similar Set of Islands -- 3.6 Ballistic Mode Islands -- 3.7 General Comments About the Islands -- Conclusions -- 4 NONLINEARITY VERSUS PERTURBATION -- 4.1 Beyond the KAM-Theory -- 4.2 Web-Tori -- 4.3 Width of the Stochastic Web -- 4.4 Transition from the KAM-Tori to Web-Tori -- Conclusions -- 5 FRACTALS AND CHAOS -- 5.1 Fractal Dynamics -- 5.2 Generalised Fractal Dimension -- 5.3 Renormalisation Group and Generalised Fractal Dimension -- 5.4 Multi-Fractal Spectra -- Conclusions -- 6 POINCAR RECURRENCES AND FRACTAL TIME -- 6.1 Poincare Recurrences -- 6.2 Poissonian Distribution of Recurrences -- 6.3 Non-Ergodicity, Stickiness and Quasi-Traps -- 6.4 Renormalisation Formulas for the Exit Time Distribution -- 6.5 Fractal Time -- 6.6 Fractal and Multi-Fractal Recurrences -- 6.7 Multi-Fractal Space-Time and Its Dimension Spectrum -- 6.8 Critical Exponent for the Poincare Recurrences -- 6.9 Rhombic Billiard -- Conclusions -- 7 CHAOS AND FOUNDATION OF STATISTICAL PHYSICS -- 7.1 The Dynamical Foundation of Statistical Physics -- 7.2 Fractal Traps and Maxwell's Demon.

7.3 Coupled Billiards -- 7.4 Contacted Cassini-Sinai Billiards -- 7.5 Weak Mixing and Stickiness -- 7.6 Persistent Fluctuations -- Conclusions -- 8 CHAOS AND SYMMETRY -- 8.1 Stochastic Webs -- 8.2 Stochastic Web with Quasi-Crystalline Symmetry -- 8.3 Stochastic Web Skeleton -- 8.4 Symmetries and Their Dynamical Generation -- 8.5 The Width of the Stochastic Web -- Conclusions -- COLOUR PLATES (C.1 C.8) -- 9 MORE DEGREES OF FREEDOM -- 9.1 General Remarks -- 9.2 Four-Dimensional Map for the Motion in Magnetic Field -- 9.3 Multi-Web Structures in the Phase Space -- 9.4 Equilibrium of the Atomic Chains -- 9.5 Discretisation -- Conclusions -- 10 NORMAL AND ANOMALOUS KINETICS -- 10.1 Fokker-Planck-Kolmogorov (FPK) Equation -- 10.2 Transport for the Standard Map and Web-Map -- 10.3 Dynamics in the Potential with q-Fold Symmetry -- 10.4 More Examples of the Anomalous Transport -- 10.5 Levy Processes -- 10.6 The Weierstrass Random Walk -- Conclusions -- 11 FRACTIONAL KINETICS -- 11.1 Fractional Generalisation of the Fokker-Planck-Kolmogorov Equation (FFPK) -- 11.2 Evolution of Moments -- 11.3 Method of the Renormalisation Group for Kinetics (RGK) -- 11.4 Complex Exponents and Log-Periodicity -- Conclusions -- 12 WEAK CHAOS AND PSEUDOCHAOS -- 12.1 Definitions -- 12.2 Billiards with Pseudochaotic Dynamics -- 12.3 Filamented Surfaces -- 12.4 Bar-in-Square Billiard -- 12.5 Renormalisation Group Equation for Recurrences -- 12.6 Recurrences in the Multi-Bar-Billiard -- Conclusions -- Appendix 1 THE NONLINEAR PENDULUM -- Appendix 2 SOLUTION TO THE RENORMALISATION TRANSFORM EQUATION -- Appendix 3 FRACTIONAL INTEGRO-DIFFERENTIATION -- Appendix 4 FORMULAS OF FRACTIONAL CALCULUS -- REFERENCES -- INDEX.