THE PHYSICS OF CHAOS IN HAMILTONIAN SYSTEMS
CONTENTS 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 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 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 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 6 POINCAR´E RECURRENCES AND FRACTAL TIME 6 1 POINCAR´E 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 POINCAR´E RECURRENCES ?6 9 RHOMBIC BILLIARD 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 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 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 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 L´EVY PROCESSES 10 6 THE WEIERSTRASS RANDOM WALK 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 ?12 WEAK CHAOS AND PSEUDOCHAOS 12 1 DE?NITIONS 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