DYNAMICAL SYSTEMS AND FRACTALS
CONTENTS OVERVIEW AND BASIC CONCEPTS INTRODUCTION THE PROGRAMS LITERATURE ON CHAOTIC DYNAMICS NONLINEAR DYNAMICS AND DETERMINISTIC CHAOS DETERMINISTIC CHAOS HAMILTONIAN SYSTEMS INTEGRABLE AND ERGODIC SYSTEMS POINCAR´E SECTIONS THE KAM THEOREM HOMOCLINIC POINTS DISSIPATIVE DYNAMICAL SYSTEMS ATTRACTORS ROUTES TO CHAOS SPECIAL TOPICS THE POINCAR´E-BIRKHO?THEOREM CONTINUED FRACTIONS THE LYAPUNOV EXPONENT FIXED POINTS OF ONE-DIMENSIONAL MAPS FIXED POINTS OF TWO-DIMENSIONAL MAPS BIFURCATIONS REFERENCES BILLIARD SYSTEMS DEFORMATIONS OF A CIRCLE BILLIARD NUMERICAL TECHNIQUES INTERACTING WITH THE PROGRAM COMPUTER EXPERIMENTS FROM REGULARITY TO CHAOS ZOOMING IN SENSITIVITY AND DETERMINISM SUGGESTIONS FOR ADDITIONAL EXPERIMENTS SUGGESTIONS FOR FURTHER STUDIES REAL EXPERIMENTS AND EMPIRICAL EVIDENCE GRAVITATIONAL BILLIARDS: THE WEDGE THE POINCAR´E MAPPING PERIODIC MOTION AND PHASE SPACE ORGANIZATION BIFURCATION PHENOMENA ‘PLANE FILLING’ WEDGE BILLIARDS THE DOUBLE PENDULUM EQUATIONS OF MOTION NUMERICAL ALGORITHMS DI?ERENT TYPES OF MOTION DYNAMICS OF THE DOUBLE PENDULUM DESTRUCTION OF INVARIANT CURVES CHAOTIC SCATTERING SCATTERING O?THREE DISKS SCATTERING FUNCTIONS AND TWO-DISK COLLISIONS TREE ORGANIZATION OF THREE-DISK COLLISIONS UNSTABLE PERIODIC ORBITS FRACTAL SINGULARITY STRUCTURE FERMI ACCELERATION FERMI MAPPING EXPLORING PHASE SPACE FOR DI?ERENT WALL OSCILLATIONS KAM CURVES AND STOCHASTIC ACCELERATION FIXED POINTS AND LINEAR STABILITY ABSOLUTE BARRIERS THE DU?NG OSCILLATOR THE DU?NG EQUATION CHAOTIC AND REGULAR OSCILLATIONS THE FREE DU?NG OSCILLATOR ANHARMONIC VIBRATIONS: RESONANCES AND BISTABILITY COEXISTING LIMIT CYCLES AND STRANGE ATTRACTORS FEIGENBAUM SCENARIO ONE-DIMENSIONAL MAPS PERIOD-DOUBLING BIFURCATIONS THE CHAOTIC REGIME LYAPUNOV EXPONENTS THE TENT MAP NONLINEAR ELECTRONIC CIRCUITS 10 1 A CHAOS GENERATOR 10 2 NUMERICAL TECHNIQUES 10 3 INTERACTING WITH THE PROGRAM 10 4 COMPUTER EXPERIMENTS 10 4 1 HOPF BIFURCATION 10 4 2 PERIOD-DOUBLING 10 4 3 RETURN MAP 10 4 4 SUGGESTIONS FOR ADDITIONAL EXPERIMENTS 10 5 REAL EXPERIMENTS AND EMPIRICAL EVIDENCE MANDELBROT AND JULIA SETS 11 1 TWO-DIMENSIONAL ITERATED MAPS 11 2 NUMERICAL METHODS 11 3 INTERACTING WITH THE PROGRAM 11 4 COMPUTER EXPERIMENTS 11 4 1 MANDELBROT AND JULIA-SETS 11 4 2 ZOOMING INTO THE MANDELBROT SET 11 4 3 GENERAL TWO-DIMENSIONAL QUADRATIC MAPPINGS 11 4 4 SUGGESTIONS FOR ADDITIONAL EXPERIMENTS 11 5 SUGGESTIONS FOR FURTHER STUDIES 11 6 REAL EXPERIMENTS AND EMPIRICAL EVIDENCE ORDINARY DI?ERENTIAL EQUATIONS 12 1 NUMERICAL TECHNIQUES 12 2 INTERACTING WITH THE PROGRAM 12 3 COMPUTER EXPERIMENTS 12 3 1 THE PENDULUM 12 3 2 A SIMPLE HOPF BIFURCATION 12 3 3 THE DU?NG OSCILLATOR REVISITED 12 3 4 HILL’S EQUATION 12 3 5 THE LORENZ ATTRACTOR 12 3 6 THE R¨OSSLER ATTRACTOR 12 3 7 THE H´ENON-HEILES SYSTEM 12 3 8 SUGGESTIONS FOR ADDITIONAL EXPERIMENTS 12 4 SUGGESTIONS FOR FURTHER STUDIES KICKED SYSTEMS 13 1 INTERACTING WITH THE PROGRAM 13 2 COMPUTER EXPERIMENTS 13 2 1 THE STANDARD MAPPING 13 2 2 THE KICKED QUARTIC OSCILLATOR 13 2 3 THE KICKED QUARTIC OSCILLATOR WITH DAMPING 13 2 4 THE H´ENON MAP 13 2 5 SUGGESTIONS FOR ADDITIONAL EXPERIMENTS 13 3 REAL EXPERIMENTS AND EMPIRICAL EVIDENCE