HIGH-DIMENSIONAL CHAOTIC AND ATTRACTOR SYSTEMS
CONTENTS INTRODUCTION TO ATTRACTORS AND CHAOS BASICS OF ATTRACTOR AND CHAOTIC DYNAMICS A BRIEF HISTORY OF CHAOS THEORY IN 5 STEPS HENRY POINCAR´E: QUALITATIVE DYNAMICS, TOPOLOGY AND CHAOS STEPHEN SMALE: TOPOLOGICAL HORSESHOE AND CHAOS OF STRETCHING AND FOLDING MITCHELL FEIGENBAUM: A CONSTANT AND UNIVERSALITY LORD ROBERT MAY: POPULATION MODELLING AND CHAOS MICHEL H´ENON: A SPECIAL 2D MAP AND ITS STRANGE ATTRACTOR SOME CLASSICAL ATTRACTOR AND CHAOTIC SYSTEMS BASICS OF CONTINUOUS DYNAMICAL ANALYSIS A MOTIVATING EXAMPLE SYSTEMS OF ODES LINEAR AUTONOMOUS DYNAMICS: ATTRACTORS & REPELLORS CONSERVATIVE VERSUS DISSIPATIVE DYNAMICS BASICS OF NONLINEAR DYNAMICS ERGODIC SYSTEMS CONTINUOUS CHAOTIC DYNAMICS DYNAMICS AND NON–EQUILIBRIUM STATISTICAL MECHANICS STATISTICAL MECHANICS OF NONLINEAR OSCILLATOR CHAINS GEOMETRICAL MODELLING OF CONTINUOUS DYNAMICS LAGRANGIAN CHAOS STANDARD MAP AND HAMILTONIAN CHAOS CHAOTIC DYNAMICS OF BINARY SYSTEMS EXAMPLES OF DYNAMICAL MAPS CORRELATION DIMENSION OF AN ATTRACTOR BASIC HAMILTONIAN MODEL OF BIODYNAMICS ED LORENZ: WEATHER PREDICTION AND CHAOS SMALE HORSESHOES AND HOMOCLINIC DYNAMICS SMALE HORSESHOE ORBITS AND SYMBOLIC DYNAMICS HORSESHOE TRELLIS TRELLIS–FORCED DYNAMICS HOMOCLINIC BRAID TYPE HOMOCLINIC CLASSES FOR GENERIC VECTOR–FIELDS LYAPUNOV STABILITY HOMOCLINIC CLASSES COMPLEX–VALUED H´ENON MAPS AND HORSESHOES COMPLEX HENON–LIKE MAPS COMPLEX HORSESHOES CHAOS IN FUNCTIONAL DELAY EQUATIONS POINCAR´E MAPS AND HOMOCLINIC SOLUTIONS STARTING VALUE AND TARGETS SUCCESSIVE MODI?CATIONS OF G TRANSVERSALITY TRANSVERSALLY HOMOCLINIC SOLUTIONS 3–BODY PROBLEM AND CHAOS CONTROL MECHANICAL ORIGIN OF CHAOS RESTRICTED 3–BODY PROBLEM SCALING AND REDUCTION IN THE 3–BODY PROBLEM PERIODIC SOLUTIONS OF THE 3–BODY PROBLEM BIFURCATING PERIODIC SOLUTIONS OF THE 3–BODY PROBLEM BIFURCATIONS IN LAGRANGIAN EQUILIBRIA CONTINUATION OF KAM–TORI PARAMETRIC RESONANCE AND CHAOS IN COSMOLOGY ELEMENTS OF CHAOS CONTROL FEEDBACK AND NON–FEEDBACK ALGORITHMS FOR CHAOS CONTROL EXPLOITING CRITICAL SENSITIVITY LYAPUNOV EXPONENTS AND KAPLAN–YORKE DIMENSION KOLMOGOROV–SINAI ENTROPY CHAOS CONTROL BY OTT, GREBOGI AND YORKE (OGY) FLOQUET STABILITY ANALYSIS AND OGY CONTROL BLIND CHAOS CONTROL JERK FUNCTIONS OF SIMPLE CHAOTIC FLOWS EXAMPLE: CHAOS CONTROL IN MOLECULAR DYNAMICS PHASE TRANSITIONS AND SYNERGETICS PHASE TRANSITIONS, PARTITION FUNCTION AND NOISE EQUILIBRIUM PHASE TRANSITIONS CLASSI?CATION OF PHASE TRANSITIONS BASIC PROPERTIES OF PHASE TRANSITIONS LANDAU’S THEORY OF PHASE TRANSITIONS PARTITION FUNCTION NOISE–INDUCED NON–EQUILIBRIUM PHASE TRANSITIONS ELEMENTS OF HAKEN’S SYNERGETICS PHASE TRANSITIONS MEZOSCOPIC DERIVATION OF ORDER PARAMETERS EXAMPLE: SYNERGETIC CONTROL OF BIODYNAMICS EXAMPLE: CHAOTIC PSYCHODYNAMICS OF PERCEPTION KICK DYNAMICS AND DISSIPATION–FLUCTUATION THEOREM SYNERGETICS OF RECURRENT AND ATTRACTOR NEURAL NETWORKS STOCHASTIC DYNAMICS OF NEURONAL FIRING STATES SYNAPTIC SYMMETRY AND LYAPUNOV FUNCTIONS DETAILED BALANCE AND EQUILIBRIUM STATISTICAL MECHANICS SIMPLE RECURRENT NETWORKS WITH BINARY NEURONS SIMPLE RECURRENT NETWORKS OF COUPLED OSCILLATORS ATTRACTOR NEURAL NETWORKS WITH BINARY NEURONS ATTRACTOR NEURAL NETWORKS WITH CONTINUOUS NEURONS CORRELATION– AND RESPONSE–FUNCTIONS PATH–INTEGRAL APPROACH FOR COMPLEX DYNAMICS 4310 HIERARCHICAL SELF–PROGRAMMING IN NEURAL NETWORKS TOPOLOGICAL PHASE TRANSITIONS AND HAMILTONIAN CHAOS PHASE TRANSITIONS IN HAMILTONIAN SYSTEMS GEOMETRY OF THE LARGEST LYAPUNOV EXPONENT EULER CHARACTERISTICS OF HAMILTONIAN SYSTEMS PATHWAYS TO SELF–ORGANIZATION IN HUMAN BIODYNAMICS PHASE SYNCHRONIZATION IN CHAOTIC SYSTEMS LYAPUNOV VECTORS AND LYAPUNOV EXPONENTS FORCED R¨OSSLER OSCILLATOR SECOND LYAPUNOV EXPONENT: PERTURBATIVE CALCULATION PHASE SYNCHRONIZATION IN COUPLED CHAOTIC OSCILLATORS OSCILLATORY PHASE NEURODYNAMICS KURAMOTO SYNCHRONIZATION MODEL LYAPUNOV CHAOTIC SYNCHRONIZATION SYNCHRONIZATION GEOMETRY GEOMETRY OF COUPLED NONLINEAR OSCILLATORS NOISY COUPLED NONLINEAR OSCILLATORS SYNCHRONIZATION CONDITION COMPLEX NETWORKS AND CHAOTIC TRANSIENTS JOSEPHSON JUNCTIONS AND QUANTUM ENGINEERING JOSEPHSON E?ECT PENDULUM ANALOG DISSIPATIVE JOSEPHSON JUNCTION JUNCTION HAMILTONIAN AND ITS EIGENSTATES TRANSITION RATE JOSEPHSON JUNCTION LADDER (JJL) UNDERDAMPED JJL SYNCHRONIZATION IN ARRAYS OF JOSEPHSON JUNCTIONS PHASE MODEL FOR UNDERDAMPED JJL COMPARISON OF LKM2 AND RCSJ MODELS ‘SMALL–WORLD’ CONNECTIONS IN JJL ARRAYS FRACTALS AND FRACTIONAL DYNAMICS FRACTALS MANDELBROT SET ROBUST STRANGE NON–CHAOTIC ATTRACTORS QUASI–PERIODICALLY FORCED MAPS 2D MAP ON A TORUS HIGH DIMENSIONAL MAPS E?ECTIVE DYNAMICS IN HAMILTONIAN SYSTEMS E?ECTIVE DYNAMICAL INVARIANTS FORMATION OF FRACTAL STRUCTURE IN MANY–BODY SYSTEMS A MANY–BODY HAMILTONIAN LINEAR PERTURBATION ANALYSIS FRACTIONAL CALCULUS AND CHAOS CONTROL FRACTIONAL CALCULUS FRACTIONAL–ORDER CHUA’S CIRCUIT FEEDBACK CONTROL OF CHAOS FRACTIONAL GRADIENT AND HAMILTONIAN DYNAMICS GRADIENT SYSTEMS FRACTIONAL DI?ERENTIAL FORMS FRACTIONAL GRADIENT SYSTEMS HAMILTONIAN SYSTEMS FRACTIONAL HAMILTONIAN SYSTEMS TURBULENCE PARAMETER–SPACE ANALYSIS OF THE LORENZ ATTRACTOR STRUCTURE OF THE PARAMETER–SPACE ATTRACTORS AND BIFURCATIONS PERIODICALLY–DRIVEN LORENZ DYNAMICS TOY MODEL ILLUSTRATION LORENZIAN DI?USION TURBULENT FLOW THE GOVERNING EQUATIONS OF TURBULENCE GLOBAL WELL-POSEDNESS OF THE NAVIER–STOKES EQUATIONS SPATIO–TEMPORAL CHAOS AND TURBULENCE IN PDES GENERAL FLUID DYNAMICS COMPUTATIONAL FLUID DYNAMICS TURBULENCE KINETICS KINETIC THEORY FILTERED KINETIC THEORY HYDRODYNAMIC LIMIT HYDRODYNAMIC EQUATIONS LIE SYMMETRIES IN THE MODELS OF TURBULENCE LIE SYMMETRIES AND PROLONGATIONS ON MANIFOLDS NOETHER THEOREM AND NAVIER–STOKES EQUATIONS LARGE–EDDY SIMULATION MODEL ANALYSIS THERMODYNAMIC CONSISTENCE STABILITY OF TURBULENCE MODELS ADVECTION OF VECTOR–FIELDS BY CHAOTIC FLOWS ADVECTIVE FLUID FLOW CHAOTIC FLOWS BROWNIAN MOTION AND DI?USION RANDOM WALK MODEL MORE COMPLICATED TRANSPORT PROCESSES ADVECTION–DI?USION BEYOND THE DI?USION COE?CIENT GEOMETRY, SOLITONS AND CHAOS FIELD THEORY CHAOTIC DYNAMICS AND RIEMANNIAN GEOMETRY CHAOS IN PHYSICAL GAUGE FIELDS SOLITIONS HISTORY OF SOLITONS IN BRIEF THE FERMI–PASTA–ULAM EXPERIMENTS THE KRUSKAL–ZABUSKY EXPERIMENTS A FIRST LOOK AT THE KDV EQUATION SPLIT–STEPPING KDV SOLITONS FROM A PENDULUM CHAIN 1D CRYSTAL SOLITON SOLITONS AND CHAOTIC SYSTEMS CHAOS FIELD THEORY REFERENCES