High-Dimensional Chaotic and Attractor Systems (eBook)

1. Introduction to Attractors and Chaos
1.1 Basics of Attractor and Chaotic Dynamics
1.2 Brief History of Chaos Theory in 5 Steps
1.2.1 Henry Poincar´e: Qualitative Dynamics, Topology and Chaos
1.2.2 Steve Smale: Topological Horseshoe and Chaos of Stretching and Folding
1.2.3 Ed Lorenz: Weather Prediction and Chaos
1.2.4 Mitchell Feigenbaum: Feigenbaum Constant and Universality
1.2.5 Lord Robert May: Population Modelling and Chaos
1.2.6 Michel H´enon: A Special 2D Map and Its Strange Attractor
1.3 Some Classical Attractor and Chaotic Systems
1.4 Basics of Continuous Dynamical Analysis
1.4.1 A Motivating Example
1.4.2 Systems of ODEs
1.4.3 Linear Autonomous Dynamics: Attractors & Repellors
1.4.4 Conservative versus Dissipative Dynamics
1.4.5 Basics of Nonlinear Dynamics
1.4.6 Ergodic Systems
1.5 Continuous Chaotic Dynamics
1.5.1 Dynamics and Non–equilibrium Statistical Mechanics
1.5.2 Statistical Mechanics of Nonlinear Oscillator Chains
1.5.3 Geometrical Modelling of Continuous Dynamics
1.5.4 Lagrangian Chaos
1.6 Standard Map and Hamiltonian Chaos
1.7 Chaotic Dynamics of Binary Systems
1.7.1 Examples of Dynamical Maps
1.7.2 Correlation Dimension of an Attractor
1.8 Basic Hamiltonian Model of Biodynamics 2. Smale Horseshoes and Homoclinic Dynamics
2.1 Smale Horseshoe Orbits and Symbolic Dynamics
2.1.1 Horseshoe Trellis
2.1.2 Trellis–Forced Dynamics
2.1.3 Homoclinic Braid Type
2.2 Homoclinic Classes for Generic Vector–Fields
2.2.1 Lyapunov Stability
2.2.2 Homoclinic Classes
2.3 Complex–Valued H´enon Maps and Horseshoes
2.3.1 Complex Henon–Like Maps
2.3.2 Complex Horseshoes
2.4 Chaos in Functional Delay Equations
2.4.1 Poincar´e Maps and Homoclinic Solutions
2.4.2 Starting Value and Targets
2.4.3 Successive Modifications of g
2.4.4 Transversality
2.4.5 Transversally Homoclinic Solutions 3. 3–BodyProblem and Chaos Control
3.1 Mechanical Origin of Chaos
3.1.1 Restricted 3–Body Problem
3.1.2 Scaling and Reduction in the 3–Body Problem
3.1.3 Periodic Solutions of the 3–Body Problem
3.1.4 Bifurcating Periodic Solutions of the 3–Body Problem
3.1.5 Bifurcations in Lagrangian Equilibria
3.1.6 Continuation of KAM–Tori
3.1.7 Parametric Resonance and Chaos in Cosmology
3.2 Elements of Chaos Control
3.2.1 Feedback and Non–Feedback Algorithms for Chaos Control
3.2.2 Exploiting Critical Sensitivity
3.2.3 Lyapunov Exponents and KY–Dimension
3.2.4 Kolmogorov–Sinai Entropy
3.2.5 Classical Chaos Control by Ott, Grebogi and Yorke
3.2.6 Floquet Stability Analysis and OGY Control
3.2.7 Blind Chaos Control
3.2.8 Jerk Functions of Simple Chaotic Flows
3.2.9 Example: Chaos Control in Molecular Dynamics 4. Phase Transitions and Synergetics
4.1 Phase Transitions, Partition Function and Noise
4.1.1 Equilibrium Phase Transitions
4.1.2 Classification of Phase Transitions
4.1.3 Basic Properties of Phase Transitions
4.1.4 Landau’s Theory of Phase Transitions
4.1.5 Partition Function
4.1.6 Noise–Induced Non–equilibrium Phase Transitions
4.2 Elements of Haken’s Synergetics
4.2.1 Phase Transitions
4.2.2 Mezoscopic Derivation of Order Parameters
4.2.3 Example: Synergetic Control of Biodynamics
4.2.4 Example: Chaotic Psychodynamics of Perception
4.2.5 Kick Dynamics and Dissipation–Fluctuation Theorem
4.3 Synergetics of Recurrent and Attractor Neural Networks
4.3.1 Stochastic Dynamics of Neuronal Firing States
4.3.2 Synaptic Symmetry and Lyapunov Functions
4.3.3 Detailed Balance and Equilibrium Statistical Mechanics
4.3.4 Simple Recurrent Networks with Binary Neurons
4.3.5 Simple Recurrent Networks of Coupled Oscillators
4.3.6 Attractor Neural Networks with Binary Neurons
4.3.7 Attractor Neural Networks with Continuous Neurons
4.3.8