Synergetic Phenomena in Active Lattices: Patterns, Waves, Solitons, Chaos (Hardcover, 2002)

'1. Introduction: Synergetics and Models of Continuous and Discrete Active Media. Steady States and Basic Motions (Waves, Dissipative Solitons, etc.).- 1.1 Basic Concepts, Phenomena and Context.- 1.2 Continuous Models.- 1.3 Chain and Lattice Models with Continuous Time.- 1.4 Chain and Lattice Models with Discrete Time.- 2. Solitary Waves, Bound Soliton States and Chaotic Soliton Trains in a Dissipative Boussinesq-Korteweg-de Vries Equation.- 2.1 Introduction and Motivation.- 2.2 Model Equation.- 2.3 Traveling Waves.- 2.3.1 Steady States.- 2.3.2 Lyapunov Functions.- 2.4 Homoclinic Orbits. Phase-Space Analysis.- 2.4.1 Invariant Subspaces.- 2.4.2 Auxiliary Systems.- 2.4.3 Construction of Regions Confining the Unstable and Stable Manifolds Wu and Ws.- 2.5 Multiloop Homoclinic Orbits and Soliton-Bound States.- 2.5.1 Existence of Multiloop Homoclinic Orbits.- 2.5.2 Solitonic Waves, Soliton-Bound States and Chaotic Soliton-Trains.- 2.5.3 Homoclinic Orbits and Soliton-Trains. Some Numerical Results.- 2.6 Further Numerical Results and Computer Experiments.- 2.6.1 Evolutionary Features.- 2.6.2 Numerical Collision Experiments.- 2.7 Salient Features of Dissipative Solitons.- 3. Self-Organization in a Long Josephson Junction.- 3.1 Introduction and Motivation.- 3.2 The Perturbed Sine-Gordon Equation.- 3.3 Bifurcation Diagram of Homoclinic Trajectories.- 3.4 Current-Voltage Characteristics of Long Josephson Junctions 54.- 3.5 Bifurcation Diagram in the Neighborhood of c = 1.- 3.5.1 Spiral-Like Bifurcation Structures.- 3.5.2 Heteroclinic Contours.- 3.5.3 The Neighborhood of Ai.- 3.5.4 The Sets {?i} and {?i}.- 3.6 Existence of Homoclinic Orbits.- 3.6.1 Lyapunov Function.- 3.6.2 The Vector Field of (3.4) on Two Auxiliary Surfaces.- 3.6.3 Auxiliary Systems.- 3.6.4 "Tunnels" for Manifolds of the Saddle Steady State O2.- 3.6.5 Homoclinic Orbits.- 3.7 Salient Features of the Perturbed Sine-Gordon Equation.- 4. Spatial Structures, Wave Fronts, Periodic Waves, Pulses and Solitary Waves in a One-Dimensional Array of Chua's Circuits.- 4.1 Introduction and Motivation.- 4.2 Spatio-Temporal Dynamics of an Array of Resistively Coupled Units.- 4.2.1 Steady States and Spatial Structures.- 4.2.2 Wave Fronts in a Gradient Approximation.- 4.2.3 Pulses, Fronts and Chaotic Wave Trains.- 4.3 Spatio-Temporal Dynamics of Arrays with Inductively Coupled Units.- 4.3.1 Homoclinic Orbits and Solitary Waves.- 4.3.2 Periodic Waves in a Circular Array.- 4.4 Chaotic Attractors and Waves in a One-Dimensional Array of Modified Chua's Circuits.- 4.4.1 Modified Chua's Circuit.- 4.4.2 One-Dimensional Array.- 4.4.3 Chaotic Attractors.- 4.5 Salient Features of Chua's Circuit in a Lattice.- 4.5.1 Array with Resistive Coupling.- 4.5.2 Array with Inductive Coupling.- 5. Patterns, Spatial Disorder and Waves in a Dynamical Lattice of Bistable Units.- 5.1 Introduction and Motivation.- 5.2 Spatial Disorder in a Linear Chain of Coupled Bistable Units.- 5.2.1 Evolution of Amplitudes and Phases of the Oscillations.- 5.2.2 Spatial Distributions of Oscillation Amplitudes.- 5.2.3 Phase Clusters in a Chain of Isochronous Oscillators.- 5.3 Clustering and Phase Resetting in a Chain of Bistable Nonisochronous Oscillators.- 5.3.1 Amplitude Distribution along the Chain.- 5.3.2 Phase Clusters in a Chain of Nonisochronous Oscillators.- 5.3.3 Frequency Clusters and Phase Resetting.- 5.4 Clusters in an Assembly of Globally Coupled Bistable Oscillators.- 5.4.1 Homogeneous Oscillations.- 5.4.2 Amplitude Clusters.- 5.4.3 Amplitude-Phase Clusters.- 5.4.4 "Splay-Phase" States.- 5.4.5 Collective Chaos.- 5.5 Spatial Disorder and Waves in a Circular Chain of Bistable Units.- 5.5.1 Spatial Disorder.- 5.5.2 Space-Homogeneous Phase Waves.- 5.5.3 Space-Inhomogeneous Phase Waves.- 5.6 Chaotic and Regular Patterns in Two-Dimensional Lattices of Coupled Bistable Units.- 5.6.1 Methodology for a Lattice of Bistable Element