Synergetic Phenomena in Active Lattices

Within nonlinear spatio-temporal dynamics, active lattice systems are of relevance to the study of multi-dimensional dynamical systems and the theory of nonlinear waves and dis- sipative structures of extended systems. In this book, the authors deal with basic concepts and models, with methodolo- gies for studying the existence and stability of motions, understanding the mechanisms of formation of patterns and waves, their propagation and interactions in active lattice systems, and about how much cooperation or competition bet- ween order and chaos is crucial for synergetic behavior and evolution. The results described in the book have both in- ter- and trans-disciplinary features and a fundamental cha- racter. It is a textbook for graduate courses in nonlinear sciences, including physics, biophysics, biomathematics, bioengineering, neurodynamics, electrical and electronic engineering, mathematical economics, and computer sciences.

Texte du rabat

In this book, the authors deal with basic concepts and models, with methodologies for studying the existence and stability of motions, understanding the mechanisms of formation of patterns and waves, their propagation and interactions in active lattice systems, and about how much cooperation or competition between order and chaos is crucial for synergetic behavior and evolution.

Contenu

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 Elements.- 5.6.2 Stable Steady States.- 5.6.3 Spatial Disorder and Patterns in the FitzHugh-Nagumo-Schlögl Model.- 5.6.4 Spatial Disorder and Patterns in a Lattice of Bistable Oscillators.- 5.7 Patterns and Spiral Waves in a Lattice of Excitable Units.- 5.7.1 Pattern Formation.- 5.7.2 Spiral Wave Patterns.- 5.8 Salient Features of Networks of Bistable Units.- 6. Mutual Synchronization, Control and Replication of Patterns and Waves in Coupled Lattices Composed of Bistable Units.- 6.1 Introduction and Motivation.- 6.2 Layered Lattice System and Mutual Synchronization of Two Lattices.- 6.2.1 Bistable Elements or Units.- 6.2.2 Bistable Oscillators.- 6.2.3 System of Two Coupled Fibers.- 6.2.4 Excitable Units.- 6.3 Controlled Patterns and Replication of Form.- 6.3.1 Bistable Oscillators and Replication.- 6.3.2 Excitable Units.- 6.4 Salient Features of Replication Processes via Synchronization of Patterns and Waves with Interacting Bistable Units.- 7. Spatio-Temporal Chaos in Bistable Coupled Map Lattices.- 7.1 Introduction and Motivation.- 7.2 Spectrum of the Linearized Operator.- 7.2.1 Linear Operator.- 7.2.2 A Finite-Dimensional Approximation of the Linear Operator.- 7.2.3 Methodology to Obtain the Linear Spectrum.- 7.2.4 Gershgorin Disks.- 7.2.5 An Alternative Way to Obtain the Stability Criterion.- 7.3 Spatial Chaos in a Discrete Version of the One-Dimensional FitzHugh-Nagumo-Schlögl Equation.- 7.3.1 Spatial Chaos.- 7.3.2 A Discrete Version of the One-Dimensional FitzHugh-Nagumo-Schlögl Equation.- 7.3.3 Steady States.- 7.3.4 Stability of Spatially Steady Solutions.- 7.4 Chaotic Traveling Waves in a One-Dimensional Discrete FitzHugh-Nagumo-Schlögl Equation.- 7.4.1 Traveling Wave Equation.- 7.4.2 Existence of Traveling Waves.- 7.4.3 Stability of Traveling Waves.- 7.5 Two-Dimensional Spatial Chaos.- 7.5.1 Invariant Domains.- 7.5.2 Existence of Steady Solutions.- 7.5.3 Stability of Steady Solutions.- 7.5.4 Two-Dimensional Spatial Chaos.- 7.6 Synchronization in Two-Layer Bistable Coupled Map Lattices.- 7.6.1 Layered Coupled Map Lattices.- 7.6.2 Dynamics of a Single Lattice (Layer).- 7.6.3 Global Interlayer Synchronization.- 7.7 Instability of the Synchronization Manifold.- 7.7.1 Instability of the Synchronized Fixed Points.- 7.7.2 Instability of Synchronized Attractors and On-Off Intermittency.- 7.8 Salient Features of Coupled Map Lattices.- 8. Conclusions and Perspective.- Appendices.- A. Integral Manifolds of Stationary Points.- D. Instability of Spatially Homogeneous States.- E. Topological Entropy and Lyapunov Exponent.- F. Multipliers of the Fixed Point of the Coupled Map Lattice (7.55).- G. Gershgorin Theorem.- References.