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Synergetic phenomena in active lattices : patterns, waves, solitons, chaos

Author: Vladimir I Nekorkin; Manuel G Velarde
Publisher: Berlin ; New York : Springer, ©2002.
Series: Springer series in synergetics (Unnumbered)
Edition/Format:   Print book : EnglishView all editions and formats
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Deals 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  Read more...

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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Vladimir I Nekorkin; Manuel G Velarde
ISBN: 3540427155 9783540427155
OCLC Number: 51637809
Description: xvii, 357 pages : illustrations ; 25 cm.
Contents: 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-Schlogl 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-Schlogl Equation.- 7.3.1 Spatial Chaos.- 7.3.2 A Discrete Version of the One-Dimensional FitzHugh-Nagumo-Schlogl 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-Schlogl 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.
Series Title: Springer series in synergetics (Unnumbered)
Responsibility: Vladimir I. Nekorkin, Manuel G. Velarde.
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"The book may serve as an invaluable guide to all those interested in the rich phenomenology of spatio-temporal dynamic phenomena in active lattices, providing the reader with a variety of analytical Read more...

 
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