Synergetic Phenomena in Active Lattices: Patterns, Waves, Solitons, Chaos

ISBN-13: 9783540427155 / Angielski / Twarda / 2002 / 359 str.

V. I. Nekorkin; Vladimir I. Nekorkin; M. G. Velarde

Opis

In recent years there has been growing interest in the study of the nonlinear spatio-temporal dynamics of problems appearing in various ?elds of science and engineering. In a wide class of such systems an important place is - cupied by active lattice dynamical systems. Active lattice systems are, e. g., networks of identical or almost identical interacting units ordered in space. The activity of lattices is provided by the activity of units in them that possess energy or matter sources. In real (1D, 2D or 3D) space, processes develop by means of various types of connections, the simplest being di?usion. The uniqueness of lattice systems is that they represent spatially extended systems while having a ?nite-dimensional phase space. Therefore, active lattice s- tems are of interest for the study of multidimensional dynamical systems and the theory of nonlinear waves and dissipative structures of extended systems as well. The theory of nonlinear waves and dissipative structures of spatially distributed systems demands using theoretical methods and approaches of the qualitative theory of dynamical systems, bifurcation theory, and numerical methods or computer experiments. In other words, the investigation of spat- temporal dynamics in active lattice systems demands a multitool, synergetic approach, which we shall use in this book.

Recenzje

"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 and numerical methods that can be used in the study of concrete applications. In sum, it is a highly enjoyable, well-written book by two leading scientists in the field, with carefully chosen material, which is highly recommended to anyone wanting a good introduction to the subject." (Mathematical Reviews 2003b)

"In an applied, descriptive manner, the authors present and corroborate results through a mixture of heuristics, numerical investigations and mathematical analysis guided by general methods from geometric dynamical systems theory. [...] Each of the well written chapters starts with a motivation and ends with a summary; general conclusions and perspectives are given in the final chapter. [...] This book provides an easily accessible introduction and overview of phenomena and the current state of understanding in the field of waves and synchronization in spatially discrete systems." (Zentralblatt MATH, 1006, 2003)

"This book gives a comprehensive account of synergetic phenomena in active lattices. ... throughout the book insights on the possible use of the results in applications such as computer architecture or neuronal science are given. ... The book may serve as an invaluable guide to all those interested in the rich pheonomenology of spatio-temporal dynamic phenomena in active lattices ... . In sum, it is a highly enjoyable, well-written book ... which is highly recommended to anyone wanting a good introduction to the subject." (Athanasios Yannacopoulos, Mathematical Reviews, Issue 2003 b)

"The present textbook concerns pattern formation in lattices of coupled cells with oscillatory or excitable dynamics. In an applied, descriptive manner, the authors present and corroborate results through a mixture of heuristics, numerical investigations and mathematical analysis guided by general methods from geometric dynamical systems theory. ... This book provides an easily accessible introduction and overview of phenomena and the current state of understanding in the field of waves and synchronization in spatially discrete systems." (Jens Rademacher, Zentralblatt MATH, Vol. 1006, 2003)

Spis treści

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.

Blurb

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.