Synergetic Phenomena in Active Lattices

by Nekorkin, Vladimir I.; Velarde, Manuel G.

• ISBN13: 9783540427155
• ISBN10: 3540427155
• Format: Hardcover
• Copyright: 2002-06-01
• Publisher: Springer Nature

Summary

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.

Table of Contents

Introduction: Synergetics and Models of Continuous and Discrete Active Media. Steady States and Basic Motions (Waves, Dissipative Solitons, etc.)	1	(18)
Basic Concepts, Phenomena and Context	1	(7)
Continuous Models	8	(4)
Chain and Lattice Models with Continuous Time	12	(3)
Chain and Lattice Models with Discrete Time	15	(4)
Solitary Waves, Bound Soliton States and Chaotic Soliton Trains in a Dissipative Boussinesq--Korteweg--de Vries Equation	19	(30)
Introduction and Motivation	19	(2)
Model Equation	21	(2)
Traveling Waves	23	(3)
Steady States	24	(1)
Lyapunov Functions	25	(1)
Homoclinic Orbits. Phase-Space Analysis	26	(5)
Invariant Subspaces	26	(1)
Auxiliary Systems	27	(1)
Construction of Regions Confining the Unstable and Stable Manifolds Wu and Ws	28	(3)
Multiloop Homoclinic Orbits and Soliton-Bound States	31	(8)
Existence of Multiloop Homoclinic Orbits	31	(3)
Solitonic Waves, Soliton-Bound States and Chaotic Soliton-Trains	34	(1)
Homoclinic Orbits and Soliton-Trains. Some Numerical Results	35	(4)
Further Numerical Results and Computer Experiments	39	(9)
Evolutionary Features	40	(3)
Numerical Collision Experiments	43	(5)
Salient Features of Dissipative Solitons	48	(1)
Self-Organization in a Long Josephson Junction	49	(28)
Introduction and Motivation	49	(1)
The Perturbed Sine--Gordon Equation	50	(1)
Bifurcation Diagram of Homoclinic Trajectories	51	(3)
Current--Voltage Characteristics of Long Josephson Junctions	54	(2)
Bifurcation Diagram in the Neighborhood of c = 1	56	(11)
Spiral-Like Bifurcation Structures	56	(2)
Heteroclinic Contours	58	(3)
The Neighborhood of Ai	61	(4)
The Sets {γi} and {γi}	65	(2)
Existence of Homoclinic Orbits	67	(7)
Lyapunov Function	68	(1)
The Vector Field of (3.4) on Two Auxiliary Surfaces	69	(1)
Auxiliary Systems	69	(1)
``Tunnels'' for Manifolds of the Saddle Steady State O2	70	(1)
Homoclinic Orbits	71	(3)
Salient Features of the Perturbed Sine--Gordon Equation	74	(3)
Spatial Structures, Wave Fronts, Periodic Waves, Pulses and Solitary Waves in a One-Dimensional Array of Chua's Circuits	77	(88)
Introduction and Motivation	77	(2)
Spatio-Temporal Dynamics of an Array of Resistively Coupled Units	79	(27)
Steady States and Spatial Structures	80	(6)
Wave Fronts in a Gradient Approximation	86	(8)
Pulses, Fronts and Chaotic Wave Trains	94	(12)
Spatio-Temporal Dynamics of Arrays with Inductively Coupled Units	106	(31)
Homoclinic Orbits and Solitary Waves	106	(17)
Periodic Waves in a Circular Array	123	(14)
Chaotic Attractors and Waves in a One-Dimensional Array of Modified Chua's Circuits	137	(24)
Modified Chua's Circuit	137	(2)
One-Dimensional Array	139	(1)
Chaotic Attractors	139	(22)
Salient Features of Chua's Circuit in a Lattice	161	(4)
Array with Resistive Coupling	162	(1)
Array with Inductive Coupling	162	(3)
Patterns, Spatial Disorder and Waves in a Dynamical Lattice of Bistable Units	165	(62)
Introduction and Motivation	165	(1)
Spatial Disorder in a Linear Chain of Coupled Bistable Units	166	(6)
Evolution of Amplitudes and Phases of the Oscillations	166	(2)
Spatial Distributions of Oscillation Amplitudes	168	(3)
Phase Clusters in a Chain of Isochronous Oscillators	171	(1)
Clustering and Phase Resetting in a Chain of Bistable Nonisochronous Oscillators	172	(7)
Amplitude Distribution along the Chain	173	(2)
Phase Clusters in a Chain of Nonisochronous Oscillators	175	(1)
Frequency Clusters and Phase Resetting	176	(3)
Clusters in an Assembly of Globally Coupled Bistable Oscillators	179	(16)
Homogeneous Oscillations	180	(1)
Amplitude Clusters	181	(5)
Amplitude-Phase Clusters	186	(5)
``Splay-Phase'' States	191	(3)
Collective Chaos	194	(1)
Spatial Disorder and Waves in a Circular Chain of Bistable Units	195	(11)
Spatial Disorder	195	(2)
Space-Homogeneous Phase Waves	197	(4)
Space-Inhomogeneous Phase Waves	201	(5)
Chaotic and Regular Patterns in Two-Dimensional Lattices of Coupled Bistable Units	206	(10)
Methodology for a Lattice of Bistable Elements	206	(3)
Stable Steady States	209	(2)
Spatial Disorder and Patterns in the FitzHugh--Nagumo--Schlogl Model	211	(1)
Spatial Disorder and Patterns in a Lattice of Bistable Oscillators	212	(4)
Patterns and Spiral Waves in a Lattice of Excitable Units	216	(7)
Pattern Formation	217	(2)
Spiral Wave Patterns	219	(4)
Salient Features of Networks of Bistable Units	223	(4)
Mutual Synchronization, Control and Replication of Patterns and Waves in Coupled Lattices Composed of Bistable Units	227	(52)
Introduction and Motivation	227	(1)
Layered Lattice System and Mutual Synchronization of Two Lattices	228	(24)
Bistable Elements or Units	228	(7)
Bistable Oscillators	235	(2)
System of Two Coupled Fibers	237	(13)
Excitable Units	250	(2)
Controlled Patterns and Replication of Form	252	(24)
Bistable Oscillators and Replication	252	(18)
Excitable Units	270	(6)
Salient Features of Replication Processes via Synchronization of Patterns and Waves with Interacting Bistable Units	276	(3)
Spatio-Temporal Chaos in Bistable Coupled Map Lattices	279	(46)
Introduction and Motivation	279	(1)
Spectrum of the Linearized Operator	280	(4)
Linear Operator	280	(1)
A Finite-Dimensional Approximation of the Linear Operator	281	(1)
Methodology to Obtain the Linear Spectrum	282	(1)
Gershgorin Disks	283	(1)
An Alternative Way to Obtain the Stability Criterion	284	(1)
Spatial Chaos in a Discrete Version of the One-Dimensional FitzHugh--Nagumo--Schlogl Equation	284	(8)
Spatial Chaos	284	(1)
A Discrete Version of the One-Dimensional FitzHugh--Nagumo--Schlogl Equation	285	(1)
Steady States	285	(4)
Stability of Spatially Steady Solutions	289	(3)
Chaotic Traveling Waves in a One-Dimensional Discrete FitzHugh--Nagumo--Schlogl Equation	292	(5)
Traveling Wave Equation	292	(1)
Existence of Traveling Waves	293	(2)
Stability of Traveling Waves	295	(2)
Two-Dimensional Spatial Chaos	297	(5)
Invariant Domains	297	(3)
Existence of Steady Solutions	300	(1)
Stability of Steady Solutions	300	(1)
Two-Dimensional Spatial Chaos	301	(1)
Synchronization in Two-Layer Bistable Coupled Map Lattices	302	(15)
Layered Coupled Map Lattices	302	(5)
Dynamics of a Single Lattice (Layer)	307	(5)
Global Interlayer Synchronization	312	(5)
Instability of the Synchronization Manifold	317	(5)
Instability of the Synchronized Fixed Points	317	(2)
Instability of Synchronized Attractors and On--Off Intermittency	319	(3)
Salient Features of Coupled Map Lattices	322	(3)
Conclusions and Perspective	325	(4)
Appendices	329	(14)
A. Integral Manifolds of Stationary Points	329	(2)
B. Relative Location of the Manifolds Ws(O) and Wu(P+)	331	(1)
C. Flow Trajectories on the Manifolds Wμs(O) and Wμu(P+)	332	(2)
D. Instability of Spatially Homogeneous States	334	(3)
E. Topological Entropy and Lyapunov Exponent	337	(2)
F. Multipliers of the Fixed Point of the Coupled Map Lattice (7.55)	339	(2)
G. Gershgorin Theorem	341	(2)
References	343	(12)
Subject Index	355