Synergetic Phenomena in Active Lattices: Patterns, Waves, Solitons, Chaos(Springer Series in Synergetics)

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.

Table of 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.- 2.4 Homoclinic Orbits. Phase-Space Analysis.- 2.5 Multiloop Homoclinic Orbits and Soliton-Bound States.- 2.6 Further Numerical Results and Computer 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.- 3.6 Existence of 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.- 4.3 Spatio-Temporal Dynamics of Arrays with Inductively Coupled Units.- 4.4 Chaotic Attractors and Waves in a One-Dimensional Array of Modified Chua’s Circuits.- 4.5 Salient Features of Chua’s Circuit in a Lattice.- 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.- 5.3 Clustering and PhaseResetting in a Chain of Bistable Nonisochronous Oscillators.- 5.4 Clusters in an Assembly of Globally Coupled Bistable Oscillators.- 5.5 Spatial Disorder and Waves in a Circular Chain of Bistable Units.- 5.6 Chaotic and Regular Patterns in Two-Dimensional Lattices of Coupled Bistable Units.- 5.7 Patterns and Spiral Waves in a Lattice of Excitable Units.- 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.- 6.3 Controlled Patterns and Replication of Form.- 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.- 7.3 Spatial Chaos in a Discrete Version of the One-Dimensional FitzHugh-Nagumo-Schlögl Equation.- 7.4 Chaotic Traveling Waves in a One-Dimensional Discrete FitzHugh-Nagumo-Schlögl Equation.- 7.5 Two-Dimensional Spatial Chaos.- 7.6 Synchronization in Two-Layer Bistable Coupled Map Lattices.- 7.7 Instability of the Synchronization Manifold.- 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.

Review :
"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, numericalinvestigations 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)