1 Introduction: Synergetics and Models of Continuous
and Discrete Active Media. Steady States and Basic
Motions (Waves, Dissipative Solitons, etc.) .................. 1
1.1 Basic Concepts, Phenomena and Context ................... 1
1.2 Continuous Models ....................................... 8
1.3 Chain and Lattice Models with Continuous Time .......... 12
1.4 Chain and Lattice Models with Discrete Time ............ 15
2 Solitary Waves, Bound Soliton States and Chaotic Soliton
Trains in a Dissipative Boussinesq—Korteweg—de Vries
Equation .................................................... 19
2.1 Introduction and Motivation ............................ 19
2.2 Model Equation ......................................... 21
2.3 Traveling Waves ........................................ 23
2.3.1 Steady States ................................... 24
2.3.2 Lyapunov Functions .............................. 25
2.4 Homoclinic Orbits. Phase-Space Analysis ................ 26
2.4.1 Invariant Subspaces ............................. 26
2.4.2 Auxiliary Systems ............................... 27
2.4.3 Construction of Regions Confining the Unstable
and Stable Manifolds Wu and Ws .................. 28
2.5 Multiloop Homoclinic Orbits and Soliton-Bound States ... 31
2.5.1 Existence of Multiloop Homoclinic Orbits ........ 31
2.5.2 Solitonic Waves, Soliton-Bound States and
Chaotic Soliton-Trains .......................... 34
2.5.3 Homoclinic Orbits and Soliton-Trains. Some
Numerical Results ............................... 35
2.6 Further Numerical Results and Computer Experiments ..... 39
2.6.1 Evolutionary Features ........................... 40
2.6.2 Numerical Collision Experiments ................. 43
2.7 Salient Features of Dissipative Solitons ............... 48
3 Self-Organization in a Long Josephson Junction .............. 49
3.1 Introduction and Motivation ............................ 49
3.2 The Perturbed Sine-Gordon Equation ..................... 50
3.3 Bifurcation Diagram of Homoclinic Trajectories ......... 51
3.4 Current-Voltage Characteristics of Long Josephson
Junctions .............................................. 54
3.5 Bifurcation Diagram in the Neighborhood of с = 1 ....... 56
3.5.1 Spiral-Like Bifurcation Structures .............. 56
3.5.2 Heteroclinic Contours ........................... 58
3.5.3 The Neighborhood of At .......................... 61
3.5.4 The Sets {γi} and {Ỹi} .......................... 65
3.6 Existence of Homoclinic Orbits ......................... 67
3.6.1 Lyapunov Function ............................... 68
3.6.2 The Vector Field of (3.4) on Two Auxiliary
Surfaces ........................................ 69
3.6.3 Auxiliary Systems ............................... 69
3.6.4 "Tunnels" for Manifolds of the Saddle Steady
State 02 ........................................ 70
3.6.5 Homoclinic Orbits ............................... 71
3.7 Salient Features of the Perturbed Sine-Gordon
Equation ............................................... 74
4 Spatial Structures, Wave Fronts, Periodic Waves, Pulses
and Solitary Waves in a One-Dimensional Array of Chua's
Circuits .................................................... 77
4.1 Introduction and Motivation ............................ 77
4.2 Spatio-Temporal Dynamics of an Array of Resistively
Coupled Units .......................................... 79
4.2.1 Steady States and Spatial Structures ............ 80
4.2.2 Wave Fronts in a Gradient Approximation ......... 86
4.2.3 Pulses, Fronts and Chaotic Wave Trains .......... 94
4.3 Spatio-Temporal Dynamics of Arrays with Inductively
Coupled Units ......................................... 106
4.3.1 Homoclinic Orbits and Solitary Waves ........... 106
4.3.2 Periodic Waves in a Circular Array ............. 123
4.4 Chaotic Attractors and Waves in a One-Dimensional
Array of Modified Chua's Circuits ..................... 137
4.4.1 Modified Chua's Circuit ........................ 137
4.4.2 One-Dimensional Array .......................... 139
4.4.3 Chaotic Attractors ............................. 139
4.5 Salient Features of Chua's Circuit in a Lattice ....... 161
4.5.1 Array with Resistive Coupling .................. 162
4.5.2 Array with Inductive Coupling .................. 162
5 Patterns, Spatial Disorder and Waves in a Dynamical
Lattice of Bistable Units .................................. 165
5.1 Introduction and Motivation ........................... 165
5.2 Spatial Disorder in a Linear Chain of Coupled
Bistable Units ........................................ 166
5.2.1 Evolution of Amplitudes and Phases of the
Oscillations ................................... 166
5.2.2 Spatial Distributions of Oscillation
Amplitudes ..................................... 168
5.2.3 Phase Clusters in a Chain of Isochronous
Oscillators .................................... 171
5.3 Clustering and Phase Resetting in a Chain of Bistable
Nonisochronous Oscillators ............................ 172
5.3.1 Amplitude Distribution along the Chain ......... 173
5.3.2 Phase Clusters in a Chain of Nonisochronous
Oscillators .................................... 175
5.3.3 Frequency Clusters and Phase Resetting ......... 176
5.4 Clusters in an Assembly of Globally Coupled Bistable
Oscillators ........................................... 179
5.4.1 Homogeneous Oscillations ....................... 180
5.4.2 Amplitude Clusters ............................. 181
5.4.3 Amplitude-Phase Clusters ....................... 186
5.4.4 "Splay-Phase" States ........................... 191
5.4.5 Collective Chaos ............................... 194
5.5 Spatial Disorder and Waves in a Circular Chain of
Bistable Units ........................................ 195
5.5.1 Spatial Disorder ............................... 195
5.5.2 Space-Homogeneous Phase Waves .................. 197
5.5.3 Space-Inhomogeneous Phase Waves ................ 201
5.6 Chaotic and Regular Patterns in Two-Dimensional
Lattices of Coupled Bistable Units .................... 206
5.6.1 Methodology for a Lattice of Bistable
Elements ....................................... 206
5.6.2 Stable Steady States ........................... 209
5.6.3 Spatial Disorder and Patterns in the
FitzHugh-Nagumo-Schlцgl Model .................. 211
5.6.4 Spatial Disorder and Patterns in a Lattice
of Bistable Oscillators ........................ 212
5.7 Patterns and Spiral Waves in a Lattice of Excitable
Units ................................................. 216
5.7.1 Pattern Formation .............................. 217
5.7.2 Spiral Wave Patterns ........................... 219
5.8 Salient Features of Networks of Bistable Units ........ 223
6 Mutual Synchronization, Control and Replication of
Patterns and Waves in Coupled Lattices Composed of
Bistable Units ............................................. 227
6.1 Introduction and Motivation ........................... 227
6.2 Layered Lattice System and Mutual Synchronization
of Two Lattices ....................................... 228
6.2.1 Bistable Elements or Units ..................... 228
6.2.2 Bistable Oscillators ........................... 235
6.2.3 System of Two Coupled Fibers ................... 237
6.2.4 Excitable Units ................................ 250
6.3 Controlled Patterns and Replication of Form ........... 252
6.3.1 Bistable Oscillators and Replication ........... 252
6.3.2 Excitable Units ................................ 270
6.4 Salient Features of Replication Processes via
Synchronization of Patterns and Waves with
Interacting Bistable Units ............................ 276
7 Spatio-Temporal Chaos in Bistable Coupled Map Lattices ..... 279
7.1 Introduction and Motivation ........................... 279
7.2 Spectrum of the Linearized Operator ................... 280
7.2.1 Linear Operator ................................ 280
7.2.2 A Finite-Dimensional Approximation of the
Linear Operator ................................ 281
7.2.3 Methodology to Obtain the Linear Spectrum ...... 282
7.2.4 Gershgorin Disks ............................... 283
7.2.5 An Alternative Way to Obtain the Stability
Criterion ...................................... 284
7.3 Spatial Chaos in a Discrete Version of the One-
Dimensional FitzHugh-Nagumo-Schlцgl Equation .......... 284
7.3.1 Spatial Chaos .................................. 284
7.3.2 A Discrete Version of the One-Dimensional
FitzHugh-Nagumo-Schlögl Equation ............... 285
7.3.3 Steady States .................................. 285
7.3.4 Stability of Spatially Steady Solutions ........ 289
7.4 Chaotic Traveling Waves in a One-Dimensional
Discrete FitzHugh-Nagumo-Schlögl Equation ............. 292
7.4.1 Traveling Wave Equation ........................ 292
7.4.2 Existence of Traveling Waves ................... 293
7.4.3 Stability of Traveling Waves ................... 295
7.5 Two-Dimensional Spatial Chaos ......................... 297
7.5.1 Invariant Domains .............................. 297
7.5.2 Existence of Steady Solutions .................. 300
7.5.3 Stability of Steady Solutions .................. 300
7.5.4 Two-Dimensional Spatial Chaos .................. 301
7.6 Synchronization in Two-Layer Bistable Coupled Map
Lattices .............................................. 302
7.6.1 Layered Coupled Map Lattices ................... 302
7.6.2 Dynamics of a Single Lattice (Layer) ........... 307
7.6.3 Global Interlayer Synchronization .............. 312
7.7 Instability of the Synchronization Manifold ........... 317
7.7.1 Instability of the Synchronized Fixed Points ... 317
7.7.2 Instability of Synchronized Attractors
and On-Off Intermittency ....................... 319
7.8 Salient Features of Coupled Map Lattices .............. 322
8 Conclusions and Perspective ................................ 325
Appendices
A Integral Manifolds of Stationary Points
В Relative Location of the Manifolds Wsμ(O) and Wuμ(P+) .... 331
C Flow Trajectories on the Manifolds Wsμ(O) and Wuμ(P+) .... 332
D Instability of Spatially Homogeneous States ............. 334
E Topological Entropy and Lyapunov Exponent ............... 337
F Multipliers of the Fixed Point of the Coupled Map
Lattice (7.55) .......................................... 339
G Gershgorin Theorem ...................................... 341
References
Subject Index
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