Understanding the spontaneous formation and dynamics of spatiotemporal patterns in dissipative nonequilibrium systems is one of the major challenges in nonlinear science. This collection of expository papers and advanced research articles, written by leading experts, provides an overview of the state of the art. The topics include new approaches to the mathematical characterization of spatiotemporal complexity, with special emphasis on the role of symmetry, as well as analysis and experiments of patterns in a remarkable variety of applied fields such as magnetoconvection, liquid crystals, granular media, Faraday waves, multiscale biological patterns, visual hallucinations, and biological pacemakers. The unitary presentations, guiding the reader from basic fundamental concepts to the most recent research results on each of the themes, make the book suitable for a wide audience.
Author(s): Gerhard Dangelmayr, Iuliana Oprea
Series: World Scientific Series on Nonlinear Science, Series B: Special Theme Issues and Proceedings
Publisher: World Scientific Publishing Company
Year: 2004
Language: English
Pages: 405
PREFACE......Page 6
CONTENTS......Page 12
PART I Instabilities, Bifurcation, and the Role of Symmetry......Page 16
1. Introduction to Geometric Visual Hallucinations......Page 18
2. Models, Symmetry, and Planforms......Page 21
3. A Brief Outline of Local Equivariant Steady-State Bifurcation Theory......Page 28
4. Square Lattice Planforms......Page 31
References......Page 33
1. Introduction......Page 35
2. A Dynamical System Setting......Page 36
3. Numerical Continuation......Page 37
4. Implementing the action of H' (u, T, )......Page 38
5. Calculating Special Points......Page 39
6. Simple Bifurcation Points......Page 41
7. Period-Doubling Bifurcation......Page 42
8. Torus Bifurcation......Page 43
9. Cellular Exclusion Algorithms......Page 44
10. Construction of Dominant Functions......Page 46
11. Local Expansions to Obtain Exclusion Tests......Page 47
12. Exclusion Algorithms for Unconstrained Optimization......Page 48
13. Application to Nematic Electroconvection......Page 51
References......Page 53
1. Introduction......Page 54
2. The Model......Page 56
3. Some Preparations......Page 61
4. Derivation of the Ginzburg-Landau Equation......Page 64
5. The Non Autonomous Case......Page 66
6. Comparison of the Ginzburg-Landau Equations......Page 68
7. The Final Step......Page 70
References......Page 71
1. Introduction......Page 73
2. Relative Equilibria......Page 77
3. Relative Periodic Orbits......Page 86
4. Conclusions......Page 90
References......Page 91
1. Introduction......Page 93
2. The Dispersion Relation......Page 94
3. Numerical Results: The Full Dispersion Relation......Page 96
4. Asymptotic Analysis......Page 104
5. Discussion......Page 115
References......Page 116
1. Introduction......Page 117
2. Spherical Harmonics and their Properties......Page 119
3. Groups, Symmetries and Patterns......Page 120
4. Solution Branches of (7) for Even l......Page 124
5. Solution Branches for Even l including Cubic Terms......Page 132
6. Discussion......Page 137
References......Page 138
1. Introduction......Page 139
2. Model Equations......Page 144
3. Small Divisors......Page 148
4. The Question of Convergence......Page 151
5. Discussion and Speculation......Page 152
References......Page 154
PART II Localized Patterns, Waves, and Weak Turbulence......Page 156
1. Introduction......Page 158
2. Phase Instabilities, Phase Equations and Phase Turbulence......Page 159
3. Weak Turbulence in the Complex Ginzburg-Landau Equation......Page 163
4. Conclusions......Page 168
References......Page 170
1. Introduction......Page 173
2. The Mathieu Equation......Page 174
3. The Mathieu PDE......Page 177
4. The Nonlinear Schrödinger Equation......Page 180
References......Page 187
1. Introduction......Page 189
2. Basic Equations......Page 191
3. Derivation of Model Equations......Page 192
4. Mean Flow Equations......Page 195
5. Bifurcations of Periodic Solutions......Page 198
6. Mean Flow Generated by Defects......Page 203
7. Conclusions......Page 206
References......Page 207
1. Introduction......Page 209
2. Rogue Wave Solutions of the 2D (1+1) NLS Equation......Page 212
3. Rogue Waves in 2D (1+1) higher order NLS Equations......Page 215
4. Rogue Waves in the 3D (2 + 1) NLS Equation......Page 218
5. Melnikov Analysis......Page 221
6. Conclusions......Page 226
Appendix: Statistical Diagnostics......Page 227
References......Page 228
1. Introduction......Page 229
2. The Complex Ginzburg-Landau Equation......Page 230
3. Pacemakers in the Phase Dynamics Approximation......Page 231
4. Stable, Extended and Localized Wave Patterns......Page 232
5. Unstable Wave Patterns and Phase Slips......Page 236
6. Discussion......Page 241
References......Page 243
PART III Modelling and Characterization of Spatio-Temporal Complexity......Page 244
1. Introduction......Page 246
2. Binary Fluid Convection......Page 247
3. Natural Doubly Diffusive Convection......Page 263
4. Fast-slow Systems: Faraday Oscillations......Page 269
5. Conclusions......Page 283
References......Page 287
1. Introduction......Page 289
2. Lattice Gas Models......Page 291
3. Representing Cell Shape......Page 293
4. A Model for Myxobacteria Aggregation......Page 294
5. A Model for Chondrogenic Patterning......Page 299
References......Page 304
1. Introduction......Page 307
2. Reduction Methodology......Page 308
3. Results......Page 311
Acknowledgments......Page 322
References......Page 323
1. Background......Page 324
2. Experiments......Page 328
References......Page 333
1. Introduction......Page 334
2. Labyrinthine Patterns......Page 335
3. Domain Growth......Page 339
References......Page 342
1. Introduction......Page 344
2. Observations of Heteroclinic Cycles......Page 346
3. Imperfect Heteroclinic Behavior......Page 348
4. The 1:2 Resonance: Novel Heteroclinic Behavior......Page 349
5. 1:2 Resonance with Broken O(2) SO(2) Symmetry......Page 357
6. Conclusion......Page 367
References......Page 369
1. Introduction......Page 372
2. Internal Dynamics of Intermittency......Page 373
3. Internal Dynamics of Intermittency for a 2-d Mapping......Page 375
4. Markov Model......Page 380
5. Non-Ergodic Intermittency......Page 384
6. Discussion and Conclusions......Page 386
References......Page 387
1. Introduction......Page 388
2. Experimental Setup......Page 390
3. Experimental Results......Page 391
4. Discussion......Page 397
5. Conclusions......Page 398
References......Page 399
INDEX......Page 402
