Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Google eBook)

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Jean-René Chazottes, Bastien Fernandez
Springer Science & Business Media, Jul 6, 2005 - Science - 361 pages
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This book is about the dynamics of coupled map lattices (CML) and of related spatially extended systems. It will be useful to post-graduate students and researchers seeking an overview of the state-of-the-art and of open problems in this area of nonlinear dynamics. The special feature of this book is that it describes the (mathematical) theory of CML and some related systems and their phenomenology, with some examples of CML modeling of concrete systems (from physics and biology). More precisely, the book deals with statistical properties of (weakly) coupled chaotic maps, geometric aspects of (chaotic) CML, monotonic spatially extended systems, and dynamical models of specific biological systems.

  

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Contents

The CML2004 Project
1
1 Statistical Properties of Coupled Chaotic Maps
4
2 Geometric Aspects of Lattice Dynamical Systems
5
3 Spatially Extended Systems with Monotone Dynamics
6
4 Specific Lattice Dynamical Systems
7
at the Age of Maturity
9
2 Multicomponent Dynamical Systems MDS
13
3 Deterministic DCA and Probabilistic PCA Cellular Automata
17
4 Anisotropic Riddling in Coupled System
195
5 Some Open Problems
200
References
204
The FrenkelKontorova Model
209
2 Equilibrium States
214
3 Dissipative Dynamics
219
4 Exercises
221
5 Ratchet Effect
223

4 Relevant Measures for Dynamical Systems
21
5 Phase Transitions in Multicomponent Dynamical Systems
24
References
29
On Phase Transitions in Coupled Map Lattices
33
2 Symbolic Dynamics of Coupled Maps
40
3 Coupled Map Lattices and Kinetic Ising Models
55
References
62
Indecomposable Coupled Map Lattices with Nonunique Phase
65
2 A Selection of Examples
67
3 Motivation for Gibbs Phases
79
4 Some Challenges
88
References
92
SRBMeasures for Coupled Map Lattices
95
2 Projection Results
102
3 Counterexample to BricmontKupiainen Conjecture
106
References
111
A Spectral Gap for a Onedimensional Lattice of Coupled Piecewise Expanding Interval Maps
115
2 Dynamics at a Single Site
118
3 Finite Systems
125
4 Infinite Systems over L Z
133
References
150
Some Topological Properties of Lattice Dynamical Systems
153
3 Fixed Points of LDS
156
4 Spatiallyhomogeneous Solutions
164
5 Traveling Waves
167
6 Weakly Coupled Systems
173
7 From Infinite to Finite Lattices Concluding Remarks and Problems
174
References
177
Riddled Basins and Coupled Dynamical Systems
181
2 Riddled Sets and Riddled Basins
187
3 Symmetry Transverse Stability and Riddling
190
6 Collective Ratchet Effects in FK Model
225
7 Discommensuration Theory
231
8 Exercises
234
References
240
Spatially Extended Systems with Monotone Dynamics Continuous Time
241
2 First Order Local Dynamics
242
3 Gradient Dynamics of the FrenkelKontorova Model
246
4 Second Order Local Dynamics
253
5 Overdamped Inertial Dynamics of FrenkelKontorova Models
258
6 Further Discussion and Open Problems
261
References
262
Spatially Extended Monotone Mappings
265
2 Bistable Extended Maps
266
3 Extended Circle Maps
276
References
283
Desynchronization and Chaos in the Kuramoto Model
285
2 FrequencySplitting Bifurcation
291
3 Symmetric Kuramoto Model
297
4 Conclusion
305
Continuous and Discrete Approaches
307
2 Genetic Regulatory Networks
309
3 Ordinary Differential Equation Models
312
4 Coupled Map Networks
326
5 Discussion
334
References
336
Waves and Oscillations in Networks of Coupled Neurons
341
2 PRC Theory and Coupled Oscillators
344
3 Waves in Spiking Models
348
References
356
Index
359
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