## Dynamics of Coupled Map Lattices and of Related Spatially Extended SystemsJean-René Chazottes, Bastien Fernandez 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 |

356 | |

359 | |