Synergetic phenomena in active lattices. Patterns, waves, solitons, chaos.

*(English)*Zbl 1006.37002
Springer Series in Synergetics. Berlin: Springer. xvii, 357 p. (2002).

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, numerical investigations and mathematical analysis guided by general methods from geometric dynamical systems theory.

In the first four chapters major techniques and language are introduced by way of examining paradigmatic examples: KdV type model, long Josephson junction and in particular Chua’s circuit. The next two chapters concern coupled bistable cells and patterns replication in coupled layers of arrays of bistable cells. A more or less independent and more formal chapter on travelling waves, their (in)stability and chaos in coupled map lattices concludes the studies. 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.

Chapter one introduces into the basic concepts and phenomena to be investigated. The second chapter contains studies of a dissipative Boussinesq-KdV model using geometric phase space analysis in terms of homoclinic and heteroclinic connections as well as numerical results. In a similar manner chapter three considers the long Josephson junction with more emphasis on pathfollowing and bifurcations. In the longest chapter these techniques are used to study pulses, fronts, wave trains and spatio-temporal chaos in a chain of Chua’s circuits. Chapter five focuses on coupled bistable cells. Examinations of amplitude and phase evolution, clustering, phase resetting culminate in two dimensional arrays with chaotic, regular and spiral wave pattern. The sixth chapter is devoted to the phenomenon of “self-organized” replication in coupled bistable cells. Symmetrically coupled layers of two dimensional arrays can copy an ordered, clustered state in one layer into a previously unordered second or third layer. Finally, in chapter seven, the authors look at similar phenomena in coupled map lattices. The stability and instability of travelling waves in such system is studied more formally and results on spatial chaos, synchronization and intermittency in two dimensional arrays are presented.

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.

In the first four chapters major techniques and language are introduced by way of examining paradigmatic examples: KdV type model, long Josephson junction and in particular Chua’s circuit. The next two chapters concern coupled bistable cells and patterns replication in coupled layers of arrays of bistable cells. A more or less independent and more formal chapter on travelling waves, their (in)stability and chaos in coupled map lattices concludes the studies. 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.

Chapter one introduces into the basic concepts and phenomena to be investigated. The second chapter contains studies of a dissipative Boussinesq-KdV model using geometric phase space analysis in terms of homoclinic and heteroclinic connections as well as numerical results. In a similar manner chapter three considers the long Josephson junction with more emphasis on pathfollowing and bifurcations. In the longest chapter these techniques are used to study pulses, fronts, wave trains and spatio-temporal chaos in a chain of Chua’s circuits. Chapter five focuses on coupled bistable cells. Examinations of amplitude and phase evolution, clustering, phase resetting culminate in two dimensional arrays with chaotic, regular and spiral wave pattern. The sixth chapter is devoted to the phenomenon of “self-organized” replication in coupled bistable cells. Symmetrically coupled layers of two dimensional arrays can copy an ordered, clustered state in one layer into a previously unordered second or third layer. Finally, in chapter seven, the authors look at similar phenomena in coupled map lattices. The stability and instability of travelling waves in such system is studied more formally and results on spatial chaos, synchronization and intermittency in two dimensional arrays are presented.

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.

Reviewer: Jens Rademacher (Berlin)

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

37L60 | Lattice dynamics and infinite-dimensional dissipative dynamical systems |

37N25 | Dynamical systems in biology |

35Q51 | Soliton equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |