Dynamics Of Coupled Map Lattices Research Articles

J.-R. Chazottes and B. Fernandez (eds): Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems

The book presents a collection of lecture notes written for the Coupled Map Lattices (CML) summer school/conference held at the Institute Henri Poincare in Paris in the beginning of the summer 2004. By a CML one usually understands a finite or countable collection of ‘sites’, a simple, usually low-dimensional, dynamical system placed at each site and some ‘coupling’ to systems at other sites. The sites are often located at points in a regular lattice (whence the name) but this need not be so. A series of 15 lectures gives the reader an overview of the state of art in the subject. Instead of reviewing each lecture let me give a list of some of the important questions discussed in the book (more or less in this order):

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling

We construct a mixing continuous piecewise linear map on [−1, 1] with the property that a two-dimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borrowed from Gielis and MacKay (2000 Nonlinearity 13 867–88); namely, we compare the dynamics of the CML with those of a probabilistic cellular automaton of Toom's type; see MacKay (2005 Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Lecture Notes in Physics vol 671) ed J-R Chazottes and B Fernandez (Berlin: Springer) pp 65–94) for a detailed discussion.

Symbolic Dynamics of Coupled Map Lattices

We present a method to reduce the [FORMULA: SEE TEXT] dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given.