On Phase Transitions in Coupled Map Lattices

Abstract

Coupled map lattices are a paradigm for studying fundamental questions in spatially extended dynamical systems. Within this tutorial we focus on qualitative changes of the motion which are intimately related with the limit of large system size. Similar to equilibrium phase transitions, such qualitative changes are an ubiquitous feature of dynamical systems with a large number of degrees of freedom. Within the first section of this chapter we present an overview and some phenomenological facts of phase transitions in coupled map lattices. The following two sections describe in some details analytical tools which are useful for understanding phase transition behaviour in dynamical systems beyond plain numerical simulations. In Sect. 2 we explain how coupled map lattices are linked with the canonical equilibrium physics of spin systems when techniques of symbolic dynamics are applied. Using a simple model we explain how coupled map lattices are linked with phase transitions in equilibrium spin models. In the third section we describe an alternative approach in terms of kinetic spin models linking the dynamics of coupled map lattices with equilibrium and nonequilibrium statistical mechanics. We keep our presentation throughout this tutorial entirely elementary and confine the presentation to some basic concepts which are useful for tackling the analysis of phase transitions in extended dynamical systems.