Mean-field theory and critical behavior of coupled map lattices

Abstract

Methods of statistical physics are used to study properties of a one-dimensional lattice of coupled tent maps as a model for spatiotemporal intermittency. Ensemble-averaged quantities are calculated from a large number of random initial values. In a coarse-grained description, the dynamical phase transition that occurs in the model can be characterized as a directed percolation process. Two approximations of mean-field type are introduced. The first method assumes a uniform probability density for the steady-state distribution of phase-space variables. In the second method, this invariant probability measure is calculated self-consistently. The mean-field predictions are compared to direct numerical iterations of finite lattices. The critical behavior is characterized by finite-size scaling expressions that allow us to extract critical exponents from small lattices. The scaling indices depend on the parameters of the model.