Statistical cycling in coupled map lattices

Abstract

This work describes a class of phase transitions observed in large coupled map lattices (CML's) with chaotic local maps. The phase transitions are seen to occur (1) when the coupling is fixed as other parameters are changed and (2) when the coupling is changed with other parameters fixed. The different phases of the lattices of tent and quadratic maps are interpreted as reflecting different spectral properties of the Perron-Frobenius operator for the CML's. The spectral characteristics of this operator induced by piecewise linear CML's are investigated analytically using general results from the theory of linear operators. Sufficient conditions for the cyclical evolution of phase space densities are given. This implies that the CML's under consideration can reach equilibrium states in which ensemble statistics are not time independent (stationary), but time periodic.