Spatio-temporal chaos: A solvable model

Abstract

A solvable coupled map lattice model exhibiting spatio-temporal chaos is studied. Exact expressions are obtained for the spectra of Lyapunov exponents as a function of the model parameters. Although the model has spatio-temporal structure, the time series measured at a single lattice site are shown to consist of independent, identically distributed samples for several values of the model parameters. For these parameter values, the spatial series measured at a fixed time also consist of independent, identically distributed samples. In these cases, the information dimension density is 1, but the information entropy density depends on the model parameters. Thus, the model is an example where the information entropy density can be obtained neither from a time series measured at a single lattice site nor from a spatial series measured at a fixed time. We conclude that in studying only a time series or a spatial series without any knowledge of the system, one could be easily led into thinking that there is no spatio-temporal structure. For a full characterization of the system, structure in time and space will have to be considered simultaneously.