Solvable model for spatiotemporal chaos

The study of temporal chaos in low-dimensional systems, some of which can be described by low-dimensional maps @1,2#, was extremely beneficial for the understanding of turbulence. In 1984 coupled map lattices ~CMLs! were introduced into the physical literature as a tool for studying spatiotemporal chaos in spatially extended, i.e., highdimensional systems @3#. They consist of spatially coupled low-dimensional maps and represent dynamical systems that are discrete in space and time, but continuous in the state variable. They serve as models for coupled Josephson junctions, excitable media, population dynamics, neural dynamics, and turbulence @4#. Although a number of statements regarding the appearance of coherent structures from spatiotemporal chaos were proved analytically ~see, for instance, the works by Bunimovich and Sinai @5#!, most results in the field have been obtained by numerical simulations @4,6#. The study of temporal chaos has greatly profited from the existence of simple maps such as the Bernoulli shift map and the cat map @2,7#, which can be solved explicitly ~for integer expansion rates!, thereby making the mechanisms of mixing and temporal chaos understandable. Unfortunately, no in-depth investigation of this type has been provided up to now for the problem of spatiotemporal chaos and coherent structures. The biggest progress was achieved for the class of coupled maps @3#