The Distribution of periodic and Aperiodic Pattern Evolutions in Rings of Diffusively Coupled Maps

This work reports a systematic investigation of the distribution of periodic and aperiodic patternevolution in rings of diffusively coupled quadratic maps. Our main motivation here is the wish to investigate whether lattices of coupled maps can be effectively used to simulate a plethora of phenomena associated with climatic variability and change. Presently, simulations of these phenomena reduce, essentially, to the solutions of large arrays of coupled differential equations, a quite computerdemanding task. As may be recognized from the contributions in books edited by Kaneko [1993b] and Schuster [1999], coupled map lattices (CMLs) are known to reproduce many interesting dynamical aspects associated with spatio-temporal complexity for large classes of phenomena. Some 30 years ago, meteorological studies awakened great interest in the investigation of using discrete maps as simple models of complex temporal dynamics. It is our hope that lattices of coupled maps might turn out to be an efficient framework for a variety of applications, from the interpolation of missing-field data, or the generation of synthetic series of climate elements, to the actual simulation of large-scale phenomena involving several time-scales, e.g. the El-Niño Southern Oscillation (ENSO) and the North Atlantic Oscillation. What is presently known about the dynamics of CMLs that could be interesting for meteorological and climatic applications? A survey of the literature shows that most of the work so far concentrates on the so-called globally coupled maps, where the essential contribution driving locally each oscillator arises from the mean-field of all other oscillators, the environment. Such approach is strongly influenced by the methodology of statistical mechanics. For the application that we have in mind, however, diffusively coupled maps seem to be a proper