Chapter 6 - Graphical identification of spatio-temporal chaos

Abstract

Although the possibility of chaotic behavior has been known in principle since the work of Poincaire, the current enthusiasm for chaos is based on the availability of computers for numerical experiments, and computer graphics for displaying the results of these numerical and laboratory experiments. These methods can be applied to time series, but most processes of interest are not simply time series––but spatially extensive. Thus, they are spatio-temporal patterns and can be represented in discrete space by discrete time, discrete state (cellular automata), or discrete time-continuous state (coupled map lattice) models. In continuous space, a continuous time, continuous state model is a partial differential equation. These spatially extensive systems and their models can exhibit periodic, patterned, or irregular behaviors. A major question is whether spatio-temporal irregularity (in a model or in experimental observations) is due todeterministic nonlinear processes, and hence is spatio-temporal chaos. In discrete space models, spatio-temporal chaos can be considered as a pattern that does not repeat itself as long as the model is iterated: a simple model guarantees the determinism. This is not applicable to experimental observations.