Study of spatially extended dynamical systems using probabilistic cellular automata

Spatially extended dynamical systems are ubiquitous and include such things as insect and animal populations; complex chemical, technological, and geochemical processes; humanity itself, and much more. It is clearly desirable to have a certain universal tool with which the highly complex behaviour of nonlinear dynamical systems can be analyzed and modelled. For this purpose, cellular automata seem to be good candidates. In the present review, emphasis is placed on the possibilities that various types of probabilistic cellular automata (PCA), such as DSMC (direct simulation Monte Carlo) and LGCA (lattice-gas cellular automata), offer. The methods are primarily designed for modelling spatially extended dynamical systems with inner fluctuations accounted for. For the Willamowskii–Roessler and Oregonator models, PCA applications to the following problems are illustrated: the effect of fluctuations on the dynamics of nonlinear systems; Turing structure formation; the effect of hydrodynamic modes on the behaviour of nonlinear chemical systems (stirring effects); bifurcation changes in the dynamical regimes of complex systems with restricted geometry or low spatial dimension; and the description of chemical systems in microemulsions.