Spatiotemporal pattern formation occurring in discretely-coupled nonlinear dynamical systems has been studied numerically. Reaction-diffusion systems can be viewed as an assembly of a large number of identical local subsystems which are coupled to each other by diffusion. Here, the local subsystems are defined by a system of nonlinear ordinary differential equations. While for continuous systems, the characteristic time scale corresponding to the diffusion is slower than that corresponding to the local subsystems, in discretely-coupled systems, both time scales can be of the same order of magnitude. Discrete systems can exhibit behaviors different from those exhibited by their equivalent continuous model: the wave propagation failure phenomenon occurring in nerve-pulse propagation due to transmission blockage is a case in point. In this case, it is found that the wave fails to propagate at or below some critical value of the coupling coefficient. Systems of coupled cells can be found to occur in the transformation and transport processes in living cells, tissues, neuron networks, physiological systems and ecosystems, as well as in all forms of chemical, biochemical reactors and combustion systems. In this paper, we review the possibilities of using arrays of discretely-coupled nonlinear electronic circuits to study these systems. The possibility of building large arrays of these circuits via VLSI technology makes this approach a unique tool for real time applications. Classical examples occurring in other continuous media, such as spiral wave initiation and propagation, and Turing pattern formation, are depicted here. Because of the discrete nature of our system, the influence of inhomogeneities arising from damaged cells, or from an anisotropic media, is analyzed for spiral wave propagation. Spiral wave initiation and vulnerability effects are considered and compared with their corresponding effects in continuous media. More complex spatiotemporal structures are also studied via three-dimensional arrays of discretely-coupled circuits. Straight and twisted scroll waves, as well as scroll ring waves, are shown to exist in these arrays, where their properties can be easily measured. Sidewall forcing of Turing patterns is shown to be capable of driving the system into a perfect spatial organization, namely, a rhombic pattern, where no defects occur. The dynamics of the two layers supporting Turing and Hopf modes, respectively, is analyzed as a function of the coupling strength between them. The competition between these two modes is shown to increase with the diffusion between layers.
Multidimensional Approximation of Nonlinear Dynamical Systems
