Formation of Periodic and Aperiodic Waves in Reaction-Diffusion Systems

Abstract

It is well known that spatially distributed excitable media can conduct excitations (e.g., pulses) initiated by a sufficiently large perturbations. Neurones, Purkinje fibers and other excitable tissues in biological systems belong to the most often discussed examples of excitable media [4,8]. However, more simple chemical systems with reaction and diffusion can be also excitable and may thus serve as model systems [15]. Let us consider an S-component, spatially one-dimensional system

$$ \frac{{\partial {\text{X}}_{\text{i}}}} {{\partial {\text{t}}}}\; = \;{\text{D}}_{\text{i}} \;\frac{{\partial ^2 {\text{X}}_{\text{i}}}} {{\partial {\text{Z}}^{\text{2}}}}\; + \;{\text{f}}_{\text{i}} ({\text{X}}_1 ,...{\text{X}}_{\text{s}} ),{\text{ i = 1,}}...{\text{,S ,}} $$

(1)

where D_i denote diffusion coefficients and f_i kinetic functions of a suitable form. The spatial coordinate z is from the interval [0,L] and proper boundary conditions (for example, describing zero flux of the components X₁ at the boundaries) are also defined. We shall assume such conditions (kinetic and diffusion parameter values) that if the system is unperturbed, it stays in a stationary state without spatial gradients.