ON THE PRESENCE OF SPATIAL PATTERN-FORMATION IN A BISTABLE DYNAMICAL SYSTEM

Abstract

The aim is to investigate whether in a structural bistable reaction-diffusion system pattern formation may emerge simultaneously from both steady states. Therefore, a dynamical system is modelled by three coupled nonlinear differential equations from which synergetic ordering may arise. In addition, the nonlinear terms are chosen such that the homogeneous system is governed by the canonical form of a cusp bifurcation in a two-dimensional control space. Thus, structural bistability is established. Based on a linear stability analysis the region of bistability is decomposed into four different domains in the control plane. It is shown that in one of these domains self-organization can lead to pattern formation from both steady states simultaneously. In two other domains self-organization can arise from only one steady state and finally in one domain patterning is impossible. An expression for the wavelength of a spatial structure is derived and discussed in terms of parameters of the system. As a possible application of the present results a crystal under irradiation with particles of high energy is considered. It is demonstrated for the case of steel that the parameters of the system can be chosen such that a two-fold spatial instability for irradiation induced cavities may emerge.