Pattern Formation in Dissipative Nonvariational Systems: The Effects of Front Bifurcations

Abstract

Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally develop in bistable and excitable systems, but may also appear far beyond Hopf and Turing bifurcations. The global behavior of domain patterns strongly depends on the fronts' inner structures. In this paper we study a symmetry breaking front bifurcation expected to occur in a wide class of reaction-diffusion systems, and the effects it has on pattern formation and pattern dynamics. We extend previous works on this type of front bifurcation and clarify the relations among them. We show that the appearance of front multiplicity beyond the bifurcation point allows the formation of persistent patterns rather than transient ones. In a different parameter regime, we find that the front bifurcation outlines a transition from oscillating (or breathing) patterns to traveling ones. Near a boundary we find that fronts beyond the bifurcation can reflect, while those below it either bind to the boundary or disappear.