Transition from stationary to traveling localized patterns in a two-dimensional reaction-diffusion system.

Abstract

The bifurcation from localized stationary solutions to traveling patterns is studied in two-dimensional reaction-diffusion systems. In the case of local kinetics, which is of the Bonhoeffer--van der Pol type, the bifurcation point can be computed directly in terms of the stationary solution for arbitrary parameters. A stripelike pattern, which is the extension of a one-dimensional localized pattern into the second spatial dimension, is considered as a concrete example of such a pattern. The branching mode is computed for piecewise linear kinetics in a singular limit. The bifurcation is predominantly subcritical. Eventually the linear stability analysis of the branching traveling pattern is performed. Following the bifurcating branch, the last mode to become stable is a laterally wavy perturbation with an arbitrarily long wavelength.