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

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.