Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions

Abstract

We consider a diffusion equation with time delay having a stable spatially homogeneous periodic solution bifurcating from a steady state. We show that under certain circumstances the bifurcating periodic solution loses its stability very near the bifurcation point if the diffusion coefficients are sufficiently small. Such a destabilization phenomenon also occurs when in place of the diffusion coefficients, the shape of the domain is varied instead. Sufficient conditions for the occurrence of such phenomena, along with some specific examples, will be presented.