Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems

Abstract

We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter ε by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.