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.
ASJC Scopus subject areas
- Applied Mathematics