The uniform bifurcation of N-front traveling waves in the singularly perturbed Fitzhugh-Nagumo equations

In this dissertation we consider traveling wave solutions of the FitzHugh-Nagumo equations, [special characters omitted] In phase space, the FitzHugh-Nagumo equations possess n-front traveling wave solutions that correspond to n-front heteroclinic orbits. These solutions bifurcate from a heteroclinic loop. However, the FitzHugh-Nagumo equations are singularly perturbed by the parameter &epsis;. The bifurcation does not occur when the FitzHugh-Nagumo equations are in their singular state, it only occurs on the set [special characters omitted] This gives rise to the problem that in the cγ parameter space, where c is the propagation speed of the traveling waves, the domains of definition of the bifurcation curves are dependent on &epsis;, the singular parameter. We show that these domains can be made uniform in the parameter &epsis;, that is, the set of parameters over which the bifurcation occurs is of uniform size for all [special characters omitted] This work brings together the areas of geometric singular perturbation, the Melnikov method, and Shil\u27nikov solutions. In doing so we present a new proof of the existence of Shil\u27nikov solutions in a neighborhood of a compact manifold of equilibria. We also examine the problem of heteroclinic and homoclinic orbits bifurcations in the specific setting of a singularly perturbed equations