A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems

Abstract

Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how e-expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.