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 (the Exchange Lemma). 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 fast transition are treated and it is shown how epsilon-expansions can be extracted rigorously from this approach. The result is applied to a singularly perturbed Bogdanov point in the FitzHugh-Nagumo system.
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DFG-Projekt Verbindungsorbits (Bielefeld)
DFG-Schwerpunktprogramm Dynamik (Berlin)