Connecting orbits in highdimensional dynamical systems
15/97 Wolf-Jürgen Beyn, Matthias Stiefenhofer.
A direct approach to homoclinic orbits in the fast
dynamics of singularly perturbed systems
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.
