Singular perturbation with Limit points in the fast dynamics

Abstract

A singular perturbation problem in ordinary differential equations is investigated without assuming hyperbolicity of the associated slow manifold. More precisely, the slow manifold consists of a branch of stationary points which lose their hyperbolicity at a stationary point with eigenvalues \(\pm i\). Thus, it is impossible to reduce the dynamics near the Hopf point to the slow manifold. This situation is examined within a generic one parameter unfolding. It leads to two bifurcating curves of Hopf points and associated to these are two manifolds of periodic orbits and possibly another manifold of invariant tori, all three of which intersect in the central Hopf point. The proof employs a suitable Ansatz resulting in a Hopf bifurcation theorem which determines precisely the bifurcation structure near a certain Hopf point with an additional zero eigenvalue.