Singular perturbation with Limit points in the fast dynamics

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 or a branch of periodic orbits which lose their hyperbolicity at a limit point. Thus, in a neighbourhood of this point a reduction to the slow manifold is not possible. This situation is examined within a generic one parameter unfolding which leads in case of a stationary or periodic limit point to a curve of Hopf or Naimark-Sacker bifurcation points with associated periodic orbits or invariant tori respectively. The stationary case is examined in detail with the aim of characterizing the domain in parameter space yielding small periodic orbits as precisely as possible. Moreover, the shape and stability of the periodic orbits is determined. The paper examines one case of a not finite determined Bogdanov point with properties partly motivated by formal results in [3]. As an application we consider the van der Pol oscillator treated as a cusp unfolding. This is done for the space independent case as well as for the travelling wave problem of the space dependent case.