Preprint of the project: DFG Priority Research Program: DANSEConnecting orbits in highdimensional dynamical systems10/98 Matthias Stiefenhofer. Bogdanov points that occur in the fast dynamics of singular perturbation problems are often encountered in applications; e.g. in the van der PolDuffing oscillator [Koper 95] or in the FitzHughNagumo equation [Beyn Stiefenhofer 97]. We analyze these points using a generic unfolding which ensures that the typical phenomena near a regularly perturbed Bogdanov point (saddlenodes, Hopf points, periodic orbits, homoclinic orbits) carry over to Bogdanov points viewed in the context of singular perturbations. We combine analytical and numerical results to study the relations between these structures in the 3dimensional unfolding space. In particular, the singularly perturbed homoclinic orbits can be analyzed after an appropriate blow of the singularly perturbed Bogdanov point using a technique from [Beyn Stiefenhofer 97]. As indicated by numerical calculations the homoclinic orbits turn into homoclinic orbits of Shilnikov type that vanish presumably by a duck like explosion process [Diener 81], [Eckhaus 93], [Arnold 94], [Dumortier 96].
