Preprint of the project: DFG Priority Research Program: DANSE

Connecting orbits in highdimensional dynamical systems

10/98 Matthias Stiefenhofer.
Unfolding singularly perturbed Bogdanov points

Bogdanov points that occur in the fast dynamics of singular perturbation problems are often encountered in applications; e.g. in the van der Pol-Duffing oscillator [Koper 95] or in the FitzHugh-Nagumo equation [Beyn Stiefenhofer 97]. We analyze these points using a generic unfolding which ensures that the typical phenomena near a regularly perturbed Bogdanov point (saddle-nodes, 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 3-dimensional 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].