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].
preprint_10_98.ps.gz (627KB, includes 14 illustrations)
preprint_10_98.ps (15MB (!), includes 14 illustrations)
DFG-Projekt Verbindungsorbits (Bielefeld)
DFG-Schwerpunktprogramm Dynamik (Berlin)