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 +i, -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.
preprint_59_96.ps.gz (615KB, includes 10 illustrations)
preprint_59_96.ps (16MB (!), includes 10 illustrations)
DFG-Project Connecting Orbits (Bielefeld)
DFG-Schwerpunktprogramm Dynamik (Berlin)