Henk W. Broer, Heinz Hanßmann, Jiangong You
Umbilical Torus Bifurcations in Hamiltonian Systems
(623K, PostScript, gzipped and uuencoded)
ABSTRACT. We consider perturbations of integrable Hamiltonian systems
in the neighbourhood of normally umbilic invariant tori. These lower
dimensional tori do not satisfy the usual non--degeneracy conditions
that would yield persistence by an adaption of KAM theory, and there
are indeed regions in parameter space with no surviving torus. We
assume appropriate transversality conditions to hold so that the tori
in the unperturbed system bifurcate according to a (generalized)
umbilical catastrophe. Combining techniques of KAM theory and
singularity theory we show that such bifurcation scenarios of
invariant tori survive the perturbation on large Cantor sets.
Applications to gyrostat dynamics are pointed out.