- 08-196 Henk W. Broer, Heinz Hanßmann, Jiangong You
- On the destruction of resonant Lagrangean tori in Hamiltonian systems
Oct 23, 08
(auto. generated ps),
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Abstract. Starting from Poincar\'e's fundamental problem of
dynamics, we consider perturbations of integrable Hamiltonian
systems in the neighbourhood of resonant maximal invariant
tori with a single resonance. We investigate how such a torus
disintegrates when the action variables vary in the resonant
surface. For open subsets of this surface the resulting lower
dimensional tori are either hyperbolic or elliptic. For a
better understanding of the dynamics, both qualitatively and
quantitatively, we also investigate the singular tori and the
way in which they are being unfolded by the action variables.
In fact, if $N$ is the number of degrees of freedom,
singularities up to codimension $N-1$ cannot be avoided. In
the case of Kolmogorov non-degeneracy the singular tori are
parabolic, while under the weaker non-degeneracy condition of
R\"ussmann the lower dimensional tori may also undergo e.g.
umbilical bifurcations. We emphasise that this application of
Singularity Theory only uses internal (or distinguished)
parameters and no external ones.