08-196 Henk W. Broer, Heinz Hanßmann, Jiangong You

On the destruction of resonant Lagrangean tori in Hamiltonian systems (454K, PostScript) Oct 23, 08

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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.

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