On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems [chapter]

Henk W. Broer, Heinz Hanßmann, Jiangong You
2013 Recent Trends in Dynamical Systems  
Starting from Poincaré's fundamental problem of dynamics, we consider perturbations of integrable Hamiltonian systems in the neighbourhood of resonant Lagrangean (i.e. maximal) invariant tori with a single (internal) resonance. Applying KAM Theory and Singularity Theory 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
more » ... ding 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 co-dimension 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 Rüssmann 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.
doi:10.1007/978-3-0348-0451-6_13 fatcat:atwfyvillbc6darjm3hwllbbxi