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A Survey on Bifurcations of Invariant Tori

In: New Advances in Celestial Mechanics and Hamiltonian Systems

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  • Heinz Hanßmann

    (Princeton University, Program for Applied and Computational Mathematics)

Abstract

Invariant tori of dynamical systems occur both in the dissipative and in the conservative context. I focus here on the latter, where the tori are intrinsically parametrised by the actions y 1,…, y n conjugate to the angles x 1,…,x n on the torus. The distribution of maximal tori in a nearly integrable Hamiltonian system is governed by the invariant tori of co-dimension one. The different Cantor families of maximal tori shrink down to normally elliptic tori and are separated by the web formed by stable and unstable manifolds of normally hyperbolic tori. The lower dimensional invariant tori form Cantor families themselves, and occurring bifurcations in turn organize the distribution of normally elliptic and hyperbolic tori.

Suggested Citation

  • Heinz Hanßmann, 2004. "A Survey on Bifurcations of Invariant Tori," Springer Books, in: J. Delgado & E. A. Lacomba & J. Llibre & E. Pérez-Chavela (ed.), New Advances in Celestial Mechanics and Hamiltonian Systems, pages 109-121, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-9058-7_7
    DOI: 10.1007/978-1-4419-9058-7_7
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