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Journal of Nonlinear Science

, Volume 10, Issue 1, pp 49–68 | Cite as

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

  • F. Cong
  • T. Küpper
  • Y. Li
  • J. You
Article

Summary.

In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors.

Key words. Hamiltonian systems, resonant invariant tori, KAM-type theorem 

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Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • F. Cong
    • 1
  • T. Küpper
    • 2
  • Y. Li
    • 1
  • J. You
    • 3
  1. 1.Department of Mathematics, Jilin University, Changchun 130023, People's Republic of ChinaChina
  2. 2.Mathematisches Institut, Universität Köln, D-50931 Köln, GermanyGermany
  3. 3.Department of Mathematics, Nanjing University, Nanjing 210093, People's Republic of ChinaChina

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