Communications in Mathematical Physics

, Volume 203, Issue 2, pp 385–419 | Cite as

Lower Dimensional Invariant Tori in the Regions of Instability for Nearly Integrable Hamiltonian Systems

  • Chong-Qing Cheng


Consider a Hamiltonian system of KAM type, H(p,q)=N(p)+P(p,q), with n degrees of freedom (n>2), where the Hessian of N is nondegenerate. For one resonance condition <I,N p >=0, \ (I∈ℤ n ), there is an immersed (n−1) dimensional submanifold ? in action variable space, where almost every point corresponds to a resonant torus for the unperturbed system, which is foliated by (n−1) dimensional ergodic components. It is shown in this paper that there is a subset of ? with positive (n−1)-dim Lebesgue measure, such that for each resonant torus corresponding to a point in this set at least two (n−1)-dimensional tori can survive perturbations. Generically, one is hyperbolic and the other one is elliptic.


Hamiltonian System Resonance Condition Variable Space Unperturbed System Lower Dimensional 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Chong-Qing Cheng
    • 1
  1. 1.Department of Mathematics, Nanjing University, Nanjing 210093, China.¶E-mail:

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