Lower Dimensional Invariant Tori in the Regions of Instability for Nearly Integrable Hamiltonian Systems
Abstract
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∈^{n}), there is an immersed (n1) dimensional submanifold ? in action variable space, where almost every point corresponds to a resonant torus for the unperturbed system, which is foliated by (n1) dimensional ergodic components. It is shown in this paper that there is a subset of ? with positive (n1)dim Lebesgue measure, such that for each resonant torus corresponding to a point in this set at least two (n1)dimensional tori can survive perturbations. Generically, one is hyperbolic and the other one is elliptic.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 1999
 DOI:
 10.1007/s002200050618
 Bibcode:
 1999CMaPh.203..385C