The mechanism of destruction of resonance tori of Hamiltonian systems

By the KAM theory most of the nonresonant invariant tori of a nondegenerate integrable Hamiltonian system are preserved under a small perturbation of the Hamiltonian. On the other hand, Poincaré's theorem states that the invariant tori of the unperturbed system, foliated by periodic solutions, are not completely scattered under a perturbation: as a rule, several periodic solutions are preserved and become nondegenerate. This paper fills out the gap between these two results. Namely, it is shown here that the resonant tori of an integrable Hamiltonian system that are fibered in more than one-dimensional ergodic components are not completely destroyed under a small perturbation of the Hamiltonian: as a rule, several of their nonresonant subtori are preserved and become hyperbolic.
In concrete systems it is shown that there exist a large number of hyperbolic systems, and that this effect is an obstruction for the integrability of these systems.
Bibliography: 16 titles.