Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems
The starting point for this analysis is a perturbation of an integrable Hamiltonian system, whose frequencies at a point \(p_0\) satisfy a resonance condition. The torus \(p=p_0\) of the unperturbed system being foliated into a family of \((n-1)\)-dimensional submanifolds, the idea is to investigate whether some of these submanifolds survive and still carry a quasi-periodic flow. The main theorem of the paper states that such lower dimensional tori indeed survive small perturbations under the following assumptions: the perturbed Hamiltonian is real analytic in some neighbourhood; the unperturbed part is convex; the frequency of the unperturbed flow on the \((n-1)\)-dimensional submanifold satisfy a KAM-type inequality. The surviving tori are generically of hyperbolic type. Proving this theorem, as usual in this area, is a very long and technical matter.
scientific article; zbMATH DE number 906584