1209.2893 (Livia Corsi et al.)

Lower-dimensional invariant tori for perturbations of a class of
non-convex Hamiltonian functions    [PDF]

Livia Corsi, Roberto Feola, Guido Gentile

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.

View original: http://arxiv.org/abs/1209.2893