FM 09-2012 (arXiv:1209.2893; mp_arc 12-100)

Author: Livia Corsi, Roberto Feola and Guido Gentile

Title: Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions
 
Abstract: 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.

Keywords: Quasiperiodic motions; Lower-dimensional tori; Renormalization group; Bryuno vectors; Small divisors.

Livia Corsi
Dipartimento di Matematica
UniversitÓ di Napoli "Federico II"
Monte Sant'Angelo, Via Cinthia, 80126 Napoli, Italy
e-mail: livia.corsi@unina.it

Roberto Feola
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: roberto_feola@hotmail.com

Guido Gentile
Dipartimento di Matematica
UniversitÓ di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@mat.uniroma3.it