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. © 2013 Springer Science+Business Media New York.
Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions / Livia, Corsi; Feola, Roberto; Guido, Gentile. - In: JOURNAL OF STATISTICAL PHYSICS. - ISSN 0022-4715. - STAMPA. - 150:1(2013), pp. 156-180. [10.1007/s10955-012-0682-8]
Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions
FEOLA, ROBERTO;
2013
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. © 2013 Springer Science+Business Media New York.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.