Optimal stability and instability results for a class of nearly integrable Hamiltonian systems

Berti, Massimiliano ; Biasco, Luca ; Bolle, Philippe

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 13 (2002), p. 77-84 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We consider nearly integrable, non-isochronous, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(µ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_{d} = O((1/ \mu) \log(1/ \mu))$ by a variational method which does not require the existence of «transition chains of tori» provided by KAM theory. We also prove that our estimate of the diffusion time $T_{d}$ is optimal as a consequence of a general stability result proved via classical perturbation theory.

Résumé

In questa Nota consideriamo sistemi Hamiltoniani quasi-integrabili, non-isocroni, a-priori instabili soggetti ad una perturbazione di ordine $\mu$ (un polinomio trigonometrico) che non preserva i tori imperturbati. Facendo uso di tecniche variazionali che NON richiedono l’esistenza di «catene di tori KAM di transizione», dimostriamo l’esistenza di orbite di diffusione con un tempo di diffusione $T_{d} = O((1/ \mu) \log(1/ \mu))$. Proviamo inoltre che la nostra stima sul tempo di diffusione è ottimale, a seguito di un risultato generale di stabilità per le variabili di azione dimostrato mediante la teoria classica delle perturbazioni.

Détail

Publié le : 2002-06-01

MR 1949480 | Zbl 1072.37060

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