We consider a Hamiltonian system in a neighbourhood of an elliptic equilibrium which is a minimum for the Hamiltonian. With appropriate non-resonance conditions we prove that in the neighbourhood of the equilibrium there exist low dimensional manifolds that are exponentially stable in Nekhoroshev’s sense. This generalizes the theorem of Lyapounov on the existence of periodic orbits. The result may be meaningful for, e.g., the dynamics of non-linear chains of the Fermi-Pasta-Ulam (FPU) type.
Si considera un sistema Hamiltoniano nell’intorno di un punto di equilibrio che sia punto di minimo per l'Hamiltoniana. Si dimostra, sotto condizioni opportune di non-risonanza, che nell’intorno del punto di equilibrio esistono varietà di dimensione bassa che presentano caratteristiche di stabilità esponenziale nel senso di Nekhoroshev. Il risultato costituisce una generalizzazione del teorema di Lyapounov sull’esistenza di orbite periodiche. Questo può essere interessante, ad esempio, per la comprensione della dinamica di catene non-lineari del tipo di Fermi, Pasta e Ulam (FPU).
@article{BUMI_2006_8_9B_1_1_0,
author = {Antonio Giorgilli and Daniele Muraro},
title = {Exponentially stable manifolds in the neighbourhood of elliptic equilibria},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {9-A},
year = {2006},
pages = {1-20},
zbl = {1178.70084},
mrnumber = {2204898},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2006_8_9B_1_1_0}
}
Giorgilli, Antonio; Muraro, Daniele. Exponentially stable manifolds in the neighbourhood of elliptic equilibria. Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006) pp. 1-20. http://gdmltest.u-ga.fr/item/BUMI_2006_8_9B_1_1_0/
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