A note on micro-instability for Hamiltonian systems close to integrable

In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of “micro-diffusion”: under generic assumptions on hh and ff, there exists an orbit of the system...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2016-04, Vol.144 (4), p.1553-1560
Main Authors: Bounemoura, Abed, Kaloshin, Vadim
Format: Article
Language:English
Subjects:
B. ANALYSIS
Dynamical Systems
Mathematics
Research article
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Summary:In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of “micro-diffusion”: under generic assumptions on hh and ff, there exists an orbit of the system for which the drift of its action variables is at least of order ε\sqrt {\varepsilon }, after a time of order ε−1\sqrt {\varepsilon }^{-1}. The assumptions, which are essentially minimal, are that there exists a resonant point for hh and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/12796