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

Bounemoura, Abed; Kaloshin, Vadim

doi:10.1090/proc/12796

A note on micro-instability for Hamiltonian systems close to integrable
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Proc. Amer. Math. Soc. 144 (2016), 1553-1560 Request permission

Abstract:

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 $h$ and $f$, there exists an orbit of the system for which the drift of its action variables is at least of order $\sqrt {\varepsilon }$, after a time of order $\sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $h$ 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.

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