A note on micro-instability for Hamiltonian systems close to integrable
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- by Abed Bounemoura and Vadim Kaloshin PDF
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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.References
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Additional Information
- Abed Bounemoura
- Affiliation: CNRS-IMPA UMI, Rio de Janeiro AC 22460-320, Brazil
- MR Author ID: 853363
- Email: abedbou@gmail.com
- Vadim Kaloshin
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 624885
- Email: vadim.kaloshin@gmail.com
- Received by editor(s): December 19, 2014
- Received by editor(s) in revised form: March 30, 2015
- Published electronically: December 21, 2015
- Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1553-1560
- MSC (2010): Primary 37J25, 37J40
- DOI: https://doi.org/10.1090/proc/12796
- MathSciNet review: 3451232