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

Bounemoura A, Kaloshin V. 2015. A note on micro-instability for Hamiltonian systems close to integrable. Proceedings of the American Mathematical Society. 144(4), 1553–1560.

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*Journal Article*|

*Published*|

*English*

Author

Bounemoura, Abed;
Kaloshin, Vadim

^{ISTA}^{}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.

Publishing Year

Date Published

2015-12-21

Journal Title

Proceedings of the American Mathematical Society

Publisher

American Mathematical Society

Volume

144

Issue

4

Page

1553-1560

IST-REx-ID

### Cite this

Bounemoura A, Kaloshin V. A note on micro-instability for Hamiltonian systems close to integrable.

*Proceedings of the American Mathematical Society*. 2015;144(4):1553-1560. doi:10.1090/proc/12796Bounemoura, A., & Kaloshin, V. (2015). A note on micro-instability for Hamiltonian systems close to integrable.

*Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/12796Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.”

*Proceedings of the American Mathematical Society*. American Mathematical Society, 2015. https://doi.org/10.1090/proc/12796.A. Bounemoura and V. Kaloshin, “A note on micro-instability for Hamiltonian systems close to integrable,”

*Proceedings of the American Mathematical Society*, vol. 144, no. 4. American Mathematical Society, pp. 1553–1560, 2015.Bounemoura, Abed, and Vadim Kaloshin. “A Note on Micro-Instability for Hamiltonian Systems Close to Integrable.”

*Proceedings of the American Mathematical Society*, vol. 144, no. 4, American Mathematical Society, 2015, pp. 1553–60, doi:10.1090/proc/12796.