On diffusion in high-dimensional Hamiltonian systems

J. Bourgain, V. Kaloshin, Journal of Functional Analysis 229 (2005) 1–61.

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Journal Article | Published | English
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Abstract
The purpose of this paper is to construct examples of diffusion for E-Hamiltonian perturbations of completely integrable Hamiltonian systems in 2d-dimensional phase space, with d large. In the first part of the paper, simple and explicit examples are constructed illustrating absence of ‘long-time’ stability for size E Hamiltonian perturbations of quasi-convex integrable systems already when the dimension 2d of phase space becomes as large as log 1/E . We first produce the example in Gevrey class and then a real analytic one, with some additional work. In the second part, we consider again E-Hamiltonian perturbations of completely integrable Hamiltonian system in 2d-dimensional space with E-small but not too small, |E| > exp(−d), with d the number of degrees of freedom assumed large. It is shown that for a class of analytic time-periodic perturbations, there exist linearly diffusing trajectories. The underlying idea for both examples is similar and consists in coupling a fixed degree of freedom with a large number of them. The procedure and analytical details are however significantly different. As mentioned, the construction in Part I is totally elementary while Part II is more involved, relying in particular on the theory of normally hyperbolic invariant manifolds, methods of generating functions, Aubry–Mather theory, and Mather’s variational methods.
Keywords
Publishing Year
Date Published
2005-12-01
Journal Title
Journal of Functional Analysis
Volume
229
Issue
1
Page
1-61
ISSN
IST-REx-ID

Cite this

Bourgain J, Kaloshin V. On diffusion in high-dimensional Hamiltonian systems. Journal of Functional Analysis. 2005;229(1):1-61. doi:10.1016/j.jfa.2004.09.006
Bourgain, J., & Kaloshin, V. (2005). On diffusion in high-dimensional Hamiltonian systems. Journal of Functional Analysis, 229(1), 1–61. https://doi.org/10.1016/j.jfa.2004.09.006
Bourgain, Jean, and Vadim Kaloshin. “On Diffusion in High-Dimensional Hamiltonian Systems.” Journal of Functional Analysis 229, no. 1 (2005): 1–61. https://doi.org/10.1016/j.jfa.2004.09.006.
J. Bourgain and V. Kaloshin, “On diffusion in high-dimensional Hamiltonian systems,” Journal of Functional Analysis, vol. 229, no. 1, pp. 1–61, 2005.
Bourgain J, Kaloshin V. 2005. On diffusion in high-dimensional Hamiltonian systems. Journal of Functional Analysis. 229(1), 1–61.
Bourgain, Jean, and Vadim Kaloshin. “On Diffusion in High-Dimensional Hamiltonian Systems.” Journal of Functional Analysis, vol. 229, no. 1, Elsevier, 2005, pp. 1–61, doi:10.1016/j.jfa.2004.09.006.

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