An Example of Instability in High-dimensional Hamiltonian Systems

In this article, we use a mechanism first introduced by Herman, Marco, and Sauzin to show that if a Gevrey or an analytic perturbation of a quasi-convex integrable Hamiltonian system is not too small with respect to the number of degrees of freedom, then the classical exponential stability estimates...

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Bibliographic Details
Published in:International mathematics research notices 2012, Vol.2012 (3), p.685-716
Main Author: Bounemoura, Abed
Format: Article
Language:English
Subjects:
Dynamical Systems
Mathematics
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Summary:In this article, we use a mechanism first introduced by Herman, Marco, and Sauzin to show that if a Gevrey or an analytic perturbation of a quasi-convex integrable Hamiltonian system is not too small with respect to the number of degrees of freedom, then the classical exponential stability estimates do not hold. Indeed, we construct an unstable solution whose drifting time is polynomial with respect to the inverse of the size of the perturbation. A different example was already given by Bourgain and Kaloshin, with a linear time of drift but with a perturbation which is larger than ours. As a consequence, we obtain a better upper bound on the threshold of validity of exponential stability estimates.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnr043