On diffusion in high-dimensional Hamiltonian systems
Bourgain, Jean
Kaloshin, Vadim
Analysis
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.
Elsevier
2005
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https://research-explorer.app.ist.ac.at/record/8516
Bourgain J, Kaloshin V. On diffusion in high-dimensional Hamiltonian systems. <i>Journal of Functional Analysis</i>. 2005;229(1):1-61. doi:<a href="https://doi.org/10.1016/j.jfa.2004.09.006">10.1016/j.jfa.2004.09.006</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jfa.2004.09.006
info:eu-repo/semantics/altIdentifier/issn/0022-1236
info:eu-repo/semantics/closedAccess