On diffusion in high-dimensional Hamiltonian systems

Abstract

The purpose of this paper is to construct examples of diffusion for ε -Hamiltonian perturbations of completely integrable Hamiltonian systems in 2 d -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 ε Hamiltonian perturbations of quasi-convex integrable systems already when the dimension 2 d of phase space becomes as large as log 1 ε . 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 ε -Hamiltonian perturbations of completely integrable Hamiltonian system in 2 d -dimensional space with ε -small but not too small, | ε | > 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. Part I is due to Bourgain and Part II due to Kaloshin.