Topological methods in the instability problem of Hamiltonian systems

Document Type

Article

Publication Date

8-1-2017

Abstract

We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems. In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold ?, so that: (a) the manifold ? is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds Ws? , Wu ? intersect transversally, (c) the systems satisfy some explicit non-degeneracy assumptions, which hold generically. In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount. As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability. Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.

Publication Title

Discrete and Continuous Dynamical Systems- Series A

Volume Number

37

Issue Number

8

First Page

295

Last Page

328

ISSN

10780947

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