05-93 Marian Gidea, Rafael de la Llave

Topological methods in the instability problem of Hamiltonian systems (1082K, pdf) Mar 7, 05

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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 $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\st_\Lambda$, $W^\un_\Lambda$ intersect transversally, (c) the systems satisfies 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 to produce concrete estimates on the time to move, which were not considered in the previous papers.

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