Topological methods in the instability problem of Hamiltonian systems
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 diferent topology), (b) the manifolds \(W_\Lambda^s, W_\Lambda^u\) 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 diferentiability. Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
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