DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 867-877

Zhihong Xia

Title: Arnold Diffusion: A Variational Construction

We use variational method to study Arnold diffusion and instabilities in high dimensional Hamiltonian systems. Our method is based on a generalization of Mather's theory on twist maps and their connecting orbits to a higher dimensional setting. Under some generic nondegeneracy conditions, we can construct transition chains of arbitrary fixed length, crossing gaps of any size between invariant KAM (lower dimensional) tori. One of notable features of our result is that, instead of using transition tori alone for diffusion as in Arnold's construction, we also use cantori from Aubry-Mather theory in our mechanism for diffusion. Other results, such as shadowing properties, symbolic dynamics and transitivity, etc., can also be obtained by our method. Our nondegeneracy condition is a condition on the splitting of separatrix and in the so-called {\it a priori}\/ unstable systems, this condition can be verified by the so-called Poincar\'e-Melnikov integrals. In Arnold's original example for the instability, the perturbation is carefully chosen so that it does not touch any invariant tori on the normally hyperbolic invariant manifold. As an application of our results, we can choose arbitrary perturbations and are able to conclude the same results (in fact stronger), as long as the Poincar\'e-Melnikov integrals are in some sense non-degenerate.

1991 Mathematics Subject Classification:

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