97-494 Rudnev M., Wiggins S.
On the Use of the Melnikov Integral in the Arnold Diffusion Problem (66K, LaTeX) Sep 15, 97
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Abstract. In this note we want to point out a number of difficulties of arithmetic nature with the the so-called Melnikov integral (i.e., first order perturbation theory) as a measure of the splitting distance between the stable and unstable manifolds of tori in perturbations of a-priori stable integrable Hamiltonian systems with three or more degrees-of-freedom. We do this by considering a specific example which illustrates a number of the issues. We show that it is possible to introduce additional assumptions on the frequencies of the tori so that the Melnikov integral is the dominant term in the perturbation series for the distance between the stable and unstable manifolds of the torus. However, even when the Melnikov integral can be used to estimate the splitting distance, we show that even more difficulties arise when one uses it to determine if the manifolds intersect transversely, which is a key ingredient for constructing transition chains in the Arnold diffusion problem.

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