97-502 Rudnev M., Wiggins S.

On the Dominant Fourier Modes in the Separatrix Splitting Distance Function for an A-Priori Stable, Three Degree-of-Freedom Hamiltonian System (531K, LaTeX) Sep 16, 97

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Abstract. This note is devoted to the analysis of the Fourier series that one obtains for the splitting distance function in an a-priori stable Hamiltonian system with three degrees-of-freedom, lately studied in connection with the so-called ``Arnold diffusion''. We give a summary of the theory, developed in Rudnev and Wiggins [1997], and compare it with a number of numerical experiments. These experiments not only illustrate and confirm the former theoretical concepts, but suggest that beyond the aforementioned theory, the same direction of reasoning shall prove useful. In particular, it suggests that for the vast majority of frequencies, and most of the values of the perturbation parameter $\epsilon$, the Fourier series in question has two components, which dominate the rest of it, whose index increases as $\varepsilon\rightarrow 0$. These components come from a certain sub-series of the Fourier series under consideration, which corresponds to the sequence of the best approximations to the frequency vector. We believe that the latter situation is generic, no matter what the approximation properties of the frequency vector. Therefore, in the case of three degrees of freedom, the arithmetic issues, which seemed so far one of the major obstructions to dealing quantitatively with exponentially small quantities, can be handled using the more refined approach developed in Rudnev and Wiggins [1997].

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