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)
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].