Rudnev M., Wiggins S.
On the Use of the Melnikov Integral in the Arnold Diffusion
Problem
(66K, LaTeX)
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.