G.Gentile
A proof of existence of whiskered tori with
quasi flat homoclinic
intersections in a class of almost integrable hamiltonian systems
(149K, Plain Tex)
ABSTRACT. Rotators interacting with a pendulum via small,
velocity independent, potentials are considered: the invariant tori
with diophantine rotation numbers are unstable and have stable and
unstable manifolds ({\it ``whiskers''}), whose intersections define
a set of homoclinic points. The homoclinic splitting can be introduced
as a measure of the splitting of the stable and unstable manifolds near
to any homoclinic point. In a previous paper, [G1], cancellation
mechanisms in the perturbative series of the homoclinic splitting have
been investigated. This led to the result that, under suitable conditions,
if the frequencies of the quasi periodic motion on the tori are large,
the homoclinic splitting is smaller than any power in the frequency of the
forcing (``quasi flat homoclinic intersections"). In the case $l=2$ the
result was uniform in the twist size: for $l>2$ the discussion
relied on a recursive proof, of KAM type, of the whiskers
existence, (so loosing the uniformity in the twist size). Here we extend
the non recursive proof of existence of whiskered tori to the more than
two dimensional cases, by developing some ideas illustrated in the quoted
reference.}