INTRODUCTION ix
depending on the method, has the advantage to provide control on the remainders in
the Poincare-Melnikov asymptotic formula. In general, if p is small, the Melnikov
function does not give the right asymptotics in the case of exponentially small
splitting. In [HMS], Holmes et al are able to give upper and lower bounds for
the splitting of separatrices for quite general systems and for values of p 8. The
situation improves when dealing with specific systems. The most studied example
is the pendulum. In [Gel] and in [DS1], asymptotic expressions are given for the
separatrix splitting of the equation
x -f sin x = fj,ep sin t/e
for p 5 and p 0 respectively. Later on, Delshams and Seara in [DS2], could
get an asymptotic expression of the separatrix splitting for more general systems
given that p is bigger than a certain quantity which depends on the perturbation
and of the singularity order of the homoclinic orbit. Gelfreich in [Ge2] also gives
an asymptotic expression for the separatrix splitting, but it is difficult to find out
which p is needed in order to apply it. Finally, in [Ge3] Gelfreich studies in some
specific examples the p 0 case. The proposed method is the use of an auxiliary
system whose invariant manifolds are a good approximation near the singularities of
the invariant manifolds of the initial system. In [An] Angenent studies the splitting
using variational methods. Treshev [Tr] studies a more general perturbation of the
pendulum which includes the equation considered by Poincare. He uses a different
method based on the continuous averaging procedure developed by himself. The
asymptotic formula he obtains for the area, in his example, differs from the one
predicted by the Poincare-Melnikov integral. It is worth noting that there are
examples for which the asymptotic expressions are not of the form
sre~~al£^
but
instead involve infinitely many terms of the form
e~ne~a/£,
n 0, [SMH].
In all these cases, one deals with Hamiltonian systems of one and a half degrees
of freedom or area-preserving maps such that the origin is a hyperbolic fixed point
of the non-perturbed Hamiltonian. Another situation where the separatrix splitting
phenomenon appears is when one considers quasi-periodic perturbations. We refer
to [DG], [DGJS1], [DGJS2] and [GGM] for such case.
Exponentially small phenomena are also found by Fiedler and Scheurle [FS] in
one step discretizations of autonomous equations.
This memoir is devoted to study the splitting for one and a half degrees of
freedom Hamiltonian systems of the form (3) such that the origin is a parabolic
fixed point. Specifically we assume that the linear part of the vector field at (0,0)
is
(Si)-
We consider the case of fast frequency perturbation. The paper [CFN] deals with
the case of constant frequency. The first point is to put sufficient conditions such
that the perturbed system also has invariant manifolds.
We have followed basically the structure of [DS2]. However, due to the fact
that many of their arguments strongly rely on the hyperbolic character of the fixed
point, we have had to introduce new techniques to deal with the parabolic case. To
this end we have also used tools introduced by Lazutkin [La2] [Lai]. It is worth
remarking that most of our arguments are can be adapted for the hyperbolic case.
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