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.