1. Notation and main results

In this chapter we present the main problem we consider, the hypotheses we

assume, and the rigorous statement of the main results. For that we have to begin

by introducing some notation.

At the end we present an example where the above mentioned results apply.

1.1. Notation and hypotheses

We study the splitting of separatrices in the case which we call the parabolic

case. Next we describe the settings of this case and the hypotheses we will need.

We consider Hamiltonian systems of one and a half degrees of freedom with

Hamiltonian

H(x, y, t/e, /i, E) = h0(x, y) +

{isvhx

(x, y, t/e, /i, e)

where

h0{x,y) = ^+V(x),

V(x) is an analytic function of order n, that is

V{x) =

anxn

H

with n 3. With these assumptions, for the unperturbed system (i.e. the sys-

tem when [i — 0) the origin is a parabolic fixed point and the derivative of the

Hamiltonian vector field at (0,0) is

0 1

0 0

The differential equations associated to the Hamiltonian are

(1.1) x — y -f

fj,epdyhi(x,y1t/e,/j,,e)

y = -V'{x) - nepdxhi(x,y,t/e,n,e).

We will assume the following hypotheses related to the unperturbed system.

Note that the unperturbed system is autonomous and independent on e.

1.1.1. Hypothesis for the unperturbed system.

HP1 We assume that, ho(x,y) =

y2/2

+ V(x) is analytic and V(x) =

anxn-\

with an 0 and n 3. Moreover we assume that ho has a homoclinic

orbit, associated to the equilibrium point (0,0)

We denote the time parameterization of the homoclinic orbit by

7o (u) = (a0(u),p0(u))

with some chosen (fixed) initial condition 70(0) = (#0,2/0) o n t n e n o m o -

clinic orbit.

We assume that 70(^) is analytic in a complex strip | Imu\ a with

branching points at u — ±ia, i.e., there exists p 0 such that for u G C

1