1.2. MAIN RESULTS 3

Consider the terms aij(0, /i, s)xlyi of h\ evaluated on 70. We define £ to be the

greatest order of the branching points ztia corresponding to CLij(6, \i, e)al0(u)f3Q(u).

That is:

(1.2) ^ = m a x { i ( r - 1) + jr : aij{6,ii,e) ^ 0 } .

Also we define

v — p — £.

H P 5 The constant v is greater or equal than 0.

R E M A R K 1.4. Hypothesis HP5 controls the growth of the perturbation term

lis9 h\(x ,y ,t / e, n, e)

evaluated at the homoclinic orbit, near the singularities. In fact, if hypothesis HP 5

is assumed:

/x£ p ||M7o(u),£/e,/i,£)||oo = 0(/i),

for I Imu | a — e.

R E M A R K 1.5. According to Hypotheses HP1-HP5, if p 1, then dyh\ = 0.

Indeed, if £ 1, then by hypothesis HP5, p 1. Therefore, we consider the case

£ 1. By definition of £ and using that r 1, we have that for any pair of positive

integers, i, j such that aij(8,iJL,e) ^ 0,

1 £ i{r - 1) + jr jr j .

Therefore, j = 0 and this implies that h\ has no terms depending on the variable

y. Therefore dyh\ — 0.

1.2. Mai n results

It is a well known fact that Poincare maps associated to periodic Hamiltonian

perturbations of one degree of freedom Hamiltonian systems having a homoclinic

connection, have either primary homoclinic points or a homoclinic connection, the

latter possibility being non-generic. These points are related to the zeros of the

Melnikov function M(s,e) defined by

M(S,E)

/

oo

{h0,h1}(-f0{t + s),t/e)dt.

-OO

Let Pto be the Poincare map from to to to + 2ns. We denote by A the area of the

lobe generated by the stable and the unstable manifolds between two homoclinic

points and by # the angle between the stable and unstable invariant manifolds at a

homoclinic point. We observe that, since the Poincare map is area preserving, the

area A will not depend on the homoclinic points.

The main results are:

T H E O R E M 1.1. Under hypotheses HP 1-HP5, for e — 0 + , /x -» 0, and for any

to G M, the following formulae hold:

A = nep M(v:£)dv + 0{ii2s2l/+r,n2£l,+p+\n£p+2)e-a/*,

sintf =

f

,£p]].M'{S^£\].2+0^2£2^r-2^2£^-\^)e-^,