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