4
I. BALDOMA AND E. FONTIC H
where so $o
are
the two zeros of the Melnikov function (associated to two consecu
tive homoclinic points), closest to zero, which depend on to Furthermore to + so(to)
is 2TT£ periodic.
R E M A R K 1.6. Since to + so(^o)
^s
2neperiodic, the expression for the angle $
is 2TT£periodic in to
We define the function
J(x,y,0) = {h0,hi}(x,y,6).
By hypothesis H P 2 on hi, J is 27rperiodic in 9 and has zero average with respect
to 0. Then we can consider its Fourier expansion
J(x,y,9)^^2jk(x,y)eik9.
Moreover, for all k G Z, J^ (70(11)) has a branching point of order at most £ + 1 at
u = ±ia . Therefore, near the singularity u = ia,
Jfc(7o(w))
has the form
Jfc(7o(M))=
(u  Lv+i f
Jfc"°+
S
J*~
"
i a
)
m / 9
^ \ ral
and, near the singularity u — —ia, Jk(lo(u)) has the form
v
\ ral J
We note that J^0 — J_k0
We further consider the following hypothesis:
H P 6 The Fourier coefficients J±\ evaluated on 70(u), that is J±i(7o(^))? have
singularities of order exactly I + 1 at the points u = zbai.
R E M A R K 1.7. Hypothesis HP 6 is generic because it is equivalent to suppose
that the coefficients J±±
0
of the Laurent expansion of J± 1(70(1*)) are different from
zero.
We can obtain an asymptotic expression of the Melnikov function and conse
quently of the area of the lobe and of the angle.
COROLLARY 1.1. If HP 1HP6 hold, then for e —
0+,
\x » 0 and for any
t0 eR,
M(s,e) =
e~l
^ 1 J^QI R e ( e ^  ^
+ 1
^
2
e  * * /
£
) ) e 
Q
/
£
+ 0 (
g

m
e ~
a
/
£
) ,
8TT
,
+0(/x
2
£
2 l
'
+ r

2
,/x
2
£"
+ p

1
,M£
l /
)e
a / £
,
where J±0 =
\J^o\e*e
and T is the Gamma function.