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
2ne-periodic, 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 27r-periodic 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 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 1-HP6 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.
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