2 I. BALDOMA AND E. FONTICH
satisfying \u ia\ p, a,ig(u ia) G (—37r/2,7r/2), 70 (W) can be expressed
as
a°^
= 1 ^ w J
1
+ ° (
u
-
i a
)
1 / 9
)' A)(«) = 7 ^ T T ^
1
+
°(M
-
ia)1/9)-
(w ia)c'q [u ia)l+c/q
and for ii G C such that \u -f za| p, arg(i/, + ia) G (—7r/2, 37r/2), 70(W)
can be expressed as
(w + za) 0 / 9 [u + ia)l+c/q
where c, g G Z, g ^ 0. Moreover onw = ±za there are no other singularities
of 70. We define
c
r = 1 + - 1.
Of course, poles are included in this definition of branching points.
R E M A R K 1.1. According to Proposition 2.3 (Chapter 2) always exists a 0
such that 7o(ii) is analytic on the strip { w G C : | lvau\ a}.
R E M A R K 1.2. According to Proposition 24 (Chapter 2), ifV(x) = anxn H h
a
m
x m is a polynomial and we assume that ao(u) has a singularity at u u* G C,
then for u in a neighborhood of u* we have that
«oM =
(
, _ ^
2 / ( m
-
2 )
( i + Q(^^) 2 / ( m " 2 ) )
&«) = -
( M
_ ^ l
/ ( T O
-
2
) (
1
+
Q(M-^)2/(T O"2))-
i s a consequence, the exponents of u it* in £/ie expressions of a^ and J3Q are
rational numbers.
1.1.2. H y p o t h e s e s over th e perturbation.
H P 2 The function hi(x,y,6,fjL,e) is defined for (x,y) C U C C 2 , 0 G R, /i G
),/io)? £ £ (0, So), it is and 27r-periodic in #, has zero mean:
r
2ir
hi(x,y,9, fi,e) dO 0
/o
and it is real analytic with respect to (x,y,fi).
H P 3 The function hi(x,y,0,n,s) is a polynomial of degree K and order k (i.e.
the lowest degree of the monomials in h±) in the (x,y) variables. That is
hi(x,y,0,fj,,£)= ] T atj(0,n,£)xlyJ.
H P 4 The order k of the perturbation satisfies
2fc - 2 n.
R E M A R K 1.3. VFe observe that HP4 implies that the origin also is a parabolic
fixed point of the perturbed system and the derivative of the vector field evaluated
at this point is the same as the one of the unperturbed system.
f
Jo
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