Ricardo Nicasso Benito A Reduç˜ao de Liapunov-Schmidt ... - Unesp

A BIFURCAÇÃO DE HOPF 60
〈v ∗ 2, v ∗ 2〉 = 1
2π
= 1
2π
2π
0
2π
0
(d t 2cos(s) + d t 1sen(s))(d2cos(s) + d1sen(s))ds
[d t 2d2cos 2 (s) + d t 1d1sen 2 (s) + d t 2d1cos(s)sen(s)
+ d t 1d2cos(s)sen(s)]ds
= 1
2π
1 + cos(2s)
d
2π 0 2
t 1 − cos(2s)
2d2 + d
2
t 1d1
+ cos(s)sen(s)(d t 2d1 + d t
1d2) ds
= 1
2π
(d
2π 0
t 2d2 + d t 1d1) 1
2 + (dt2d2 − d t 1d1) cos(2s)
2
+ (d t 2d1 + d t 1d2) sen(2s)
ds
2
= 1
2π
(d
4π 0
t 2d2 + d t 1d1) + (d t 2d2 + d t 1d1)cos(2s)
+ (d t 2d1 + d t
1d2)sen(2s) ds
= 1
(d
4π
t 2d2 + d t 2π
1d1) ds + (d
0
t 2d2 + d t 2π
1d1) cos(2s)ds
0
+ (d t 2d1 + d t 2π
1d2) sen(2s)ds
0
= 1
(d
4π
t 2d2 + d t 1d1)s 2π 0 + (dt2d2 + d t 1d1) sen(2s)
2
− (d t 2d1 + d t 1d2) cos(2s)
2
2π
0
= 1 t
(d2d2 + d
4π
t 1d1)2π
= 1
2 (dt 2d2 + d t 1d1) = dtd 2 .
Para o item (b) usaremos as seguintes relações:
2π
0