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

A BIFURCAÇÃO DE HOPF 59
〈v ∗ 1, v ∗ 2〉 = 1
2π
= 1
2π
2π
0
2π
0
(d t 1cos(s) − d t 2sen(s))(d2cos(s) + d1sen(s))ds
[d t 1d2cos 2 (s) + d t 1d1cos(s)sen(s) − d t 2d2cos(s)sen(s)
− d t 2d1sen 2 (s)]ds
= 1
2π
1 + cos(2s)
d
2π 0 2
t 1d2 + d t sen(2s)
1d1 − d
2
t sen(2s)
2d2
2
−d t
1 − cos(2s)
2d1
ds
2
= 1
2π t
d1d2 − d
4π 0
t 2d1 + (d t 1d2 + d t 2d1)cos(2s) + d t 1d1sen(2s)
−d t 2d2sen(2s) ds
= 1
d
4π
t 2π
1d2
2π
ds − d
0
t 2d1
0
+ d t 2π
1d1
0
sen(2s)ds − d t 2π
2d2
0
−d t 1d1
ds + (d t 1d2 + d t 2π
2d1) cos(2s)ds
0
sen(2s)ds
= 1
d
4π
t 1d2s 2π 0 − dt2d1s 2π 0 + [dt1d2 + d t 2d1] sen(2s)
2
cos(2s)
2
2π
0 + dt cos(2s)
2d2
2
2π
0
= 1
4π [2π(dt1d2 − d t 2d1)] = 0.
2π
0