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

A BIFURCAÇÃO DE HOPF 62
〈v ∗ 2, v2〉 = 1
2π
= 1
2π
2π
0
2π
0
(d t 2cos(s) + d t 1sen(s))(c2cos(s) + c1sen(s))ds
[d t 2c2cos 2 (s) + d t 2c1cos(s)sen(s)
+ d t 1c2cos(s)sen(s) + d t 1c1sen 2 (s)]ds
= 1
2π
2π
0
[d t 2c2cos 2 (s) + d t 2c2sen 2 (s)
+ (d t 2c1 + d t 1c2)cos(s)sen(s)]ds
= 1
2π
= 1
2π
〈v ∗ 1, v2〉 = 1
2π
= 1
2π
2π
0
2π
0
= 1
(2π) = 1.
2π
2π
0
2π
0
[d t 2c2(cos 2 (s) + sen 2 (s)) + 0.cos(s)sen(s)]ds
ds = 1
2π (s) 2π
0
(d t 1cos(s) − d t 2sen(s))(c2cos(s) + c1sen(s))ds
[d t 1c2cos 2 (s) + d t 1c1cos(s)sen(s)
− d t 2c2cos(s)sen(s) − d t 2c1sen 2 (s)]ds
= 1
2π
2π
0
[d t 1c2cos 2 (s) − d t 1c2sen 2 (s)
+ d t 1c1cos(s)sen(s) − d t 1c1cos(s)sen(s)]ds
= 1
2π
= 1
2π
2π
0
2π
0
[d t 1c2(cos 2 (s) − sen 2 (s))]ds
0ds = 0.