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

A BIFURCAÇÃO DE HOPF 64
< v2, v2 > = 1
2π
= 1
2π
= 1
2π
2π
0
2π
0
+ c t 1c1sen 2 (s)]ds
= 1
2π
2π
0
(c t 2cos(s) + c t 1sen(s))(c2cos(s) + c1sen(s))ds
[c t 2c2cos 2 (s) + c t 2c1cos(s)sen(s) + c t 1c2cos(s)sen(s)
[c t 2c2cos 2 (s) + (2 − c t 2c2)sen 2 (s)
+ (c t 2c1 + c t 2c1)cos(s)sen(s)]ds
2π
0
[c t 2c2cos 2 (s) + 2sen 2 (s) − c t 2c2sen 2 (s)
+ 2c t 2c1cos(s)sen(s)]ds
= 1
2π
+c t 2c1
= 1
2π
2π
0
c t 2c2
[c t 2c2cos(2s) + 1 − cos(2s) + c t 2c1sen(2s)]ds
2π
2π
= 1
cos(2s)ds + (1 − cos(2s))ds
2π 0
0
2π
sen(2s)ds
0
c t sen(2s)
2c2 |
2
2π
0 + s| 2π
0 − sen(2s)
|
2
2π
0 − c t cos(2s)
2c1 |
2
2π
0
= 1 t
2π − c2c1 + c
2π
t
2c1 = 1.