multiple time scale dynamics with two fast variables and one slow ...

with F(z)=O(z 2 ) and M= (mi j). Taylor expanding F yields
z ′ = Mz+ 1 1
B(z, z)+ C(z, z, z)
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where the multilinear functions B and C are given by:
Bi(u, v) =
Ci(u, v, w) =
N ∂2
Fi(ξ)
∂ξ j∂ξk
N ∂3
Fi(ξ)
∂ξ j∂ξk∂ξl
j,k=1
j,k,l=1
u jvk
ξ=0
u jvkwl
ξ=0
The matrix M has eigenvaluesλ1,2=±iω0 forω0> 0. Let q∈C N be the eigenvec-
tor ofλ1 and p∈C N the corresponding eigenvector of the transpose M T i.e.
Mq=iω0q, M ¯q=−iω0 ¯q, M T p=−iω0p, M ¯p=iω0 ¯p
where the the overbar denotes componentwise complex conjugation. We can
always normalize p so that the standard complex inner product with q satisfies
¯p T q= N
j=1 ¯p jq j= 1. The first Lyapunov coefficient of the Hopf bifurcation can
then be defined by (Kuznetsov [88], p.180):
l Ku
1
1
T T −1 T
= ¯p C(q, q, ¯q)−2 ¯p B(q, L B(q, ¯q))+ ¯p B(¯q, (2iω0IN− M)
2ω0
−1 B(q, q)) (6.9)
In the case of a two-dimensional vector field F= (F 1 , F 2 ) the formula (6.9) can
be expressed in the simpler form (Kuznetsov [88], p.98):
where
l Ku
1
= 1
2ω 2
0
Re(ig20g11+ω0g21) (6.10)
g20= ¯p T B(q, q), g11= ¯p T B(q, ¯q), g21= ¯p T C(q, q, ¯q)
It is important to note that l Ku
1 is not uniquely defined until we choose a normal-
ization of the eigenvector q. We adopt the convention using unit norm ¯q T q=1.
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