Appendix 2: Change of Variable for Invariant Manifolds
267
dx(t)
dτ
= α
y(t) − k y − x(t) + k x
−Rf (x(t) − k x + ϕ M 0 ) + Rf (ϕ M 0 ) + Rq C 1 0
dy(t)
dτ
= −(y(t) − k y − x(t) + k x ) + (z(t) − k z ) + Rq C 2 0
dz(t)
dτ
= −γ (z(t) − k z ) − β(y(t) − k y ) + βϕ L 0
that is
dx(t)
dτ
= α
y(t) − x(t) − Rf (x(t) − k x + ϕ M 0 )
+(k x − k y + Rf (ϕ M 0 ) + Rq C 1 0 )
(6.55a)
dy(t)
dτ
= x(t) − y(t) + z(t) + (−k x + k y − k z + Rq C 2 0 )
(6.55b)
dz(t)
dτ
= −γ z(t) − βy(t) + (γ k y + βk z + βϕ L 0 ).
(6.55c)
It turns out that the Eqs. (6.55) assume the form of the SEs (6.40) if and only if
k x , k y , and k z are such that
−k x + ϕ M 0 = 0
(6.56a)
−k x + k y − k z + Rq C 2 0 = 0
(6.56b)
βk y + γ k z + βϕ L 0 = 0
(6.56c)
and
X 0 = k x − k y + Rf (ϕ M 0 ) + Rq C 1 0 .
(6.57)
The solution of (6.56) permits to obtain
k x = ϕ M 0
(6.58a)
k y =
γ
β + γ
ϕ M 0 −
β
β + γ
ϕ L 0 −
γ
β + γ
Rq C 2 0
(6.58b)
k z =
β
β + γ
−ϕ M 0 + ϕ L 0 + Rq C 2 0
.
(6.58c)
The expressions (6.39) and (6.42) are readily obtained by inserting (6.58) in (6.54)
and (6.57), respectively.