4 An Introduction to Emergence Dynamics in Complex Systems
141
˜
A =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
λ 1 0 0 0 0
0 λ 2 0 0 0
0 0 λ 3 0 0
0 0 0 . . . 0
0 0 0 0 λ n
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(4.18)
To distinguish the eigenvalue λ 1 from other eigenvalues, we relabel these
eigenvalues as.
λ u = λ 1 , λ
1
s = λ 2 , λ
2
s = λ 3 , . . . , λ
(n−1)
s
= λ n ,
where Reλ s i < 0, i = 1, 2, . . . n − 1. Then the corresponding state vector x can be
transformed to.
(u, s)
T
= Tx.
(4.19)
Equations (4.14) can be rewritten as
˙
u = λ u u + ˜
B u (u, s),
(4.20a)
˙
s = −λ s s + ˜
B s (u, s),
(4.20b)
where s is an (n − 1)-dimensional vector s = (s 1 , s 2 , . . . , s n−1 )
T .
By using the above procedure, one successfully separates the slow mode u(t) from
all the variables in terms of the transformation (4.20a), and the remaining variables
{s i (t), i = 1, 2, . . . , n − 1} are fast variables that satisfy Eq. (4.20b). One can apply
the adiabatic elimination to (4.20b) based on the same reason as
˙
s = 0.
(4.21)
This leads to the following n − 1 equations:
λ s s = − ˜
B s (u, s).
(4.22)
Fast variables s can be analytically solved from the n − 1 equations of (4.22) as
the function of the slow variable u in the form s = s(u). Then by inserting s(u) into
Eq. (4.20a), one obtains
˙
u = λ u u + ˜
B u (u, s(u)).
(4.23)
This is the one-dimensional dynamical equation of the order parameter u, which
can be easier to analyze.
When there exist degenerations for the first m > 1 eigenvalues {λ 1 , λ 2 , . . . , λ m },
which means that they have the same real parts: