4 An Introduction to Emergence Dynamics in Complex Systems
181
where G α 1 (α 1 , n) = ∂G(α 1 , n)/∂α 1 . By using (4.107), one obtains
in(f 1 α n+1 + f 1 α n−1 + f 0 α n ) = iG α 1 (f 1 α 2 + f 1 α 0 + f 0 α 1 ).
(4.110)
The coefficients of the Fourier expansion terms f i or
−
f i at both sides in (4.110)
are equal, leading to the following relations:
nα n+1 = G α 1 α 2 , nα n−1 = G α 1 α 0 , nα n = G α 1 α 1 .
Because α 0 =
ρ(ω, θ, t)g(ω)d ωd θ = 1, one obtains
G α 1 =
nα n+1
α 2
= nα n−1 =
nα n
α 1
.
(4.111)
This naturally leads to
nG = G α 1 α 1 ,
(4.112)
One thus obtains the following form of the function:
α n = G(α 1 , n) = α
n
1 .
(4.113)
By inserting (4.113) to (4.107), one finally gets the equation of motion of α 1 as
˙
α 1 = i[f 1 (α 1 )α
2
1 + f 1 (α 1 ) + f 0 (α 1 )α 1 ].
(4.114)
It is interesting that (4.113) is just the ansatz recently proposed
by Ott and Antonsen [90, 91], which was called the Ott-Antonsen (OA) ansatz thereafter. Therefore, an infinite-dimensional dynamical system is reduced to a twodimensional order parameter equation. Undoubtedly this is a great reduction and
simplification of a complex system.
4.4.5.2 Poisson Invariant Manifold
In fact, Ott and Antonsen proposed the above ansatz based on the distribution
function. The distribution function can be expanded into Fourier series, where the
expansion coefficients are the generalized order parameters α n , that is,
ρ(θ, t) =
1
2π
1 +
∞
n=1
α n (t)e
inθ
+ α n (t)e
−inθ
.
(4.115)
Generally, the above summation can be executed to get the distribution function
only when all the Fourier coefficients {α n } are known. Ott and Antonsen assumed that
the coefficients {α n } are not independent of each other, and they are all determined