4 An Introduction to Emergence Dynamics in Complex Systems
179
In addition, we can see from the above two expressions that the generalized order
parameters are actually the Fourier coefficients of the distribution function ρ(θ, t).
The order parameters α n can describe different orderness of a population of
oscillators. First, it can be easily seen that |α n | ≤ 1. |α n | = 0 represents
a statistically homogeneous and random distribution of oscillators with phases
in [0, 2π ], which is denoted as the incoherent state. When |α n | = 1, oscillators are in the clustering state. For example, |α 1 | = 1 corresponds to the state
that all oscillators possess the same phase {θ i (t) = θ (t), i = 1, 2, . . . , N },
i.e. the globally synchronous state. |α 2 | = 1 refers to a two-cluster state,
where oscillators are divided into two synchronous clusters with a phase shift
π:
θ i (t) = θ (t), θ j (t) = θ (t) + π, i = 1, 2, . . . , M , j = M + 1, . . . , N
. Therefore
|α n | = 1 represents the state of n synchronous clusters with phase shift 2π/n, and
the state with order parameter 0 < |α n | < 1 is the partially synchronous state.
We first consider the case of N coupled identical phase oscillators. Our mission
is to derive the evolution dynamics of the order parameters. For a finite number of
oscillators, the dynamics of the generalized order parameters α should be equivalent
to that of the microdynamics oscillators, whose equations of motion are written as
˙
θ j (t) = F(α, θ j , β), j = 1, 2, . . . , N .
(4.102)
By taking the time derivative of the order parameter (4.96) on both sides and
inserting the equations of motion (4.102), one gets
˙
α n =
in
N
N
j=1
e
inθ j F(α, θ j , β).
(4.103)
Because F(α, θ, β) is a 2π-periodic functions of the phase θ, it can be expanded
into the Fourier series as
F(α, θ j , β) =
∞
k=−∞
f k (α, β)e
ikθ j .
(4.104)
The real function F(α, θ, β) requires the Fourier coefficients satisfy f −k (α, β) =
−
f k (α, β), where
−
f k is the complex conjugate of f k . Inserting the expansion (4.104)
into (4.103) and by using the definition of α n , one may get the equations of motion
for the order parameters:
˙
α n = in
∞
k=−∞
f k (α, β)α k+n .
(4.105)
It can be found that dynamical behavior of α n (t) depends on all the other order
parameters {α k+n }. Let us consider a truncated case of the system (4.102), i.e. only
the first-order order expansion of the coupling function (4.104) is kept, then
179
In addition, we can see from the above two expressions that the generalized order
parameters are actually the Fourier coefficients of the distribution function ρ(θ, t).
The order parameters α n can describe different orderness of a population of
oscillators. First, it can be easily seen that |α n | ≤ 1. |α n | = 0 represents
a statistically homogeneous and random distribution of oscillators with phases
in [0, 2π ], which is denoted as the incoherent state. When |α n | = 1, oscillators are in the clustering state. For example, |α 1 | = 1 corresponds to the state
that all oscillators possess the same phase {θ i (t) = θ (t), i = 1, 2, . . . , N },
i.e. the globally synchronous state. |α 2 | = 1 refers to a two-cluster state,
where oscillators are divided into two synchronous clusters with a phase shift
π:
θ i (t) = θ (t), θ j (t) = θ (t) + π, i = 1, 2, . . . , M , j = M + 1, . . . , N
. Therefore
|α n | = 1 represents the state of n synchronous clusters with phase shift 2π/n, and
the state with order parameter 0 < |α n | < 1 is the partially synchronous state.
We first consider the case of N coupled identical phase oscillators. Our mission
is to derive the evolution dynamics of the order parameters. For a finite number of
oscillators, the dynamics of the generalized order parameters α should be equivalent
to that of the microdynamics oscillators, whose equations of motion are written as
˙
θ j (t) = F(α, θ j , β), j = 1, 2, . . . , N .
(4.102)
By taking the time derivative of the order parameter (4.96) on both sides and
inserting the equations of motion (4.102), one gets
˙
α n =
in
N
N
j=1
e
inθ j F(α, θ j , β).
(4.103)
Because F(α, θ, β) is a 2π-periodic functions of the phase θ, it can be expanded
into the Fourier series as
F(α, θ j , β) =
∞
k=−∞
f k (α, β)e
ikθ j .
(4.104)
The real function F(α, θ, β) requires the Fourier coefficients satisfy f −k (α, β) =
−
f k (α, β), where
−
f k is the complex conjugate of f k . Inserting the expansion (4.104)
into (4.103) and by using the definition of α n , one may get the equations of motion
for the order parameters:
˙
α n = in
∞
k=−∞
f k (α, β)α k+n .
(4.105)
It can be found that dynamical behavior of α n (t) depends on all the other order
parameters {α k+n }. Let us consider a truncated case of the system (4.102), i.e. only
the first-order order expansion of the coupling function (4.104) is kept, then