4 An Introduction to Emergence Dynamics in Complex Systems
177
of this case. In fact, the stable synchronous branch in the bifurcation diagram can no
longer be obtained by the self-consistency method.
4.4.4 Order Parameter Dynamics: Equations of Motion
For a large number of interacting oscillators, we can deal with its dynamics at the
statistical or macroscopic level instead of tedious microscopic details. The selfconsistency approach successfully predicts the transition to synchronization at the
macroscopic level, but it requires that the order parameter is time independent. Further
studies revealed in many cases the order parameter is far from stationary. Therefore
it is necessary to study the non-stationary dynamics of the order parameter.
Another interesting issue is the emergent process of order parameters. According
to the principle of synergetics, the emergence of an order parameter is the result of
the spontaneous collaboration and competition among various degrees of freedom
with different time scales. Why does α 1 becomes dominant and acts as the order
parameter to characterize the macroscopic behavior of the system instead of other
quantities? A general coupled oscillator system is usually complicated, and it is
important to set up a theoretical framework of order parameter dynamics. In this
section, we focus on the emergence of dominant order parameters in terms of the
basic idea of slaving principle [6, 21, 100, 101].
Let us consider N fully coupled oscillators, and the equations of motion can be
written as
˙
θ j (t) = F(α, θ j , β, γ j ),
(4.95)
j = 1, 2, . . . , N . Here β = {β 1 , β 2 , . . . } represent a set of uniform control parameters, i.e. these parameters are the same for all oscillators. γ = {γ 1 , γ 2 , . . . , γ N }
is a set of non-uniform control parameters, these parameters are not the same for
different oscillators. A typical example is that the natural frequencies of oscillators
are usually different, and in this case {γ i = ω i , i = 1, 2, , . . . , N }.
We define the following set of collective parameters α = {α n }:
α n =
1
N
N
j=1
e
inθ j .
(4.96)
Obviously α 1 has the same expression as that defined in (4.60). Parameters α n with
n > 1 are high-order parameters. We call this set of parameters α the generalized
order parameters.
In thermodynamic limit N → ∞, the detailed dynamical information of an individual oscillator is no longer relevant. It is more convenient to introduce density
distribution function ρ(γ , θ, t) and study the statistical property of the coupled oscillator system, where ρ(γ , θ, t)d θ is the probability that the phase of an oscillator falls
177
of this case. In fact, the stable synchronous branch in the bifurcation diagram can no
longer be obtained by the self-consistency method.
4.4.4 Order Parameter Dynamics: Equations of Motion
For a large number of interacting oscillators, we can deal with its dynamics at the
statistical or macroscopic level instead of tedious microscopic details. The selfconsistency approach successfully predicts the transition to synchronization at the
macroscopic level, but it requires that the order parameter is time independent. Further
studies revealed in many cases the order parameter is far from stationary. Therefore
it is necessary to study the non-stationary dynamics of the order parameter.
Another interesting issue is the emergent process of order parameters. According
to the principle of synergetics, the emergence of an order parameter is the result of
the spontaneous collaboration and competition among various degrees of freedom
with different time scales. Why does α 1 becomes dominant and acts as the order
parameter to characterize the macroscopic behavior of the system instead of other
quantities? A general coupled oscillator system is usually complicated, and it is
important to set up a theoretical framework of order parameter dynamics. In this
section, we focus on the emergence of dominant order parameters in terms of the
basic idea of slaving principle [6, 21, 100, 101].
Let us consider N fully coupled oscillators, and the equations of motion can be
written as
˙
θ j (t) = F(α, θ j , β, γ j ),
(4.95)
j = 1, 2, . . . , N . Here β = {β 1 , β 2 , . . . } represent a set of uniform control parameters, i.e. these parameters are the same for all oscillators. γ = {γ 1 , γ 2 , . . . , γ N }
is a set of non-uniform control parameters, these parameters are not the same for
different oscillators. A typical example is that the natural frequencies of oscillators
are usually different, and in this case {γ i = ω i , i = 1, 2, , . . . , N }.
We define the following set of collective parameters α = {α n }:
α n =
1
N
N
j=1
e
inθ j .
(4.96)
Obviously α 1 has the same expression as that defined in (4.60). Parameters α n with
n > 1 are high-order parameters. We call this set of parameters α the generalized
order parameters.
In thermodynamic limit N → ∞, the detailed dynamical information of an individual oscillator is no longer relevant. It is more convenient to introduce density
distribution function ρ(γ , θ, t) and study the statistical property of the coupled oscillator system, where ρ(γ , θ, t)d θ is the probability that the phase of an oscillator falls