4 An Introduction to Emergence Dynamics in Complex Systems
171
The self-consistency approach is based on the assumption that the system has
a stationary state that does not change over time. In this stationary state, the order
parameter is a time-independent quantity, which can be obtained by its definition and
the equations of the motion. The self-consistency method is not limited to specific
dynamics and is one of the widely used methods in coupled oscillator systems.
When the number of oscillators N 1, the order parameter defined by (4.60)
is irrelevant to N and does not change with time. Due to the mean-field feature of
the summation in the coupling term, by using (4.60) one can rewrite the Kuramoto
model (4.44) to the following form:
d θ i /dt = ω i + KR sin(( − θ i ).
(4.61)
If one regards R as a parameter, Eq. (4.61) indicates that dynamics of all oscillators
are decoupled, i.e. the influence of other oscillators on the i-th oscillator is described
by the parameter R. If one knows R, (4.61) can be well worked out. However, R is
also an undetermined coefficient. A possible solution is to build an equation of R.
This equation is just the self-consistency equation [82, 84, 85]. Thus how to get the
self-consistency equation becomes the core task.
When N 1, we do not care about the micro-states {θ i (t)} any more, but how
are these phases distributed at any time t. Let ρ(θ, ω, t) denote a dynamical variable
representing the number density of the oscillators with natural frequency ω and phase
θ at time t. Since we are interested in the thermodynamic limit N → ∞, we expect
that an infinitely large number of oscillators will fall into an arbitrarily small but finite
interval θ . The single-oscillator distribution function ρ(θ, ω, t) depends not only
on the phase θ variable, but also on the natural frequency ω. ρ(θ, ω, t) is 2π-periodic
and satisfies the normalization condition
+∞
−∞
2π
0
ρ(θ, t)d θ d ω = 1.
(4.62)
The distribution of these oscillator phases directly determines the relative average.
The order parameter introduced in (4.60) as the summation of all oscillators can be
replaced by
α 1 = Re
i
=
+∞
−∞
2π
0
e
iθ
ρ(θ, t)d θ d ω.
(4.63)
We are mainly interested in the behavior, especially the long-term behavior of
ρ(θ, ω, t). Since dynamical Eq. (4.44) is invariant under a translation of θ i → θ i +θ 0 ,
one expects that the simplest collective behavior may be described by a uniform and
stationary distribution: ρ(θ, ω, t) = 1/2π, i.e.{θ i } are uniformly distributed in the
range 0 ∼ 2π. It can be easily verified that the case of R = 0 corresponds to
the incoherent state. This is always a solution of the system, but it is not always
171
The self-consistency approach is based on the assumption that the system has
a stationary state that does not change over time. In this stationary state, the order
parameter is a time-independent quantity, which can be obtained by its definition and
the equations of the motion. The self-consistency method is not limited to specific
dynamics and is one of the widely used methods in coupled oscillator systems.
When the number of oscillators N 1, the order parameter defined by (4.60)
is irrelevant to N and does not change with time. Due to the mean-field feature of
the summation in the coupling term, by using (4.60) one can rewrite the Kuramoto
model (4.44) to the following form:
d θ i /dt = ω i + KR sin(( − θ i ).
(4.61)
If one regards R as a parameter, Eq. (4.61) indicates that dynamics of all oscillators
are decoupled, i.e. the influence of other oscillators on the i-th oscillator is described
by the parameter R. If one knows R, (4.61) can be well worked out. However, R is
also an undetermined coefficient. A possible solution is to build an equation of R.
This equation is just the self-consistency equation [82, 84, 85]. Thus how to get the
self-consistency equation becomes the core task.
When N 1, we do not care about the micro-states {θ i (t)} any more, but how
are these phases distributed at any time t. Let ρ(θ, ω, t) denote a dynamical variable
representing the number density of the oscillators with natural frequency ω and phase
θ at time t. Since we are interested in the thermodynamic limit N → ∞, we expect
that an infinitely large number of oscillators will fall into an arbitrarily small but finite
interval θ . The single-oscillator distribution function ρ(θ, ω, t) depends not only
on the phase θ variable, but also on the natural frequency ω. ρ(θ, ω, t) is 2π-periodic
and satisfies the normalization condition
+∞
−∞
2π
0
ρ(θ, t)d θ d ω = 1.
(4.62)
The distribution of these oscillator phases directly determines the relative average.
The order parameter introduced in (4.60) as the summation of all oscillators can be
replaced by
α 1 = Re
i
=
+∞
−∞
2π
0
e
iθ
ρ(θ, t)d θ d ω.
(4.63)
We are mainly interested in the behavior, especially the long-term behavior of
ρ(θ, ω, t). Since dynamical Eq. (4.44) is invariant under a translation of θ i → θ i +θ 0 ,
one expects that the simplest collective behavior may be described by a uniform and
stationary distribution: ρ(θ, ω, t) = 1/2π, i.e.{θ i } are uniformly distributed in the
range 0 ∼ 2π. It can be easily verified that the case of R = 0 corresponds to
the incoherent state. This is always a solution of the system, but it is not always