180
Z. Zheng
F(α, θ) = f 1 (α)e
iθ
+ f −1 (α)e
−iθ
+ f 0 (α),
(4.106)
and the order parameter Eq. (4.105) can be greatly simplified. By substituting (4.106)
into the order parameter motion Eq. (4.105) one obtains
˙
α n = in
f 1 (α)α n+1 + f 1 (α)α n−1 + f 0 (α)α n
,
(4.107)
where n ≥ 0, α −n = α n .
One should point out that the solution of (4.105) possesses the same difficulty as
that of the original dynamical Eqs. (4.102). In fact, these two sets of equations are
equivalent descriptions, and the generalized order parameters can be regarded as a
set of collective variables transformed from the phase variables. This set of coupled
order parameter equations are difficult to solve. On the other hand, one can seek for
some specific solutions. One trivial solution is the incoherent/asynchronous solution
α n ≡ 0.
When the coupling strength is increased, oscillators will synchronize to each other,
and the microscopic motion will collapse into a low-dimensional phase space. In the
generalized order-parameter space, the system will also collapse to a low-dimensional
space. Therefore, according to the slaving principle, only a few (or even only one) of
these generalized order parameters will survive to be the dominant order parameters,
and other parameters will be fast variables and can be adiabatically eliminated [20,
21]. Here we will explore this interesting question from a theoretical perspective.
4.4.5 Emergence of Order Parameters
4.4.5.1 The Ott-Antonsen (OA) Ansatz
The above simple three-diagonal iterative form of the order-parameter dynamical equations implies the symmetry of the system and possible low-dimensional
dynamics. One of the simplest possibilities is the case when there exists a certain
relationship between different order parameters. A possible scenario is that all higherorder parameters {α n≥2 } depend on α 1 . This can be easily understood since our
previous self-consistency approach to the Kuramoto model is based on the study
of α 1 . By assuming a uniform form of function representing this dependence, let us
look for the following trial solution:
α n = G(α 1 , n),
(4.108)
where G(α 1 , n) is a differentiable function. The time derivative of (4.108) leads to
˙
α n = G α 1 (α 1 , n) ˙
α 1 .
(4.109)
Z. Zheng
F(α, θ) = f 1 (α)e
iθ
+ f −1 (α)e
−iθ
+ f 0 (α),
(4.106)
and the order parameter Eq. (4.105) can be greatly simplified. By substituting (4.106)
into the order parameter motion Eq. (4.105) one obtains
˙
α n = in
f 1 (α)α n+1 + f 1 (α)α n−1 + f 0 (α)α n
,
(4.107)
where n ≥ 0, α −n = α n .
One should point out that the solution of (4.105) possesses the same difficulty as
that of the original dynamical Eqs. (4.102). In fact, these two sets of equations are
equivalent descriptions, and the generalized order parameters can be regarded as a
set of collective variables transformed from the phase variables. This set of coupled
order parameter equations are difficult to solve. On the other hand, one can seek for
some specific solutions. One trivial solution is the incoherent/asynchronous solution
α n ≡ 0.
When the coupling strength is increased, oscillators will synchronize to each other,
and the microscopic motion will collapse into a low-dimensional phase space. In the
generalized order-parameter space, the system will also collapse to a low-dimensional
space. Therefore, according to the slaving principle, only a few (or even only one) of
these generalized order parameters will survive to be the dominant order parameters,
and other parameters will be fast variables and can be adiabatically eliminated [20,
21]. Here we will explore this interesting question from a theoretical perspective.
4.4.5 Emergence of Order Parameters
4.4.5.1 The Ott-Antonsen (OA) Ansatz
The above simple three-diagonal iterative form of the order-parameter dynamical equations implies the symmetry of the system and possible low-dimensional
dynamics. One of the simplest possibilities is the case when there exists a certain
relationship between different order parameters. A possible scenario is that all higherorder parameters {α n≥2 } depend on α 1 . This can be easily understood since our
previous self-consistency approach to the Kuramoto model is based on the study
of α 1 . By assuming a uniform form of function representing this dependence, let us
look for the following trial solution:
α n = G(α 1 , n),
(4.108)
where G(α 1 , n) is a differentiable function. The time derivative of (4.108) leads to
˙
α n = G α 1 (α 1 , n) ˙
α 1 .
(4.109)