184
Z. Zheng
If the natural-frequency distribution function g(ω) is the following Lorentz form
g(ω) = 1/[π(ω
2
+ 1)],
(4.127)
then by inserting it into (4.124) one has
z(t) =
∞
−∞
α 1 (ω, t)d ω
π(ω 2 + 1)
.
(4.128)
This integral can be done by extending to the complex plane of ω. Obviously the
extension should be to the upper half complex plane. By using the Cauchy residue
theorem, one obtains
z(t) = α 1 (ω = i, t).
(4.129)
By substituting (4.129) to (4.126)–(4.128), one has
˙
z(t) =
1
2
z(t)(K − 2 − K|z(t)|
2
).
(4.130)
This is a complex dynamical equation, which describes a two-dimensional realspace dynamics. It can be seen that there exists a critical point K c = 2 for the
dynamical system (4.130). When K ≤ K c , the system has the only one fixed-point
solution z ≡ 0, which denotes the asynchronously disordered state. When K ≥ K c ,
the zero solution becomes unstable, and the system experiences a bifurcation to the
non-zero branch
|z| =
(K − K c )/K.
(4.131)
This non-zero solution represents the emergence of the self-organized synchronization state. By comparing with the result (4.93) obtained in terms of the selfconsistency approach proposed in Sec. 4.3, one can find that here we obtain the same
results.
An important issue is the stability of the OA invariant manifolds [92, 93]. Recently,
we studied the stability of the OA manifold by using the analysis in the functional
space of the phase distribution function [102, 103]. We proved that the OA manifold is in fact the two-dimensional invariant manifold in the infinite-dimensional
density functional space. This greatly expands the applicability of the OA ansatz
as an effective method for analyzing the dynamics of globally coupled phase
oscillators [100], where the OA approach has its validity.
Studies on synchronization of the globally coupled phase oscillators can be
naturally extended to the case of complex networks:
Z. Zheng
If the natural-frequency distribution function g(ω) is the following Lorentz form
g(ω) = 1/[π(ω
2
+ 1)],
(4.127)
then by inserting it into (4.124) one has
z(t) =
∞
−∞
α 1 (ω, t)d ω
π(ω 2 + 1)
.
(4.128)
This integral can be done by extending to the complex plane of ω. Obviously the
extension should be to the upper half complex plane. By using the Cauchy residue
theorem, one obtains
z(t) = α 1 (ω = i, t).
(4.129)
By substituting (4.129) to (4.126)–(4.128), one has
˙
z(t) =
1
2
z(t)(K − 2 − K|z(t)|
2
).
(4.130)
This is a complex dynamical equation, which describes a two-dimensional realspace dynamics. It can be seen that there exists a critical point K c = 2 for the
dynamical system (4.130). When K ≤ K c , the system has the only one fixed-point
solution z ≡ 0, which denotes the asynchronously disordered state. When K ≥ K c ,
the zero solution becomes unstable, and the system experiences a bifurcation to the
non-zero branch
|z| =
(K − K c )/K.
(4.131)
This non-zero solution represents the emergence of the self-organized synchronization state. By comparing with the result (4.93) obtained in terms of the selfconsistency approach proposed in Sec. 4.3, one can find that here we obtain the same
results.
An important issue is the stability of the OA invariant manifolds [92, 93]. Recently,
we studied the stability of the OA manifold by using the analysis in the functional
space of the phase distribution function [102, 103]. We proved that the OA manifold is in fact the two-dimensional invariant manifold in the infinite-dimensional
density functional space. This greatly expands the applicability of the OA ansatz
as an effective method for analyzing the dynamics of globally coupled phase
oscillators [100], where the OA approach has its validity.
Studies on synchronization of the globally coupled phase oscillators can be
naturally extended to the case of complex networks: