4 An Introduction to Emergence Dynamics in Complex Systems
163
K ≥ 0 is the coupling strength. The frequencies are randomly chosen from a given
probability density g(ω), which is usually assumed to be one-humped and symmetric
about its mean ω 0 . This mean-field model was hereforth called the Kuramoto model.
The Kuramoto mean-field model can be successfully solved by using the selfconsistency approach in terms of statistical physics, which reveals that a large number
of coupled oscillators can overcome the disorder due to different natural frequencies
by interacting with each other, and the synchronized state emerges in the system.
The success of Winfree’s and Kuramoto’s works aroused extensive studies of
synchronization under more generalized cases (interested reader may refer the review
papers and the monographs [83–85]). The study of coupled phase oscillator synchronization, and the Kuramoto model on complex networks has become the focus of
research [86–89].
Apart from the self-consistency approach, recently Ott and Antonsen proposed
an approach (OA ansatz) to obtain the dynamical equations of order parameters
[90, 91]. Strogatz explained the physical meaning of the OA ansatz based on the
Watanabe-Strogatz transformation [92, 93].
In recent years, with the widespread studies of chaotic oscillations, the notion
of synchronization has been generalized to chaotic systems [76]. The study of
synchronization of coupled chaotic oscillators extended the scope of synchronization
dynamics, and different types of chaos synchronization such as complete/identical
synchronization, generalized synchronization, phase synchronization, and measure
synchronization were revealed [88, 94, 95].
4.4.2 Microdynamics of Synchronization
Let us begin with the simplest scenario to explore the synchronous dynamics. It is
very important to discuss the microscopic mechanism of synchronization, which can
make us better understand how a large number of coupled oscillators form ordered
behaviors through interaction and self-organization.
4.4.2.1 Phase-Locking of Two Limit-Cycle Oscillators
We consider two mutually coupled oscillators z 1,2 (t) that are described by Eq. (4.28)
but with different natural frequencies. They are coupled to each other and obey the
following dynamical equations of motion
˙
z 1,2 = (λ 1,2 + iω 1,2 )z 1,2 − b 1,2 z 1,2
z 1,2
2 + K 1,2 z 2,1
z 2,1
2 ,
(4.45)
where λ 1,2 > 0, K 1,2 > 0. b 1,2 are two real parameters, and ω 1 = ω 2 are the
natural frequencies of the two oscillators. The third term at the right hand side of
163
K ≥ 0 is the coupling strength. The frequencies are randomly chosen from a given
probability density g(ω), which is usually assumed to be one-humped and symmetric
about its mean ω 0 . This mean-field model was hereforth called the Kuramoto model.
The Kuramoto mean-field model can be successfully solved by using the selfconsistency approach in terms of statistical physics, which reveals that a large number
of coupled oscillators can overcome the disorder due to different natural frequencies
by interacting with each other, and the synchronized state emerges in the system.
The success of Winfree’s and Kuramoto’s works aroused extensive studies of
synchronization under more generalized cases (interested reader may refer the review
papers and the monographs [83–85]). The study of coupled phase oscillator synchronization, and the Kuramoto model on complex networks has become the focus of
research [86–89].
Apart from the self-consistency approach, recently Ott and Antonsen proposed
an approach (OA ansatz) to obtain the dynamical equations of order parameters
[90, 91]. Strogatz explained the physical meaning of the OA ansatz based on the
Watanabe-Strogatz transformation [92, 93].
In recent years, with the widespread studies of chaotic oscillations, the notion
of synchronization has been generalized to chaotic systems [76]. The study of
synchronization of coupled chaotic oscillators extended the scope of synchronization
dynamics, and different types of chaos synchronization such as complete/identical
synchronization, generalized synchronization, phase synchronization, and measure
synchronization were revealed [88, 94, 95].
4.4.2 Microdynamics of Synchronization
Let us begin with the simplest scenario to explore the synchronous dynamics. It is
very important to discuss the microscopic mechanism of synchronization, which can
make us better understand how a large number of coupled oscillators form ordered
behaviors through interaction and self-organization.
4.4.2.1 Phase-Locking of Two Limit-Cycle Oscillators
We consider two mutually coupled oscillators z 1,2 (t) that are described by Eq. (4.28)
but with different natural frequencies. They are coupled to each other and obey the
following dynamical equations of motion
˙
z 1,2 = (λ 1,2 + iω 1,2 )z 1,2 − b 1,2 z 1,2
z 1,2
2 + K 1,2 z 2,1
z 2,1
2 ,
(4.45)
where λ 1,2 > 0, K 1,2 > 0. b 1,2 are two real parameters, and ω 1 = ω 2 are the
natural frequencies of the two oscillators. The third term at the right hand side of