Synchronization is collective behaviors of periodic motions, which is easily observed in many biologiclal, social and physical systems. Y. Kuramoto proposed a mathmatical model for synchronization in 1975. Kuramoto model describes the periodic motion as a rotation of phase on the unit circle. By using polar coordinate, the dynamics can be expressed in the following form:

\begin{equation}\label{Ku-1}

\dot\theta_i = \Omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j - \theta_i) \quad \mbox{for} \quad i=1, \cdots, N

\end{equation}

and Kare called the natural frequency and the coupling strength, respectively.

Kuramoto model is derived from the dynamcis of Landau-Stuart oscillators on complex plane.

\begin{equation}\label{LS-1}

\dot z = (1-|z|^2 + {\mathrm i} \Omega)z

\end{equation}

By employing polar coordinate , \eqref{LS-1} can be separated into the dynamics on and :

\begin{equation}

\dot r = (1-r^2)r, \qquad \dot \theta = \Omega.

\end{equation}

Then, we can see that the unit circle is a stable limit cycle and the origin is an unstable equilibrium. We now plug linear couplings on \eqref{LS-1}:

\begin{equation}\label{LS-2}

\dot z_i = (1-|z_i|^2 + {\mathrm i} \Omega)z_i + \frac{K}{N}\sum_{j=1}^N (z_j - z_i)

\end{equation}

Assume that the oscillators are on the limit cycle to satisfy , i.e., and compare the imaginary parts on both sides of \eqref{LS-2}, we can attain \eqref{Ku-1}

We define synchronization for the Kuramoto model.

(1) Asymptotic complete phase synchronization:

(2) Phase-locked state:

(3) Asymptotic complete frequency synchronization:

Since sine function is odd, by summing up for all on both sides of \eqref{Ku-1}, the coupling term is cancelled out and we have

Therefore, once the synchronization occurs, the frequency of the synchronized oscillators will be the average of natural frequencies.

By assuming , we may focus on the dynamics of fluctuations.

By numerical simulations, it is known that the Kuramoto model show synchronization for sufficiently large coupling strength . The following two movies are numrical simulation of Kuramoto model with particles and the same initial configurations. The natural freqeuncies are chosen from randomly.

We set the coupling strength : the movements of oscillators are affected by others, but it doesn't show any synchronous phenomenon.

We set the coupling strength : the oscillators gather and show the phase-locked state.

Pingback: Remarks on the complete synchronization of the Kuramoto model – 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Practical synchronization of Kuramoto system with an intrinsic dynamics – 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Synchronization of the Kuramoto oscillators with adaptive couplings | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Collective synchronization of classical and quantum oscillators | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Remarks on the complete synchronization of the Kuramoto model | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Practical synchronization of Kuramoto system with an intrinsic dynamics | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Collective synchronization of classical and quantum oscillators | 박진영(Jinyeong Park, 朴 鎭永)

Pingback: Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators | 박진영(Jinyeong Park, 朴 鎭永)