Kuramoto model

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}
\Omega_i and K 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 z=r e^{\mathrm i \theta}, \eqref{LS-1} can be separated into the dynamics on r and \theta:
\begin{equation}
\dot r = (1-r^2)r, \qquad \dot \theta = \Omega.
\end{equation}
Then, we can see that the unit circle r=1 is a stable limit cycle and the origin r=0 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 r=1, i.e., z=e^{\mathrm i \theta} 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:

 \lim_{t \to \infty} |\theta_i - \theta_j| = 0.


(2) Phase-locked state:

 \lim_{t\to\infty} |\theta_i - \theta_j| = \theta_{ij}^\infty \quad \mbox{for some constant} \theta_{ij}^\infty


(3) Asymptotic complete frequency synchronization:

 \lim_{t\to\infty} |\dot\theta_i - \dot\theta_j| = 0.

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

 \sum_{i=1}^N \dot \theta_i = \sum_{i=1}^N \Omega_i.


Therefore, once the synchronization occurs, the frequency of the synchronized oscillators will be the average of natural frequencies.
By assuming \sum_{i=1}^N \Omega_i=0, 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 K. The following two movies are numrical simulation of Kuramoto model with N=50 particles and the same initial configurations. The natural freqeuncies are chosen from [-1, 1] randomly.

We set the coupling strength K=0.8: the movements of oscillators are affected by others, but it doesn't show any synchronous phenomenon.

We set the coupling strength K=2.2: the oscillators gather and show the phase-locked state.