# 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:
\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

$\Omega_i$ 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.
\label{LS-1}
\dot z = (1-|z|^2 + {\mathrm i} \Omega)z

By employing polar coordinate $z=r e^{\mathrm i \theta}$, \eqref{LS-1} can be separated into the dynamics on $r$ and $\theta$:

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

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}:
\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)

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:

(2) Phase-locked state:

(3) Asymptotic complete frequency synchronization:

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

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.