Kuramoto model

Example

Let $N \ge 2$, let $K \ge 0$ be a coupling strength, and let $\omega_1, \ldots, \omega_N \in \R$ be natural frequencies. The Kuramoto model is the autonomous system for phases $\theta_1, \ldots, \theta_N \in \R / 2\pi\Z$: $$\theta_i' = \omega_i + \frac{K}{N} \sum_{j=1}^{N} \sin(\theta_j - \theta_i), \qquad i = 1, \ldots, N.$$ Each oscillator turns at its own speed and is pulled towards the phases of the others.

For $N = 2$ the phase difference $\psi = \theta_2 - \theta_1$ satisfies $$\psi' = (\omega_2 - \omega_1) - K \sin \psi.$$ If $K > 0$ and $K \ge \abs{\omega_2 - \omega_1}$, this equation has an equilibrium $\psi^*$ with $\sin \psi^* = (\omega_2 - \omega_1)/K$. Then the two oscillators lock their phases at the difference $\psi^*$ and turn at the common speed $(\omega_1 + \omega_2)/2$. If $K < \abs{\omega_2 - \omega_1}$, then $\psi' \ne 0$ always, and the phases drift apart.