Synchronization of oscillators

Synchronization of oscillators occurs when interacting or driven oscillators acquire a sustained relation between their phases or frequencies. Equal amplitudes are not required. In one simple phase model, a relative phase $\psi$ obeys $\dot{\psi }=\Delta \omega -K\sin \psi$, where $\Delta \omega$ is the uncoupled frequency mismatch and $K>0$ the coupling strength.

When $|\Delta \omega |<K$, a fixed relative phase exists; the branch with $\cos \psi >0$ is stable because the linearized restoring coefficient is $-K\cos \psi$. The Kuramoto model extends this competition between mismatch and coupling to a population of phase oscillators.^[ref]