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The Kuramoto model couples $N$ phase oscillators through their phase differences:Here ${\omega }_i$ is the natural frequency and $K\ge 0$ measures attractive coupling. Equal coupling between every pair gives the exact collective form ${\dot{\theta }}_i={\omega }_i+Kr\sin (\psi -{\theta }_i)$ in terms of the Kuramoto order parameter.^[ref]

For example, with two phase oscillators, put $\delta ={\theta }_2-{\theta }_1$. Subtracting their equations gives $\dot{\delta }={\omega }_2-{\omega }_1-K\sin \delta$. A stable fixed phase difference exists when $|{\omega }_2-{\omega }_1|<K$, on the branch with $\cos \delta >0$. Thus synchronization of oscillators can give equal observed frequencies even when their uncoupled natural frequencies differ. Equality is the marginal locking threshold, rather than a hyperbolic stable state.

For a large population, the Kuramoto self-consistency equation tests stationary collective states. The finite two-oscillator threshold and the continuum threshold concern different frequency populations and should not be substituted for each other.

Steven H. Strogatz is an applied mathematician studying nonlinear dynamical systems. His research includes synchronization of oscillators: how interacting periodic units can develop coordinated rhythms, as in flashing fireflies or biological oscillations.^[ref] With Duncan J. Watts, he studied small-world networks, connecting strong local clustering with short paths across a network.^[ref]

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]