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The incoherent state of the Kuramoto model has a uniform conditional phase density, $\rho (\theta \mid \omega )=1/(2\pi )$, in the continuum population description. Its Kuramoto order parameter vanishes because ${\int }_0^{2\pi }{e}^{i\theta }d\theta =0$: individual phases cancel even though the oscillators keep moving.

The mean-field coupling then vanishes, so a uniform density remains uniform while each frequency class rotates. This state must be retained separately when deriving a self-consistency equation by dividing by a nonzero order parameter. A finite random sample generally has a small nonzero empirical order parameter; this fluctuation is not a contradiction of the continuum definition.^[ref]

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.

Consider the continuum Kuramoto model with an even probability density function $g(\omega )$. Seek a stationary Kuramoto order parameter in a frame with $\psi =0$. For $r>0$, phase locking gives $\sin \theta =\omega /(Kr)$ on the stable branch, so locked phase oscillators contributeDrifting phase oscillators have stationary angular density proportional to $|\omega -Kr\sin \theta {|}^{-1}$. Their real contribution integrates to zero as a periodic logarithmic derivative; their imaginary contributions cancel between $\omega$ and $-\omega$.^[ref]

Substitution $\omega =Kr\sin \theta$ and division by $r>0$ yieldThe incoherent state of the Kuramoto model must be retained separately before division. If $g$ is continuous with $0<g(0)<\infty$, any nonzero stationary branch reaching $r=0$ at a finite coupling must satisfyIndeed, the integral tends to $g(0){\int }_{-\pi /2}^{\pi /2}{\cos }^2\theta d\theta =\pi g(0)/2$. This limiting stationary calculation alone does not establish the branch's stability, its global existence, or the first instability of a multimodal frequency population.

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]