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The Kuramoto model couples phase oscillators by sine differences. With all-to-all coupling, ${\dot{\theta }}_i={\omega }_i+Kr\sin (\psi -{\theta }_i)$, where $r{e}^{i\psi }$ is the Kuramoto order parameter. Natural-frequency diversity competes with coupling-induced phase coherence. In the symmetric stationary continuum branch, oscillators with $|\omega |\le Kr$ lock to the mean phase and the remainder drift.

For uniform nearest-neighbor bonds of strength $K_0$ on a ring with $N\ge 3$, retain the normalization of the given model:with indices interpreted modulo $N$. On an open chain the interior equations are the same, whileFor nonuniform bonds, replace each $K_0$ by the corresponding $K_{ij}$. A convention without the factor $1/N$ is obtained by rescaling the bond strengths; it is not silently substituted here.

For all-to-all coupling define the Kuramoto order parameterIt is the centroid of unit phase vectors: $r$ measures phase coherence and $\psi$ the mean phase. Multiplying by $e^{-i{\theta }_i}$ and taking imaginary parts givesThus only the two components of the Kuramoto order parameter mediate interaction.

For the symmetric stationary continuum branch, choose the rotating frame with $\psi =0$. Oscillators with $|\omega |\le Kr$ lock at $\theta =\arcsin (\omega /(Kr))$, choosing $\cos \theta >0$ for stability. Those with $|\omega |>Kr$ drift, with stationary phase density proportional to $|\omega -Kr\sin \theta {|}^{-1}$. Their real contribution to the order parameter vanishes, since the integral of $\cos \theta /(\omega -Kr\sin \theta )$ over a period is the change of a periodic logarithm. Their imaginary contributions cancel between $\omega$ and $-\omega$ because $g$ is symmetric.

The locked oscillators therefore give the real self-consistency relationSet $\omega =Kr\sin \theta$ and divide by $r>0$ to obtainThis is the Kuramoto self-consistency equation for partial synchronization. The incoherent state $r=0$ must be considered before this division. If $g$ is continuous at zero with $0<g(0)<\infty$, the branch meets incoherence at $K_c=2/[\pi g(0)]$.