Kuramoto model

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.

The complex centroid of the oscillators' unit phase vectors has magnitude $r$ and argument $\psi$. A magnitude near one indicates phase coherence; zero indicates a balanced phase distribution, which need not be uniquely uniform at finite population size. Taking the imaginary part after multiplication by $e^{-i{\theta }_i}$ rewrites all-to-all sine coupling exactly in terms of these two collective variables.

For a symmetric stationary frequency distribution, locked oscillators contribute $r={\int }_{-Kr}^{Kr}\sqrt{1-{\omega }^2/(Kr{)}^2}g(\omega )d\omega$. The drifting oscillators' real contribution is zero by integrating a periodic logarithmic derivative, and imaginary contributions cancel by symmetry. Substituting $\omega =Kr\sin \theta$ and dividing by $r>0$ gives the equation. The incoherent solution must be retained before division. For finite positive continuous $g(0)$ the branch reaches it at $K_c=2/[\pi g(0)]$.