= Kuramoto self-consistency equation
{c}
{title2=$1=K\int_{-\pi/2}^{\pi/2}\cos^2\theta\,g(Kr\sin\theta)\,d\theta$}
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 $K_c=2/[\pi g(0)]$.
Back to article page
