A scoped continuum self-consistency input

In the stationary mean-field Kuramoto self-consistency calculation, let $K>0$ be the coupling and let $g$ be an even, unimodal probability density on the real frequency axis, smooth near zero, with $g(0)>0$ and $g^{\mathrm{\prime }\mathrm{\prime }}(0)<0$. Even means $g(-\omega )=g(\omega )$; unimodal means nonincreasing on $[0,\mathrm{\infty })$. The model uses all-to-all sinusoidal coupling $K\mathrm{/}n$. In a rotating frame, $r$ denotes the modulus of the mean phase phasor: $r=0$ is the incoherent branch and $r>0$ describes partial phase coherence. The stationary self-consistency branch meets $r=0$ at $K_c=2\mathrm{/}(\pi g(0))$ and is locally supercritical: its positive solutions occur for $K>K_c$ sufficiently close to $K_c$.

The cited calculation assumes a stationary population in the rotating frame, including locked oscillators and a stationary distribution of drifting oscillators. It gives the positive-branch equation $1=K{\int }_{-\pi \mathrm{/}2}^{\pi \mathrm{/}2}{\cos }^2\theta g(Kr\sin \theta )d\theta$ and the stated local branch behavior. The phasor and density terminology is as in def-expander-prereq-phasor-calculus and Empirical phase measures and Fourier order statistics. This input asserts neither a finite-$n$ long-time limit nor global stability or almost-everywhere synchronization.