Phase transitions in the Kuramoto model

We consider the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal function with the plateau section in the middle representing the maximum and symmetric tails falling off predominantly as ω-ω₀ᵐ, $m>0$, in the vicinity of the flat region. It is found that the phase transition is of first order as long as there is a finite flat region and that in the vicinity of the critical coupling the following scaling law holds r-rc∝(K-Kc)2∕(2m+3), where r is the order parameter and K is the coupling strength of the interacting oscillators.