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 $\mid \omega -{\omega }_0{\mid }^m$, $m>0$, 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 $r-r_c\propto (K-K_c{)}^{2∕(2m+3)}$, where $r$ is the order parameter and $K$ is the coupling strength of the interacting oscillators.

Figure 1

(a) Sketch of distributions [Eq. (17)] for positive frequencies $\omega$ near the end point of the plateau for $m=1∕2$, 1, and 2. (b) The order parameter $r\left(K\right)$ for the distributions given in (a).