Kuramoto model with asymmetric distribution of natural frequencies

We analyze the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal asymmetric function $g\left(\omega \right)$. It is obtained that besides a second-, also a first-order phase transition can appear if the distribution of natural frequencies possesses a sufficiently large flat section. It is derived analytically that for the first-order transitions the characteristic exponents describing the order parameter and synchronizing frequency near the critical point are equal to those for the order parameter in the corresponding symmetric case. Stability analysis of the incoherent phase shows that the synchronizing frequency at the onset of synchronization equals the perturbation rotation velocity at the border of stability. The analytic and numerical results are in agreement with numerical simulations.

Figure 1

(Color online) Order parameter $r$ and frequency shift ${\omega }_0$ versus coupling constant $K$ for distributions considered in the text: (a), (b) triangular [$a=3$, $b=1$, $\widehat{g}\left(0\right)=\sqrt{3∕2}$], (c), (d) olympic [$a=0.25$, $b=0.2$, $\widehat{g}\left(0\right)=0.55$]. The meaning of the parameters is explained in the text. Different symbols denote different results: squares, numerical solution of the system of transcendental equations (12, 18); triangles, asymptotic relationships (29, 30); circles, numerical solutions of the equations of motion (5).