Kuramoto Model of Coupled Oscillators with Positive and Negative Coupling Parameters: An Example of Conformist and Contrarian Oscillators

We consider a generalization of the Kuramoto model in which the oscillators are coupled to the mean field with random signs. Oscillators with positive coupling are “conformists”; they are attracted to the mean field and tend to synchronize with it. Oscillators with negative coupling are “contrarians”; they are repelled by the mean field and prefer a phase diametrically opposed to it. The model is simple and exactly solvable, yet some of its behavior is surprising. Along with the stationary states one might have expected (a desynchronized state, and a partially-synchronized state, with conformists and contrarians locked in antiphase), it also displays a traveling wave, in which the mean field oscillates at a frequency different from the population’s mean natural frequency.

Figure 1

The $\pi$ state and traveling wave state. Parameters: $N=25600$, $\gamma =0.05$, $K_1=-0.5$, $K_2=1.0$. (a) Phase distribution for the $\pi$ state; the mean phase difference $\delta$ between the conformist (red) and contrarian (blue) oscillators is given by $\delta =\pi$. (b) The order parameter $Z(t)$ for the $\pi$-state corresponds to a fixed point. (c) Phase distribution for the traveling wave (TW) state; $\delta \ne \pi$. (d) The order parameter for the traveling wave state traces a circle, implying a nonzero mean phase velocity $⟨\dot{\varphi }(t)⟩\ne 0$.

Figure 2

Order parameter $R$ vs $p$ for $Q=3$, computed by numerical integration (symbols) and compared to theoretical prediction from Eq. (11) (lines). Solid lines, stable states; dotted line, unstable $\pi$ state. Stable $\pi$ states were found numerically by integrating from the in-phase initial condition ${\varphi }_i(0)=0$ for all $i$, for each $p$. Stable incoherent states were found by integrating from random initial conditions. Parameters: $N=25600$, $\gamma =0.05$, $K_1=-3.0$, $K_2=1.0$.

Figure 3

Order parameter $R$ vs $p$ for $Q=0.5$, computed by numerical integration. Inset shows theoretical prediction from Eq. (11). Red solid line, stable states; blue dotted line, unstable state. Parameters: $N=25600$, $\gamma =0.05$, $K_1=-0.5$, $K_2=1.0$.

Figure 4

Wave speed $\Omega$ as a function of $\gamma$ and $p$ for $Q=0.5$. The data are also plotted as a contour map, with heights indicated by the color bar. Dotted lines show the predicted transition curves at which the traveling wave is born, as obtained from Eq. (15).