Kuramoto model of weakly conformed oscillators

Abstract

We consider a Kuramoto model with weakly conformed oscillators, for which the tendency of catching up with each other varies with the degree of synchronization, to study conformity effects. The bifurcation diagrams and dynamical behaviors of conformed oscillators are examined for unimodal and bimodal frequency distributions, respectively. For a unimodal distribution, a hysteresis loop emerges in the bifurcation diagram and the branch of unstable fixed points corresponds to the threshold, or quorum, over which systems move toward more synchronized states. For a bimodal distribution, saddle-node and homoclinic bifurcation lines shift with conformity which reflects the fact that conformity generally demotes the incoherent state. In addition, to shed light on how two groups of phase-locked oscillators reach synchronization subject to conformity, we analyze the period of limit cycles with a perturbation theory and numerics. Two groups of oscillators would reach a synchronized state when a threshold conformity parameter is met.