Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling.

Abstract

The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.