Synchronization of phase oscillators with frequency-weighted coupling

TL;DR

This paper develops a mean-field and stability analysis framework for the frequency-weighted Kuramoto model, revealing the nature of synchronization transitions, stability conditions, and bifurcation mechanisms in heterogeneous oscillator networks.

Contribution

It introduces a comprehensive analytical approach combining mean-field, linear stability, and amplitude expansion methods for the frequency-weighted Kuramoto model, advancing understanding of synchronization phenomena.

Findings

Incoherent state is neutrally stable below the critical coupling.

Order parameter amplitude decays exponentially, similar to Landau damping.

Explicit critical coupling strength is derived and bifurcation mechanisms are characterized.

Abstract

Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.