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.
Highlights
The first-order synchronization transition has been studied in systems of coupled phase oscillators
We revealed that the structural relationship between the incoherent state and the synchronous state leads to different routes to the transition of synchronization
We start by considering the frequency-weighted Kuramoto model[28,30], in which the dynamics of phase oscillators are governed by the following equations θi = ωi +
Summary
We start by considering the frequency-weighted Kuramoto model[28,30], in which the dynamics of phase oscillators are governed by the following equations θi = ωi +. The system is in the synchronous (coherent) state where the phase-locked oscillators coexist with the phase-drifting ones. One central issue in the study of synchronization is to identify all the possible asymptotic coherent states of the system as the coupling strength K increases To this end, the self-consistence method turns out to be effective. We assume that the mean-field phase Θ rotates uniformly with frequency Ω , i.e., Θ (t) = Ω t + Θ (0). Corresponding to the phase-locked oscillators entrained by the mean-field. In contrast to the phase-locked oscillators, the drifting oscillators could not be entrained by the mean-field.
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