Dynamics of phase oscillators in the Kuramoto model with generalized frequency-weighted coupling

TL;DR

This paper extends the Kuramoto model by incorporating frequency-dependent weighted coupling, providing analytical formulas for critical coupling, stability analysis, and relaxation dynamics, enhancing understanding of synchronization in heterogeneous networks.

Contribution

It introduces a generalized Kuramoto model with frequency-weighted coupling, deriving explicit critical coupling formulas and applying Ott-Antonsen reduction for dynamic analysis.

Findings

Analytical expression for critical coupling in the generalized model

Landau damping and resonance effects identified above the threshold

Ott-Antonsen reduction captures relaxation dynamics effectively

Abstract

We generalize the Kuramoto model for the synchronization transition of globally coupled phase oscillators to populations by incorporating an additional heterogeneity with the coupling strength, where each oscillator pair interacts with different coupling strength weighted by a genera; function of their natural frequency. The expression for the critical coupling can be straightforwardly extended to a generalized explicit formula analytically, and s self-consistency approach is developed to predict the stationary states in the thermodynamic limit. The landau damping effect is further revealed by means of the linear stability analysis and resonance poles theory above the critical threshold which turns to be far more generic. Furthermore, the dimensionality reduction technique of the Ott-Antonsen is implemented to capture the analytical description of relaxation dynamics of the steady states valid on a globally attracting manifold. 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.