Stability and bifurcation of mixing in the Kuramoto model with inertia

TL;DR

This paper analyzes the stability and bifurcation behavior of the second-order Kuramoto model with inertia on complex networks, identifying critical coupling thresholds for synchronization onset using spectral theory and numerical simulations.

Contribution

It provides a mathematical framework for understanding the stability of mixing and the bifurcation leading to synchronization in the second-order Kuramoto model on random and structured graphs.

Findings

Identified a critical coupling value for bifurcation.

Demonstrated loss of stability of mixing at the bifurcation point.

Validated results with numerical bifurcation diagrams on Erdős–Rényi and small-world graphs.

Abstract

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.