Mean field theory of assortative networks of phase oscillators

TL;DR

This paper develops a mean field theoretical framework for large assortative networks of phase oscillators, demonstrating its effectiveness through the Kuramoto model and capturing complex bifurcation phenomena.

Contribution

It introduces a novel application of mean field and Ott-Antonsen ansatz to analyze assortative phase oscillator networks, revealing new bifurcation behaviors.

Findings

Degree assortativity can induce transitions to oscillatory states.

The reduced ODE system accurately predicts bifurcations and attractors.

The method is applicable to a broad class of phase oscillator networks.

Abstract

Employing the Kuramoto model as an illustrative example, we show how the use of the mean field approximation can be applied to large networks of phase oscillators with assortativity. We then use the ansatz of Ott and Antonsen [Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a system of ordinary differential equations. The resulting formulation is illustrated by application to a network Kuramoto problem with degree assortativity and correlation between the node degrees and the natural oscillation frequencies. Good agreement is found between the solutions of the reduced set of ordinary differential equations obtained from our theory and full simulations of the system. These results highlight the ability of our method to capture all the phase transitions (bifurcations) and system attractors. One interesting result is that degree assortativity can induce transitions from a steady macroscopic state to a temporally oscillating macroscopic state through both (presumed) Hopf and SNIPER (saddle-node, infinite period) bifurcations. Possible use of these techniques to a broad class of phase oscillator network problems is discussed.