Ensemble order parameter equations in star network

TL;DR

This paper introduces ensemble order parameter equations to analyze finite-size star networks, revealing complex synchronization and desynchronization phase transitions through nonlinear analysis tools.

Contribution

It extends the OA ansatz to finite networks, providing a new method to study dynamics in star and star-connected networks.

Findings

Multiple phase transitions in star networks are observed.

Two-step synchronization transition in star-connected networks.

EOP dynamics reveal processes of synchronization and desynchronization.

Abstract

The OA ansatz has attracted much attention recently, infinite-dimensional Kuramoto model could collapses to a two-dimensional system of order differential equations with it. In this paper, we propose the ensemble order parameter (EOP) equations to describe the dynamics for networks with a finite size. To verify the effectiveness of this method, we apply it into the star network and star-connected network. In the star network, numerous phase transitions among different synchronous states are observed, three processes of synchronization, one process of de-synchronization and a group of hybrid phase transitions, the processes of those transitions are revealed by the EOP dynamics and other nolinear tools such as time reversibility analysis and linear stability analysis. Also in the star-connected network, the two-step synchronization transition is observed. The process of it is still be revealed by the similar methods in the single star network.