# Order parameter analysis of synchronization transitions on star networks

## TL;DR

This paper introduces an order parameter approach to analyze synchronization transitions in star networks of coupled oscillators, revealing diverse collective states and bifurcations using nonlinear dynamics tools.

## Contribution

It develops a low-dimensional dynamical framework for star network synchronization using the Watanabe-Strogatz, Ott-Antonsen, and ensemble order parameter methods.

## Key findings

- Different solutions correspond to various collective states.
- Bifurcations indicate transitions among these states.
- Analysis tools include time reversibility and stability analysis.

## Abstract

Collective behaviors of populations of coupled oscillators have attracted much attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynam- ical mechanism of collective synchronizations by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to diverse col- lective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions are revealed in the star-network model by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.