Twisted states in nonlocally coupled phase oscillators with frequency distribution consisting of two Lorentzian distributions with the same mean frequency and different widths.

Yuan Xie,Qionglin Dai,Shuangjian Guo,Junzhong Yang,Lan Zhang,Alexander N Pisarchik

doi:10.1371/journal.pone.0213471

Abstract

In globally coupled phase oscillators, the distribution of natural frequency has strong effects on both synchronization transition and synchronous dynamics. In this work, we study a ring of nonlocally coupled phase oscillators with the frequency distribution made up of two Lorentzians with the same center frequency but with different half widths. Using the Ott-Antonsen ansatz, we derive a reduced model in the continuum limit. Based on the reduced model, we analyze the stability of the incoherent state and find the existence of multiple stability islands for the incoherent state depending on the parameters. Furthermore, we numerically simulate the reduced model and find a large number of twisted states resulting from the instabilities of the incoherent state with respect to different spatial modes. For some winding numbers, the stability region of the corresponding twisted state consists of two disjoint parameter regions, one for the intermediate coupling strength and the other for the strong coupling strength.

Highlights

It is well known that natural frequency distribution g(ω) plays a critical role in displaying rich synchronous dynamics in globally coupled phase oscillators
For some winding numbers, the stability region of the corresponding twisted state consists of two disjoint parameter regions, one for the intermediate coupling strength and the other for the strong coupling strength, and all of these twisted states are related to the instabilities of the incoherent state with respect to different spatial modes
We studied a ring of nonlocally coupled phase oscillators in which the frequency distribution is made up of two Lorentzians with the same center frequency but with different half widths

Summary

Introduction

It is well known that natural frequency distribution g(ω) plays a critical role in displaying rich synchronous dynamics in globally coupled phase oscillators. The partial synchronous state steps in through a continuous transition when the coupling strength is above a critical coupling strength [1]. Further increasing the coupling strength, the partial synchronous state may transit to a global synchronization where all oscillators oscillate at a same frequency. For a bimodal frequency distribution, increasing coupling strength always first leads to a standing wave state and to a traveling wave state [2]. When the frequency distribution becomes more complicated, for example a trimodal one, the synchronous dynamics in globally coupled phase oscillators may display collective chaos through a cascade of period-doubling bifurcations [3]

Methods

Results

Conclusion