Regular Article

Frequency mismatch induces Bellerophon state and mixed explosive synchronization in a two-dimensional lattice

¹ College of Physics and Telecommunications Engineering, Yulin Normal University, Yulin 537000, P.R. China
² Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin 537000, P.R. China
³ College of Physics and Electronic Information, Huaibei Normal University, Huaibei 235000, P.R. China

^a e-mail: zhuliuhua421@sina.com

Received: 9 April 2019
Received in final form: 26 June 2019
Published online: 12 August 2019

Abstract

We study synchronization of coupled phase oscillators with linear natural frequencies in a two-dimensional lattice. The effects of two typical frequency layouts on synchronization dynamics are analyzed and discussed. The results show that the two typical frequency layouts induce two interesting dynamical behaviors. One is the Bellerophon state, the other is a mixed explosive synchronization. The Bellerophon state is a transitional state between the incoherent state and the π state, in which neither the instantaneous phases nor the instantaneous frequencies of the oscillators are locked. But the oscillators split into a series of discrete clusters and each cluster shares a unique effective frequency. The mixed explosive synchronization means that the synchronization transition is not a sudden jump, but always accompanied with continuous phase transitions. Our current work may provide a new insight into understanding of the Bellerophon state and the mixed explosive synchronization in homogenous networks.