Synchronization and Bellerophon states in conformist and contrarian oscillators.

Tian Qiu,Stefano Boccaletti,Shuguang Guan,Zonghua Liu,Jie Zhou,Ivan Bonamassa,Yong Zou

doi:10.1038/srep36713

Abstract

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency–dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean–field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.

Highlights

Synchronization is an emergent process of dynamical systems, wherein two interacting units adjust a given property of their motion to a collective behavior
Oscillators can be grouped into two populations: those with positive κi will behave like conformists, whereas those with negative κi will react as contrarians
In order to gather a better insight on Bellerophon states, we focus on the specific example of Fig. 4(b), and we further characterize in Fig. 5 the quantitative aspects of it

Summary

Introduction

Synchronization is an emergent process of dynamical systems, wherein two (or many) interacting units adjust a given property of their motion to a collective behavior. In many cases of practical interest, the connections among units of an ensemble could be inherently suppressive, or repulsive (which would correspond, instead, to negative values of the coupling strength). Both excitatory and inhibitory links are present in neural networks, in cellular interactions, or in social networks. Depending on the proportion of the conformists, the system exhibits both stationary states (the incoherent state, the fully coherent state, the partially synchronized π state with conformists and contrarians locked in anti-phase, and the traveling wave state29–33), and non-stationary (NS) states (the breathing chimera state, and the Bellerophon state (including the oscillating π state)). The stationary state refers to such an asymptotic state of the dynamical system in which the probability density function is time-independent in certain rotating frame, and non-stationary state otherwise

Methods

Results

Conclusion