**Date Published:** March 12, 2019

**Publisher:** Public Library of Science

**Author(s):** Yuan Xie, Lan Zhang, Shuangjian Guo, Qionglin Dai, Junzhong Yang, Alexander N. Pisarchik.

http://doi.org/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.**

**Partial Text**

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 unimodal frequency distribution, 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 then to a traveling wave state [2]. When the peak distance in the bimodal frequency distribution is narrow, the discontinuous transitions between different dynamical states are possible. 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]. In the above works, incoherent state is always unstable when the coupling strength is above a threshold and further increasing coupling strength always enhances synchronization among oscillators. Recently, some authors studied the synchronous dynamics when the frequency distribution is a superposition of two unimodal frequency distributions with the same mean frequency in a globally coupled oscillator system [4]. They found a non-universal synchronization transition in which the incoherent state may be restored for the coupling strength above the threshold.

We consider N phase oscillators which sit evenly on a ring with the length 2π and nonlocally interact with each other with the coupling strength K. The model is described as

θ˙j=ωj+K2M+1∑k=-Mk=Msin(θj+k-θj+α).(1)

The dynamics in Eq (1) can be reproduced by Eq (8). Therefore, we study the stability of the incoherent state in the model (1) by linearizing Eq (8) around it and the synchronous dynamics in the model (1) by performing numerical simulations on Eq (8). Traditionally, the coherence in Eqs (1) and (8) is described by the global complex order parameter defined by ReiΦ=∫02π[pu1(y,t)+(1-p)u2(y,t)]dy. However, R cannot provide correct information on twisted states with nonzero q. Considering that, in Eq (8), u1,2(x, t) is driven by spatially-dependent complex order parameter Z(x, t), we measure the coherence in Eq (8) by another global quantity 〈|Z|〉, defined as 〈|Z|〉=∫02π|Z(x,t)|dx/2π. 〈|Z|〉 ≠ 0 indicates a synchronous state while 〈|Z|〉 = 0 the incoherent state.

In summary, 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. Using OA ansatz, we derived a reduced model in the limit of infinite number of oscillators. Based on the reduced model, we studied analytically the stability of the incoherent state and found that the most unstable spatial mode to the incoherent state depends on the coupling strength, the homogeneous perturbation with the winding number qm = 0 for the weak and the strong coupling strength and inhomogeneous perturbation with nonzero winding number qm ≠ 0 for intermediate coupling strength. The critical curves of the different spatial modes to the incoherent state may be S-shaped ones, which may give rise to multiple stability islands of the incoherent state. By numerically simulating the reduced equations, we found a large number of twisted states which result from the response of the incoherent state to the perturbations in the form of different spatial modes. Especially, there are two types of twisted states, one for the intermediate coupling strength and the other for the strong coupling strength. By partitioning oscillators into two groups each of which obeys a unimodal frequency distribution, we found that, for the twisted state for the intermediate coupling strength, the group of oscillators with fat frequency distribution remains in desynchronization and, for the states for strong coupling strength, both groups are in partial synchronization.

Source:

http://doi.org/10.1371/journal.pone.0213471