Approximating the Splitting Point of the Swarmalator Model

Abstract

Swarmalator model attracts the attention of numerous scholars because of its rich dynamical behaviors. We investigate the reason for the phase transition that occurs in a 2D swarmalator model, and conclude that the limit cycle, a reflection of the coexistence of the forces of maintaining and disrupting orders, results in the various clustering phenomena. Through novel mean-field approximation and using the self-consistency argument method, we prove the clustering conditions and the influence of the number of clusters on the existence of clusters, and provide estimates of the cluster size of the splintered phase wave state and the phase transition threshold between the splintered phase wave state and active phase wave state. Due to the widespread presence of nonlinearity, our study is essential to the analysis of clustering phenomena in real physical models.