Phase transition in coupled star networks

Abstract

Collective behaviors of coupled oscillators are an important issue in the field of nonlinear dynamics and complex networks, which have attracted much attention in recent years. For some systems satisfying certain symmetries, the dynamics of high-dimensional space can be reduced to that of low-dimensional subspace in terms of the Watanabe–Strogatz theory. In this paper, the dynamics of the transition to synchronization on the coupled star networks are studied, and we identify a two-step phase transition in the system both from the macroscopic and from the microscopic viewpoints. Theoretically, the low-dimensional order-parameter dynamical equation is developed to get analytical insights, and the roles of the coupled hubs in the synchronization process are uncovered. Physically, the two-phase transition steps correspond to different bifurcations in phase space. Theoretical analysis is examined by performing the numerical simulations, and the results and analysis proposed in this paper are helpful to understand the phase transition in general heterogenous networks.