Low dimensional behavior of explosive synchronization on star graphs

Explosive synchronization (ES) on heterogenous networks is a hot topic in the study of complex networks in recent years. In this paper, we introduce an analytical framework to study the ES of Kuramoto oscillators on a star configuration, which is a simple and special heterogenous network. By using the OttAntonsen ansatz, we obtain a reduced set of low-dimensional equations for the macroscopic evolution of the system considered. Analysis of this reduced system theoretically confirms that the phase transition of the order parameter is indeed discontinuous. We also extend the problem studied to a general case with frequencies perturbation. The results show the phase transition is still discontinuous when the perturbations are small, and it degenerates to a continuous transition while the perturbation width exceeds a critical value. Numerical simulations show good agreement with the theoretical predictions. Our work provides an insightful perspective to understand the ES phenomena in large ensembles of coupled units.