Clustering and Bellerophon state in Kuramoto model with second-order coupling

In this paper, clustering in the Kuramoto model with second-order coupling is investigated under the bimodal Lorentzian frequency distribution. By linear stability analysis and the Ott-Antonsen ansatz treatment, the critical coupling strength for the synchronization transition is obtained. The theoretical results are further verified by numerical simulations. It has been revealed that various synchronization paths, including the first- and second-order transitions as well as the multiple bifurcations, exist in this system with different parameters of frequency distribution. In certain parameter regimes, the Bellerophon states are observed and their dynamical features are fully characterized.