Abnormal hybrid phase transition in the passively competing Kuramoto model

We consider a competing Kuramoto model (KM) with mixed signs of coupling constants that appear inside the sum, called passive interactions. It is known that the synchronization transition of this KM reduces to that of the ordinary KM with the mean of those coupling constants as a control parameter. In this type of KM, two order parameters can be introduced: the phase coherence R and the weighted phase coherence S. Here, we show that when the intrinsic frequency distribution g(omega) is unimodal, the order parameters R and S behave similarly with the same critical exponents beta and beta' in leading and sub-leading orders, respectively. However, when g(omega) is uniform, they behave differently, possibly because of the hybrid synchronization transition. The order parameter jumps at a transition point and then it exhibits a critical behavior. (C) 2019 Elsevier B.V. All rights reserved.