Synchronization dynamics of phase oscillator populations with generalized heterogeneous coupling

The Kuramoto model, serving as a paradigmatic tool, has been used to shed light on the collective behaviors in large ensembles of coupled dynamic agents. It is well known that the model displays a second-order(continuous) phase transition towards synchrony by increasing the homogeneous(uniform) global coupling strength. Recently, there is a great interest in investigating the effects of the heterogeneous patterns on the collective dynamics of coupled oscillator systems. Here, we consider a generalized Kuramoto model of globally coupled phase oscillators with quenched disorder in their natural frequencies and coupling strength. By correlating these two types of inhomogeneity, we systematically explore the impacts of heterogeneous structure, the correlation exponent, and the intrinsic frequency distribution on the synchronized dynamics. We develop an analytic framework for capturing the essential dynamic properties involved in synchronization transition. In particular, we demonstrate that the forward critical threshold describing the onset of synchronization is unaffected by the location of heterogeneity, which, however, does depend crucially on the correlation exponent and the form of frequency distribution. Furthermore, we reveal that the backward critical points featuring the desynchronization transition are significantly shaped by all the considered effects, thereby enhancing or weakening the ability of synchronization transition (bifurcation). Our investigation is a step forward in highlighting the importance of heterogeneous pattern presented in the complex systems, and could, thus, provide significant insights for designing the strategy of inducing and controlling synchronization.