Two-step and explosive synchronization in frequency-weighted Kuramoto model

We explore the dynamics of interacting phase oscillators in the generalized Kuramoto model with frequency-weighted couplings, focusing on the interplay of frequency distribution and network topology on the nature of transition to synchrony. We explore the impact of heterogeneity in the network topology and the frequency distribution. Our analysis includes unimodal (Gaussian, truncated Gaussian, and uniform) and bimodal frequency distributions. For a unimodal Gaussian distribution, we observe that in comparison to fully-connected network, the competition between topological and dynamical hubs hinders the transition to synchrony in the scale-free network, though explosive synchronization eventually happens. However, in the absence of very large frequencies, the transition is gradual. While uniform frequency distributions lead to explosive synchronization. In bimodal distributions, narrow distribution produce a two-step transition. In this case, central frequencies dominate the dynamics, overshadowing the topological features of the network. For wider bimodal distributions, scale-free network exhibits a gradual increase in the order parameter, whereas in fully-connected networks a first-order transition happens. These results specifically elucidate the mechanisms driving two-step and explosive synchronization in frequency-weighted Kuramoto models, offering new insights into managing synchronization phenomena in complex networks like power grids, neural systems, and social systems.