Mixed-order Oscillators and Hybrid Synchronization in the Kuramoto model

Inspired by modern power grid advancements, we examine the frequency synchronization of the Kuramoto model, incorporating a mix of first- and second-order oscillators, with proportions p and 1−p, respectively. The intrinsic frequencies follow a unimodal distribution. When p=0, the second-order Kuramoto model shows a discontinuous synchronization transition, where oscillators form synchronized clusters at distinct frequencies. We identify the origin of these clusters and determine the natural frequency ranges for each. For finite p, we uncover a complex phase diagram featuring regions dominated by either underdamped or overdamped dynamics, leading to discontinuous or continuous synchronization transitions, respectively. At the boundary between these regions, a hybrid transition occurs where the order parameter is discontinuous yet exhibits critical behavior. The critical exponent varies continuously with p, resembling a hybrid percolation transition caused by the interplay between cluster growth and suppression dynamics. These results offer valuable insights into complex dynamics in power grids and Josephson junction.