Predicting the Oscillatory Regimes of Global Synchrony Induced by Secondary Clusters

Abstract

Synchronization systems with effective inertia, such as power grid networks and coupled electromechanical oscillators, are commonly modeled by the second-order Kuramoto model. In the forward process, numerical simulations exhibit a staircase-like growth of global synchrony, reflecting temporal oscillations induced by secondary synchronized clusters of whirling oscillators. While this behavior has been observed previously, its governing conditions have not been quantitatively determined in terms of analytical criteria. Here, we develop a self-consistent theoretical framework that explicitly characterizes the secondary synchronized clusters. This analysis identifies an onset crossover mass $\tilde{m}^* \simeq 3.865$ for the emergence of secondary clusters and yields quantitative criteria for predicting both the crossover mass and the termination coupling strength at which they vanish. As a result, we determine the oscillatory regimes of coupling strengths over which global synchrony shows temporal oscillations, providing practical guidance for controlling and avoiding undesirable oscillatory behavior in inertial synchronization systems, such as power grids.