Stability of synchronization systems with inertia against frequency perturbations

Synchronization phenomena are widely observed in nature, human society, and engineering systems composed of interacting elements. Several systems, in particular power grids, require synchronization to maintain normal operational functionality, which imposes higher demands on system stability. Much effort has been devoted to exploring the underlying mechanism of such crucial phenomena, by taking the classical Kuramoto model as an ideal object of theoretical research. The introduction of an inertia term, while enriching synchronization behaviors and enabling better description of real systems, significantly increases the analytical complexity of the model and thereby requires deeper investigation. Here, we study the stability of the Kuramoto model with inertia against various types of frequency perturbation. Using the fragility performance metric as the quantitative indicator of stability, we reveal how dynamical parameters and spectral characteristics of the network interact and jointly influence the system stability against perturbations. We also show the intrinsic difference of system stability arising from network structures, which could be counteracted by a rapid vibration of the decay perturbation, an interesting phenomenon we term the "rapid vibration effect."