Inertial synchronization of networked oscillators in arbitrary dimensions

The Kuramoto model provides a paradigmatic framework for studying synchronization of interacting oscillators, and has been generalized to arbitrary dimensions to describe swarms, flocks and multi-dimensional opinion dynamics. Yet, existing formulations neglect inertia, a key mechanism known to enhance information propagation and collective responsiveness. Here, we introduce and analyze an inertial Kuramoto model in arbitrary dimensions. We show that inertia fundamentally alters the nature of the synchronization transition, inducing a crossover from continuous to discontinuous behavior, with the onset of hysteresis depending explicitly on both inertia and dimensionality parity. Our analytical theory is supported by extensive numerical simulations. These results establish inertia as a crucial ingredient of high-dimensional collective dynamics and reveal a novel universal structure in synchronization phenomena.