Fate of vortex-synchronized state in oscillator networks with node defects

We investigate synchronization behaviors of a Kuramoto oscillator network with a two-dimensional square-lattice configuration. We show that the oscillator network can reach a phase-locking vortex synchronized state in the long time limit starting from random initial oscillator phases sampled according to the von Mises distribution characterized by a zero mean and a finite concentration parameter. We further reveal that the stability of the vortex synchronized state is sensitive to the presence of local node defects, in contrast to the usual knowledge that oscillator networks should exhibit robustness against local perturbations. Moreover, we explore the behaviors of the vortex synchronized state in networks with an additional temporal white noise on the oscillator phases or a spatial noise due to randomly distributed oscillator frequencies. Interestingly, we find that the vortex synchronized state can become immune to local node defects when the variance of spatial noise is above a certain threshold, suggesting a beneficial role of usually unwanted spatial noise in protecting vortex-synchronized networks.