Glass synchronization in the network of oscillators with random phase shifts

We investigate the synchronization properties in a network of oscillators with random phase shifts. In the absence of the shift, the system reduces to the well-known Kuramoto model, which displays synchronization of phases. We introduce an additional order parameter describing glass synchronization and obtain self-consistency equations through the use of the replica method. In the presence of appropriate phase shifts, the system is found to exhibit both phase synchronization and glass synchronization. It is also pointed out that the proper scaling of the coupling strength with the system size depends on the degree of randomness.