Synchronization transition of limit-cycle system with homogeneous phase shifts

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated when phase shifts are considered. In the system of coupled oscillators, phase shifts are the same between different oscillators. Synchronization and synchronization transition are revealed with different phase shifts. Phase shifts play an important role for this kind of system. When the phase shift \alpha<0.5\pi, the synchronization state can be attained by increasing the coupling, and the system cannot reach the synchronization state while \alpha\geqslant 0.5\pi. A clear scaling between complete synchronization critical coupling strength K_pc and \alpha-0.5\pi is found.