Complete periodic synchronization in coupled systems

Recently, complete chaotic synchronization in coupled systems has been well studied. In this paper, we study complete synchronization in coupled periodic oscillators with diffusive and gradient couplings. Eight typical types of critical curve for the transverse Lyapunov exponent of standard mode, which give rise to different synchronization-desynchronization patterns, are classified. All possible desynchronous behaviors including steady state, periodic state, quasiperiodic state, low-dimensional chaotic state, and two types of high-dimensional chaotic state are identified, and two classical synchronization-desynchronizaiton bifurcations-the shortest wavelength bifurcation and Hopf bifurcation from synchronous periodic state-are classified.