Phase and frequency shifts of two nonlinearly coupled oscillators

We analyze two nonlinearly phase coupled oscillators with eigenfrequencies omega(1) and omega(2), where n omega(1) = m omega(2) + eta, with integer n and m. For eta = 0 there are up to four stable synchronized states which differ from each other only by the difference of the oscillators' phases. The number of different synchronized states depends on the coupling constants. If eta does not vanish phase shifts and frequency shifts may occur givig rise to stable synchronized states which also differ from each other due to the frequencies. By means of the center manifold theorem we calculate these shifts explicitely. Different coupling constants are investigated: symmetrical, homogenously asymmetrical and arbitrary coupling constants. Our results point out the decisive influence of the symmetry of the coupling constants upon the frequency and phase shifts. Moreover the local stability of the unperturbed synchronized state (i.e. for eta = 0) determines the magnitude of the frequency and phase shifts.