Influence of green noise on the parametric synchronization phenomenon

摘要

If a cosinusoidal (harmonic) force acts on a locking dynamic system, the system may be synchronized not only to this force but to its harmonics also. This effect refers to parametric phenomena and has been studied in many systems. If a stationary random process is, for example, a signal phase, the angular vibration of the ring laser, or the phase of a periodic potential, the additive external noise in the systems is green one when its spectral density is zero at zero frequency [Physics Letters A, 240 (1998) 43]. In this work we suppose that the green noise is the time derivative of a Ornstein-Uhlenbeck process and the locking system is the ring laser. We use a Krylov-Bogoluibov averaging method to find an effective potential which describes the system response near the locking regions located at the frequencies of the high harmonics of the force. We show that the effective Shapiro steps are well apparent but narrower then in the case of zero noise. The step size is given by the function of the external noise intensity and the harmonic force amplitude. This result is compared with that of numerical simulation accomplished by the predictor-corrector algorithm. The coincidence is excellent even if the green noise is strong enough. We also made the numeric simulation for the case of white noise. This showed that the parametric synchronization regions become ill-defined even for a very small white noise intensity.