We study a Brownian overdamped motion driven by the sequence of non-Gaussian correlated random impulses. A main characteristic of this external noise is that a following impulse has strictly opposed sign relative to the previous one. It is generated by a time derivative of stationary random jump function that may be equal or similar to a random telegraphic signal. Therefore, the noise is "green" by definition [Phys. Lett. A240 (1998) 43]. In order to find the mean drift velocity of a Brownian particle we employ two approaches: a Krylov-Bogolubov averaging method and a numerical simulation. The first method is used for the case of the jump function to be the random telegraphic signal. Then the probability dis-tribution density that describes statistics of time interval between the delta-function impulses of external noise is an ex-ponential function. The numerical calculation is performed by means of using the narrow rectangular impulse instead of the delta-function. We consider two models of such noise. In the first case the distribution density of time interval be-tween the rectangular impulses is again described by the exponential function. In other case the interval is uniformly distributed. We show that a locking effect (or a synchronization) exists even if a mean frequency of impulses is small. This effect exists with a high accuracy even if noise is strong. According to the theory an effective locking band is equal to the cosine of the amplitude of the original jump function. In particular, if the amplitude is π, the band is zero, how-ever, if it is equal to π, the band is unity as well as in the ideal case of zero noise. It is interesting that this property holds true even if the averaging method becomes inapplicable. We show also that the theory good coincide with the numerical simulation.
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. 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.
We consider a phase-locked loop for the case of an external signal with a stationary fluctuating phase. The problem reduces to the problem of a Brownian particle in a periodic potential driven by “green” noises. We numerically simulate the case in which the random phase is the Ornstain-Uhlenbeck process. The rapid irreversible transition from stationary random motion (a locked state) to a nonstationary one at a high near-constant rate (a running state) is shown to be possible for the case of the massive particle. We found that transition moments change suddenly for small variations of external parameters. We call this phenomenon the “catastrophe”. The numerical results are compared with those obtained by the Krylov-Bogoliubov averaging method. The first approximation of the method is found to be sufficiently accurate if the states coexist and the direct and backward transitions occur frequently enough.