Time-Delayed Feedback And Noise In Nonlinear Systems

Based on the Langevin equation of the nonlinear dynamical system, the effects of time delay and different kinds of noises on the statistical properties of the system are theoretically investigated through approximation methods and numerical simulations. Much attention has been paid on the steady state properties of the system, the noise-induced phase transition, the mean first passage time, the noise-induced transport of a Brownian motor, and the stochastic resonance with time-delayed feedback.Firstly, the effects of different kinds of noises on the dynamical properties of a bistable laser system are investigated.For a bistable laser system with coupling between non-Gaussian and Gaussian noise terms, the expression of the steady-state distribution function and the mean first passage time (MFPT) of the system is derived through the path integral approach and the functional approximation. The effects of the coupling between noise terms and the parameter q of the departure from the Gaussian noise on the noise-induced phase transitions and the MFPT of the bistable laser system are discussed. It is found that the couplingλbetween two noise terms can induce the reentrance-like phase transition while the parameter q can induce the first-order-like phase transition. Both the couplingλand the parameter q can change the curve of the mean first passage time from monotonically decreasing function to a peak in the curve.For a bistable laser system driven by two different kinds of colored noise, combining the unified colored noise approximation (UCNA) and the functional analysis, the analytical solutions of the stationary probability distribution (SPD) and the variance of the laser intensity are derived. The effects of the different noise correlation timesÏ„₁,Ï„₂ and the coupling strengthλbetween two noise terms on the fluctuations of the system are studied. It is found that the multiplicative noise correlation timeÏ„₁ can suppress the intensity fluctuations while the noise correlation time 12 of the coupling between two noise terms can enhance the intensity fluctuations in a laser system. The height of the peak in the normalized variance is symmetrically located at two sides ofλ= 0. The peak in the normalized variance is increased as the coupling strength |λ| is increased.Secondly, the phenomenon of stochastic resonance in nonlinear systems with time-delayed feedback is investigated.For a bistable system with non-Gaussian noise and time-delayed feedback, methods of the small time delay approximation, the path-integral approach and the unified colored noise approximation (UCNA) are applied to obtain the expression of the steady state probability distribution function. Through the two-state theory, the expression of signal-to-noise ratio (SNR) is obtained. The effects of the delay timeÏ„, the correlationtimeÏ„₀ of the non-Gaussian noise and the parameter q of the departure from the Gaussiannoise on the stationary probability distribution and the signal-to-noise ratio (SNR) are discussed. It is found that the distribution of peaks in SPD and the reentrance-like transition between one peak and two peaks and then to one peak again in the curve of SNRdepends on the delay timeÏ„, the noise correlation timeÏ„₀ and the parameter q.For a periodically driven FitzHugh-Nagumo neuronic system with time-delayed feedback and Gaussian white noise, the stochastic resonance which is characterized by theFourier coefficient Q is numerically calculated. It is found that the stochastic resonanceof the system is a non-monotonic function of the signal period and the noise strength. The period of the stochastic resonance in the system depends on the time delay. It is clear that the stochastic resonance in the system can be controlled by suitable choosing the delay time.Finally, the transport in a Brownian motor with noise and time-delayed feedback is investigated.For an inertial Brownian motor with time-delayed feedback driven by an unbiased time-periodic force, the effects of the noise intensity and the time delay on the mean velocity and the rectification efficiency are discussed. It is found that the mean velocity and the rectification efficiency are decreased when the noise intensity is increased. While both the shape and the number of peaks in the mean velocity and the rectification efficiency can be varied when the delay time is increased. Meanwhile, the symmetry in the velocity probability distribution is broken when the delay time is increased.For a spatially periodic Josephson junction with time-delayed feedback driven by coupling between two noise terms, a general equation of the current is derived according to the conservation law of the probability when the small time delay approximation is used. The effects of the additive noise intensity, the multiplicative noise intensity, the coupling strength between noise terms and the delay time on the current in Josephson junction are discussed. It is found that suitable time delay can enhance the current in the system. In the presence of the coupling between noise terms, the current direction is determined by the sign of the coupling. The negative coupling can induce positive current while the positive coupling can induce negative current. The absolute value of the current is a peaked function of the additive noise but a monotonically increasing function of the multiplicative noise.Numerical simulations are in good agreement with the approximate theoretical results.