We have investigated in the adiabatic limit the phenomenon of stochastic resonance in the gene transcriptional regulatory system subjected to an additive noise, a multiplicative noise, and a weakly periodic signal. Using the general two-state approach for the asymmetry system, the analytic expression of signal-to-noise ratio is obtained. The effects of the additive noise intensity a, the multiplicative noise intensity D and the amplitude of input periodic signal A on the signal-to-noise ratio are analysed by numerical calculation. It is found that the existence of a maximum in the RSNR a and RSNR D plots is the identifying characteristic of the stochastic resonance phenomenon in the weakened noise intensity region. The stochastic resonance phenomena are restrained with increasing a and D, and enhanced with increasing A.
This paper investigates the two-time intensity correlation function of a two-mode ring laser system subjected to both pump and quantum noises by stochastic simulation. It finds that the decay rate of the intensity correlation function of one mode gets faster with decreasing values of relevant parameters, i.e., the coupling constant ξ, the cross-correlation coefficient A, the difference of the pump parameters Aa and the pump parameter al; however, its variations get complex in the other mode when relevant parameters are changed. The investigating results also show that the effects of the mode competition on intensity correlation function are obvious.
This paper investigates the stochastic resonance （SR） induced by a multiplicative periodic signal in the gene transcriptional regulatory system with correlated noises. The expression of the signal-to-noise ratio （SNR） is derived. The results indicate that the existence of a maximum in SNR vs. the additive noise intensity α the multiplicative noise intensity D and the cross-correlated noise intensity λ is the identifying characteristic of the SR phenomenon and there is a critical phenomenon in the SNR as a function of λ, i.e., for the case of smaller values of noise intensity （α or D）, the SNR decreases as λ increases; however, for the case of larger values of noise intensity （α or D）, the SNR increases as λ increases.
Intracellular calcium ion concentration oscillation in a cell subjected to external noise and irradiated by an electromagnetic field is considered. The effects of the intensity E0, the polar angle θ and the frequency w of the external electric field on steady-state probability distribution and the mean Ca2＋ concentration, respectively, are investigated by a numerical calculation method. The results indicate that （i） variation of w cannot affect the intracellular calcium oscillation; （ii） the steady-state probability distribution presents a meaningful modification due to the variations of E0 and 0, while variation of 0 does not affect the steady-state probability distribution under the condition of a small E0, and E0 cannot affect the steady-state probability distribution either when θ=π/2; （iii） the mean Ca2＋ concentration increases as E0 increases when θ〈π/2 and, as 0 increases, it first increases and then decreases. However, it does not vary with E0 increasing when θ=π/2, but it increases with 0 increasing when E0 is small.
The effects of the time delay on the upper bound of the time derivative of information entropy are investigated in a time-delayed dynamical system driven by correlated noise. Using the Markov approximation of the stochastic delay differential equations and the Schwartz inequality principle, we obtain an analytical expression for the upper bound UB（t） of the time derivative of the information entropy. The results show that there is a critical value of T （delay time）, and UB（t） presents opposite behaviours on difference sides of the critical value. For the case of the weak additive noise, T can induce a reentrance transition. Delay time T also causes a reversal behaviour in UB（t）-λplot, where λ denotes the degree of the correlation between the two noises.
The nonequilibrium phase transition and the symmetry revival induced by time delay in a bistable system are investigated. The stationary probability distribution function （SPDF） of the bistable system with time delay and correlated noises are calculated by an analytical method and stochastic simulation respectively. The analytical and simulative results indicate that： （1） There is a certain value of λ（λ denotes the strength of correlations between the multiplicative and additive noises） to make the SPDF symmetric under some time delay; however, above or below the given value, the symmetry will be broken; （2） With the monotonic change of λ, the unimodal peak structure of SPDF becomes bimodal at the beginning, then it becomes unimodal again; this means that there is a reentrance phenomenon in the process; （3） There is a critical value of delay time, which makes the lower peak of SPDF equal to the higher one under the critical condition. This means that the symmetry revival phenomenon emerges.
Stochastic resonance （SR） of a periodically driven time-delayed linear system with multiplicative white noise and periodically modulated additive white noise is investigated. In the condition of small delay time, an approximate analytical expression of output signal-to-noise ratio （SNR） is obtained. The analytical results indicate that （1） there exists a resonance peak in the curve for SNR versus time delay; （2） the time delay will suspend the SR dramatically for SNR versus other parameters of the system, such as noise intensity, correlation intensity, and signal frequency, once a certain value is reached, the SR phenomenon disappears.
The effects of time delay on the fluctuation properties of a bistable system are investigated by simulating its normalised correlation function C（s）. Three cases including linear delay, cubic delay and global delay in the system are considered respectively. The simulation results indicate that the linear delay enhances the fluctuation of the system （reduces the stability of the system） while the cubic delay and global delay weaken it （enforce the stability of the system）, and the effect of cubic delay is more pronounced than the linear delay and global delay.
