Unified Colored Noise Approximation Research Articles

Stochastic resonance and stability for a stochastic metapopulation system subjected to non-Gaussian noise and multiplicative periodic signal

In this paper, the stability and stochastic resonance (SR) phenomenon induced by the multiplicative periodic signal for a metapopulation system driven by the additive Gaussian noise, multiplicative non-Gaussian noise and noise correlation time is investigated. By using the fast descent method, unified colored noise approximation and McNamara and Wiesenfeld’s SR theory, the analytical expressions of the stationary probability distribution function and signal-to-noise ratio (SNR) are derived in the adiabatic limit. Via numerical calculations, each effect of the addictive noise intensity, the multiplicative noise intensity and the correlation time upon the steady state probability distribution function and the SNR is discussed, respectively. It is shown that multiplicative, additive noises and the departure parameter from the Gaussian noise can all destroy the stability of the population system. However, the noise correlation time can consolidate the stability of the system. On the other hand, the correlation time always plays an important role in motivating the SR and enhancing the SNR. Under different parameter conditions of the system, the multiplicative, additive noises and the departure parameter can not only excite SR phenomenon, but also restrain the SR phenomenon, which demonstrates the complexity of different noises upon the nonlinear system.

Multidimensional stationary probability distribution for interacting active particles.

We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the “pressure” generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.

A two-phase turbulence probability density function model in unified colored-noise approximation

A unified colored-noise approximation (UCNA) method has been widely used to solve the simple one-dimensional problem, and this paper attempts to extend this method to multi-dimensional systems. Firstly, a Fokker-Planck equation is obtained by UCNA method based on the Langevin equation for particle motion, then a two-order moment trajectory model is established on this basis. The new model can be successfully used to predict the backward-facing step two-phase flow, and the simulation results agree well with the measurements. This study shows that the UCNA method is still effective in dealing with multi-dimensional two-phase turbulent systems.

Influences of correlated colored-noises on logistic model for tree growth

By virtue of Liouville Theorem and unified colored-noise approximation approach, an approximate Fokker-Planck equation for a tree growth Logistic model subjected to cross-correlated colored noises is derived, and the steady-state probability distribution (SPD) function is obtained. The steady-state properties of the Logistic model are analyzed. We find the following: (1) the position of peak of SPD moves toward left side as D increases while the position of the peak moves toward the contrary direction with Q increasing; (2) the peak of SPD becomes narrow and grows in height as |λ| increases, and for the case of λ >0, the position of peak moves toward right as D increases, but it is opposite for the case of λQ increases.

Enhancement of density divergence in an insect outbreak model driven by colored noise

The steady states and the transient properties of an insect outbreak model driven by Gaussian colored noise are studied in this paper. According to the Fokker—Planck equation in the unified colored-noise approximation, we analyse the stationary probability distribution and the mean first-passage time of this model. By numerical analysis, the effects of the self-correlation time of insect birth rate and predation rate respectively reveal a manifest population divergence on the insect density. The decrease of the mean first-passage time indicates an enhancement dynamic on the density divergency with colored noise of a large self-correlation time based on the insect outbreak model.

Stochastic resonance driven by time-delayed feedback in a bistable system with colored noise

The phenomenon of stochastic resonance (SR) driven by time-delayed feedback in a bistable system with colored noise is investigated. Combining the small time delay and unified colored noise approximation, the Fokker-Planck equation is obtained. The different effects of time delay and noise correlation time on stationary probability density and signal-to-noise ratio (SNR) are discussed respectively. It is found that time delay can markedly improve the output SNR. This method can be practically applied to many fields such as weak signal extraction, recovery and so on. Numerical simulations are presented and are in agreement with the approximate theoretical results.

Steady state characteries of FitzHugh-Nagumo neural system subjected to two different kinds of colored noises

Making use of the unified colored noise approximation, the steady sate characteristics of the one-dimension of FitzHugh-Nagumo neural system with two different colored noises are investigated. The expressions of the steady state probability distribution function and the mean value are obtained. After numerical calculation, the results show that the self-correlated time of the multiplicative noise 1, the self-correlated time of the additive noise 2,the intensity of the additive noise , and the intensity of the multiplicative noise D can induce the transition. The increases of and D are conductive to the switch from the exciting state to the resting state. However, with 1 and 2 increasing, the switch from the resting state to the exciting state becomes obvious. The noise intensity and it self-correlated time play opposite roles.

Entropic Stochastic Resonance Driven by Colored Noise

The phenomenon of entropic stochastic resonance (ESR) in a two-dimensional confined system driven by a transverse periodic force is investigated when the colored fluctuation is included in the system. Applying the method of unified colored noise approximation, the approximate Fokker–Planck equation can be derived in the absence of the periodic force. Through the escaping rate of the Brownian particle from one well to the other, the power spectral amplification can be obtained. It is found that increasing the values of the noise correlation time and the signal frequency can suppress the ESR of the system.

Stochastic resonance in an optical bistable system subjected to cross-correlated additive white noise and multiplicative colored noise

The stochastic resonance (SR) of an optical bistable system with cross-correlated additive white and multiplicative colored noises and periodic signal is studied using the unified colored noise approximation and the theory of signal-to-noise ratio (SNR). Results show that cross-correlation intensity \lambda enforces the SR of the system. The position of the peak on the SNR-\tau curves moves to the right direction along with the increase of \lambda (\tau is the self-correlation time of the multiplicative colored noise). We find the SR phenomenon in the SNR-D and SNR-Q curves (D and Q are the intensities of the additive and multiplicative noises, respectively), but not in the SNR-\lambda curves.

The steady state probability distribution and mean first passage time of FHN neural system driven by non-Gaussian noise

We investigated the FitzHugh-Nagumo neural system driven by non-Gaussian noise. The expressions of the stationary probability distribution and the mean first-passage time are obtained through the path-integral approach and the unified colored noise approximation. The results show that the intensity of additive noise can induce phase transition，while the intensity of multiplicative noise，the derivation parameter and the correlation time cannot. The non-Gaussian noise shortens transformation time between resting state and excited state and is beneficial to transmission of information in neural system.

Correlation function of an optical bistable system with cross-correlated additive white noise and multiplicative colored noise

Considering an optical bistable system with cross-correlated additive white noise and multiplicative colored noise, we study the effects of correlation between the noises on the correlation function C(s) using the unified colored noise approximation and the Stratonovich decoupling ansatz formalism. The effects of the self-correlation time τ of the multiplicative colored noise and the correlation intensity \lambda between the two noises are studied with numerical calculation. It is found that C(s) increases with the increase of the self-correlation time τ, but decreases with the increase of the correlation intensity \lambda. At large value of τ, there is almost no change for C(s) when τ changes.

Transient Properties of a Bistable System with Delay Time Driven by Non-Gaussian and Gaussian Noises: Mean First-Passage Time

The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker-Planck equation is obtained by applying the unified coloured noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects of non-Gaussian parameter (measures deviation from Gaussian character) r, the delay time τ, the noise correlation time τ0, the intensities D and α of noise on the MFPT are discussed. It is found that the escape time could be reduced by increasing the delay time τ, the noise correlation time τ0, or by reducing the intensities D and α. As far as we know, this is the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system.

Stochastic Resonance in a Bistable System with Time-Delayed Feedback Loops

The phenomenon of stochastic resonance of a bistable system subjected to linear time-delayed feedback loops driven by multiplicative Gaussian coloured noise and additive Gaussian white noise is investigated. Firstly, the analytic expression of the quasi-steady distribution function Ps(x,t) is derived by applying the unified coloured noise approximation and the Novikov Theorem; Secondly, the expression of the signal-to-noise ratio (SNR) is obtained in the adiabatic limit to quantify the stochastic resonance. Finally, the effects of the linear coefficient a, the nonlinear coefficient b, the linear time-delayed feedback coefficient c and the delay time τ on Ps(x,t) and SNR± are discussed. It is found that the effects of the linear coefficient and the nonlinear coefficient, the positive linear time-delayed feedback coefficient and the negative linear time-delayed feedback coefficient, the positive delayed time and the negative delayed time on Ps(x,t) and SNR± are different, respectively. This discussion would be helpful to the study of the system reliability and controlling stochastic resonance.

Mean First-Passage Time of an Asymmetric Kinetic Model Driven by Two Different Kinds of Colored Noises

The mean first-passage time (MFPT) of an asymmetric bistable system between multiplicative non-Gaussian noise and additive Gaussian white noise with nonzero cross-correlation time is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker–Planck equation is obtained by applying the unified colored noise approximation and the Novikov Theorem. The steady-state probability distribution (SPD) is also obtained. The basal functional analysis and simplification are employed to obtain the approximate expressions of MFPT T±. The effects of the asymmetry parameter β, the non-Gaussian parameter (measures deviation from Gaussian character) r, the noise correlation times τ and τ2, the coupling coefficient λ, the intensities D and α of noise on the MFPT are discussed. It is found that the asymmetry parameter β, the non-Gaussian parameter r and the coupling coefficient λ can induce phase transition. Moreover, the main findings are that the effect of self-existent parameters (D, α, and τ) of noise and cross-correlation parameters (λ,τ2) between noises on MFPT T± is different.

Stochastic resonance in an asymmetric bistable system driven by correlated noise

The phenomenon of stochastic resonance (SR) in a bistable system subject to correlated multiplicative colored and additive white noises and a periodic rectangular signal with a constant component is investigated in the unified colored noise approximation and by applying the two-state theory. The expression of the signal-to-noise ratio (SNR) is obtained for arbitrary signal amplitude. The SNR is a non-monotonic function of intensities of multiplicative colored and additive white noises，correlation time of multiplicative colored noise and the strength of the coupling between noises， so SR appears in the bistable system. Meanwhile， it is more effective to control SR through adjusting the additive white noise intensity than adjusting the multiplicative colored noise intensity. Moreover，the effects of potential asymmetry and the strength of the coupling between noises on SNR are opposite.

Stochastic resonance in an asymmetric bistable system driven by multiplicative colored noise and additive white noise

The phenomenon of stochastic resonance (SR) in a bistable system driven by multiplicative colored and additive white noises and a periodic rectangular signal with a constant component is studied by using the unified colored noise approximation and the theory of signal-to-noise (SNR) in the adiabatic limit. The analytic expression of the SNR is obtained for arbitrary signal amplitude without being restricted to small amplitudes. The SNR is a non-monotonic function of intensities of multiplicative colored and additive white noises and correlation time of multiplicative colored noise, so SR exhibits in the bistable system. The effects of potential asymmetry r and correlation time τ of multiplicative colored noise on SNR are opposite. Moreover, It is more sensitive to control SR through adjusting the additive white noise intensity D than adjusting the multiplicative colored noise intensity Q.

The mean first-passage time for an asymmetric bistable system driven by multiplicative colored and additive white noise with a correlated noise

We investigated the effects of asymmetry of the potential and noises on the mean first-passage time (MFPT) in two opposite directions in an asymmetric bistable system subjected to multiplicative colored and additive white noise with a correlated noise. The expression of the stationary probability distribution is obtained by means of an extended unified colored-noise approximation and the expressions of the MFPTs are derived through the steepest-descent approach. Numerical results show that the influence of the asymmetry coefficient r on T± is different and because of the correlated noise，even for r=0, T+(xs1→xs2,r) is not equal to T-(xs2→xs1,r). The curves of lnT± versus r in both cases have mono-valleys but the mono-valleys move to the opposite directions with increasing correlation intensity λ.

Stochastic resonance in a bistable system with time-delayed feedback and non-Gaussian noise

The phenomenon of stochastic resonance in a bistable system with time-delayed feedback driven by non-Gaussian noise is investigated. Combining the small time delay approximation, the path-integral approach and the unified colored noise approximation, a general approximate Fokker–Planck equation of a stochastic system is obtained. The effects of the parameter q indicating the departure from the Gaussian noise, the delay time τ, and the correlation time τ0 of the non-Gaussian noise on the quasi-steady-state probability distribution function (SPD) and the signal-to-noise ratio (SNR) are discussed. It is found that the number of peaks in SPD and the reentrant transition between one peak and two peaks and then to one peak again in the curve of SNR depends on the parameter q, the delay time τ, and the noise correlation time τ0.

Transitions in a Logistic Growth Model Induced byNoise Coupling and Noise Color

With unified colored noise approximation, the logistic growthmodel is used to analyze cancer cell population when colored noiseis included. It is found that both the coupling between noiseterms and the noise color can induce continuous first-order-likeand re-entrance-like phase transitions in the system. The couplingand the noise color can also increase tumor cell growth for smallnumber of cell mass and repress tumor cell growth for large numberof cell mass. It is shown that the approximate analyticexpressions are consistent with the numerical simulations.

Fluctuations of Single-Mode Laser Driven by TwoDifferent Kinds of Colored Noise

A single-mode laser with coupling between additive andmultiplicative noise terms is investigated when the multiplicativenoise and the coupling between two noise terms are coloredfluctuations with finite correlation times τ1 and τ2.Combining the unified colored noise approximation (UCNA) and thefunctional analysis, the stationary probability distribution (SPD)and the variance of the laser intensity is derived. It is foundthat the colored nature of multiplicative noise and the couplingstrength between two noise terms can affect both the structure andthe height of the SPD, while the colored nature of the couplingbetween two noise terms can only affect the height of the SPD. Themultiplicative noise can enhance the intensity fluctuations whilethe additive noise can reduce the fluctuations in a laser system.Numerical simulations are presented and consistent to theanalytical results.