An Integral Algorithm for Numerical Integration of One-Dimensional Additive Colored Noise Problems

We present an integral-closed algorithm for solving a Langevin equation driven by an additive colored noise. Both the mean first passage time in a bistable system and the diffusion current in a titled periodic potential are calculated and the comparison with existing algorithms is carried out. The dependence of the numerical results on the time steps is studied. Our algorithm is shown to have high accuracy and stability.     

Brownian dynamics simulation of polydisperse hard spheres

Standard algorithms used for the numerical integration of the Langevin equation require that interactions should slowly vary during the integration time-step. This in not the case for hard-body systems, where there is no clear-cut between the correlation time of the noise and the time-scale of the interactions. Starting with a short time approximation of the Smoluchowski equation, we introduce an algorithm for the simulation of the over-damped Brownian dynamics of polydisperse hard-spheres in absence of hydrodynamic interactions and briefly discuss the extension to the case of external drifts.     

Effect of time delay in FitzHugh-Nagumo neural model with correlations between multiplicative and additive noises

We study the effect of time delay in the FitzHugh-Nagumo neural model with correlations between multiplicative and additive noise terms. Based on the corresponding Fokker-Planck equation, the explicit expressions of the stationary probability distribution function (SPDF), the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) are obtained, respectively. Research results show that: (i) the system undergoes a succession of two phase transitions (i.e., the reentrance phenomenon) as the noise correlation parameter is increased and a (single) phase transition as the time delay is increased. (ii) The MFPT as a function of the multiplicative noise intensity exhibits a maximum. This maximum for MFPT identifies the noise enhanced stability (NES) effect, the noise correlation parameter intensifies the NES effect while the time delay, and the additive noise intensity weakens it. (iii) The existence of a maximum in the SNR as a function of the multiplicative noise intensity is the identifying characteristic of the stochastic resonance (SR) phenomenon, the noise correlation parameter enhances the SR while the time delay, and the additive noise intensity weaken it.     

Photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers

Quantum photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers are studied on the basis of the complex Ginzburg-Landau equation model. We find that, depending on their phase difference, the total photon-number noise of the bound soliton pair can be larger or smaller than that of a single soliton, and the two solitons in the soliton pair have a corresponding positive or negative photon-number correlation. It is predicted for the first time to our knowledge that out-of-phase soliton pairs can exhibit less noise as a result of negative correlation.     

Correlated noise-based switches and stochastic resonance in a bistable genetic regulation system

The correlated noise-based switches and stochastic resonance are investigated in a bistable single gene switching system driven by an additive noise (environmental fluctuations), a multiplicative noise (fluctuations of the degradation rate). The correlation between the two noise sources originates from on the lysis-lysogeny pathway system of the λ phage. The steady state probability distribution is obtained by solving the time-independent Fokker-Planck equation, and the effects of noises are analyzed. The effects of noises on the switching time between the two stable states (mean first passage time) is investigated by the numerical simulation. The stochastic resonance phenomenon is analyzed by the power amplification factor. The results show that the multiplicative noise can induce the switching from “on” → “off” of the protein production, while the additive noise and the correlation between the noise sources can induce the inverse switching “off” → “on”. A nonmonotonic behaviour of the average switching time versus the multiplicative noise intensity, for different cross-correlation and additive noise intensities, is observed in the genetic system. There exist optimal values of the additive noise, multiplicative noise and cross-correlation intensities for which the weak signal can be optimal amplified.     

Role of noise in a market model with stochastic volatility

We study a generalization of the Heston model, which consists of two coupled stochastic differential equations, one for the stock price and the other one for the volatility. We consider a cubic nonlinearity in the first equation and a correlation between the two Wiener processes, which model the two white noise sources. This model can be useful to describe the market dynamics characterized by different regimes corresponding to normal and extreme days. We analyze the effect of the noise on the statistical properties of the escape time with reference to the noise enhanced stability (NES) phenomenon, that is the noise induced enhancement of the lifetime of a metastable state. We observe NES effect in our model with stochastic volatility. We investigate the role of the correlation between the two noise sources on the NES effect.     

Systems under the influence of white and colored poisson noise

We deal in this paper with systems driven by white or colored Poisson noise. For a free Brownian particle under the influence of white Poisson noise an exact generalized master equation in position space is obtained. In the Gaussian and Smoluchowski limits, known results are recovered. For a general process defined by a stochastic differential equation, with colored Poisson noise, we find an approximate generalized master equation, including first order terms in the correlation time and the first correction to the gaussianity. Under a more restrictive approximation, the stationary distribution function is given. This is used to study the phase transition in the steady state for a Verhulst model. Corrections to the gaussianity are discussed in this case.     

具有时滞状态反馈的随机Van der Pol系统的动力学研究

研究了时滞及时滞反馈控制参数对Van der Pol系统极限环幅值的影响. 运用自适应的平均场近似方法给出了系统的线性化近似及系统参数Lyapunov稳定性的边界条件, 同时给出了Van der Pol系统的关联时间和功率谱密度的数值模拟结果. 通过与平均场近似下的解析结果比较后发现, 数值模拟结果和理论结果符合.进一步讨论了时滞反馈控制参数、噪声强度以及时滞对关联时间和功率谱密度的影响. 关键词： 平均场近似 关联时间 Lyapunov稳定性     

基因转录调控系统中的色噪声诱导转化研究

研究了基因转录调控系统中的色噪声诱导蛋白质浓度转换现象. 据Novikov定理和Fox近似方法得到相应的Fokker-Planck方程, 给出稳态概率分布函数、蛋白质浓度平均值及其平均首通时间的近似表达式, 分析了噪声的自关联时间对稳态概率、平均值和平均首通时间的影响. 研究表明, 分解率的噪声的自关联时间引起蛋白质的浓度经历了“关”→“开”的转换; 合成率的噪声的自关联时间引起蛋白质的浓度经历了“开”→“关”转换. 然而, 随着两种自关联时间的增大, 平均首通时间增大, 即蛋白质浓度之间的转换变得困难. 数值模拟结果与理论预测基本符合. 关键词： 色噪声 基因转录调节系统 转换     

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (τ) to the steady-state distribution as a function of (i) of the correlation strength (λ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (α). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.     

Joint Probability Distribution of the System Driven by Internal and External Noises Simultaneously

We obtain the equation for the steady state joint probability distribution of the system driven by external and internal noises simultaneously, where external noise is the dichotomous noise and internal noise is the Gaussian white noise. As to the application of our equation, we discuss two examples for which the correlation functions and associated relaxation times are calculated. We find the coupling effect of the external and the internal noises for the relaxation time.     

表面催化反应模型中关联噪声诱导非平衡相变

建立含有关联噪声的双分子-单分子(DM)表面催化反应延迟反馈模型,该模型能同时显示一级和二级非平衡动力学相变,即在一级和二级非平衡动力学相变之间的反应窗口展现.讨论双分子在DM延迟反馈模型中两种吸附机制,即局域和随机吸附模型.研究结果表明:1)外部噪声及两噪声关联性致使反应窗口的宽度收缩;2)内部噪声对非平衡动力学相变行为的影响依赖两噪声关联性,即当两噪声负关联,内部噪声致使反应窗口的宽度变宽;而当两噪声正关联时,内部噪声致使反应窗口的宽度收缩;3)关联噪声致使反应窗口变化对DM模型中一级和二级非平衡动力学相变研究具有重要的科学意义.     

色交叉关联噪声对基因转录调节系统的影响

研究了受色交叉关联噪声驱动的基因转录调节系统的瞬态性质(平均首通时间).据Novikov定理和Fox近似方法得到相应的Fokker-Planck方程,求出稳态概率分布函数的表达式.在此基础上运用最快下降法得到平均首通时间的近似表达式.经过数值计算,结果表明：在强关联,小关联时间条件下,蛋白质的浓度经历了开→关→开;在弱关联,大关联时间条件下,蛋白质的浓度经历了开→关.在基因转录过程中出现了重入现象. 关键词： 色交叉关联噪声 基因转录调节系统 平均首通时间     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

色交叉关联噪声作用下集合种群的稳定性和平均灭绝时间

在Levins模型的基础上研究了色交叉关联噪声对集合种群稳定性的影响, 应用Fokker-Plank方程得到了系统的稳态概率密度函数, 运用最快下降法得到了平均灭绝时间的解析式. 结果表明: 两噪声色关联时, 加性噪声强度和乘性噪声强度均弱化集合种群的稳定性; 噪声关联强度强化集合种群的稳定性. 两噪声之间负关联时, 平均灭绝时间是加性噪声强度和乘性噪声强度的减函数, 是噪声关联时间的增函数; 两噪声之间正关联时, 平均灭绝时间是加性噪声强度和噪声关联时间乘性噪声强度的减函数, 是乘性噪声强度的非单调函数. 关键词： 集合种群 色交叉关联噪声 稳定性 平均灭绝时间     

Dye laser model with pump and quantum fluctuations; colored noise

A single mode dye laser model with two stochastically fluctuating forces representing pump and quantum fluctuations is discussed. In extension to a previous paper we investigate the different influences ofcolored pump noise andwhite quantum fluctuations on the laser light statistics. The corresponding two-variable Fokker-Planck equation is solved by means ofmatrix continued fractions. A comparison to the model with two white noise forces is also included. We focus especially on the intensity correlation time enhancement observed for low noise strengths which seems to be the result of a competition between (multiplicative) pump noise and (additive) quantum noise. An increase in correlation time of the colored pump noise causes the intensity correlation time enhancement to disappear.     

Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications

We construct in this paper two efficient spectral-Galerkin algorithms for solving systems of n coupled second-order equations. The computational complexity of these algorithms is essentially n times the cost of solving one second-order equation. We present numerical results which illustrate the accuracy and flexibility of these algorithms, as well as several interesting and challenging applications, including in particular a number of high-order nonlinear parabolic type equations.     

Additive Temporal Coloured Noise Induced Eckhaus Instability in Complex Ginzburg-Landau Equation System

The effect of additive coloured noises, which are correlated in time, on one-dimensional travelling waves in the complex Ginzburg-Landau equation is studied by numerical simulations. We found that a small coloured noise with temporal correlation could considerably influence the stability of one-dimensional wave trains. There exists an optimal temporal correlation of noise where travelling waves are the most vulnerable. To elucidate the phenomena, we statistically calculated the convective velocities Vg of the wave packets, and found that the coloured noise with an appropriate temporal correlation can decrease Vg, making the system convectively more unstable.     

基于小关联时间两相湍流色噪声的近似模型及数值模拟

从一般高斯型色噪声模型出发,通过泛函导数,应用小关联时间,近似计算多维色噪声,得到有效Fokker-Planck方程.将其应用到两相湍流中得到颗粒相的概率密度函数输运方程,从而得到颗粒相的二阶矩模型.将颗粒应力方程简化成代数方程,建立代数应力模型.将对流扩散方程的有限分析法运用到求解两相流模型中,对壁面两相射流进行数值模拟,并将求解结果与实验结果进行对比分析.