The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time _c is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for the noise can be considered as white for all intents and purposes, (2) for the noise can be considered white for some purposes, (3) for the correlated nature of the noise must be considered in an analysis, and (4) for the noise can be considered as deterministic.     

准单色噪声驱动的耗散动力系统的信息熵演化

准单色噪声是一类所谓真正有色的噪声.本文对准单色噪声驱动的耗散动力系统的信息熵演化进行研究,文中首先运用线性变换的方法给出了所研究系统的Fokker-Planck方程,然后结合Shannon信息熵定义推导了该系统随时间演化信息熵的精确表达式, 最后分析了系统耗散参数和准单色噪声对系统信息熵的显著影响.     

准单色噪声驱动的耗散动力系统的信息熵演化

准单色噪声是一类所谓真正有色的噪声.本文对准单色噪声驱动的耗散动力系统的信息熵演化进行研究,文中首先运用线性变换的方法给出了所研究系统的Fokker-Planck方程,然后结合Shannon信息熵定义推导了该系统随时间演化信息熵的精确表达式,最后分析了系统耗散参数和准单色噪声对系统信息熵的显著影响.     

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.     

Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach

A recent calculation, in the weak-noise limit, of the rate of escape of a particle over a one-dimensional potential barrier is extended by including an inertial term in the Langevin equation. Specifically, we consider a system described by the Langevin equation , where is a Gaussian colored noise with mean zero and correlator (t)(t')=(D/)exp(–|t–t'|/). A pathintegral formulation is augmented by a steepest descent calculation valid in the weak-noise (D0) limit. This yields an escape rateexp(–S/D), where the actionS is the minimum, over paths characterizing escape over the barrier, of a generalized Onsager-Machlup functional, the extremal path being an instanton of the theory. The extremal actionS is calculated analytically for smallm and for general potentials, and numerical results forS are displayed for various ranges ofm and for the typical case of the quartic potentialV(x)=–x ₂/2+x ₄/4.     

色噪声驱动的非对称双稳系统的平均首次穿越时间

研究了由关联乘性色噪声及加性白噪声驱动的非对称双稳系统中势阱的非对称性及噪声对系统两个方向平均首次穿越时间的影响. 首先利用一致有色噪声近似推导了系统的稳态概率密度的表达式，根据最速下降法推导了平均首次穿越时间的表达式. 数值结果表明：势阱的非对称性对两个方向的平均首次穿越时间的影响是不同的；由于噪声的关联性，即使对于关联乘性色噪声及加性白噪声驱动的对称双稳系统，两个方向的平均首次穿越时间也不再相等；在lnT₊-r和lnT_{-关键词： 平均首次穿越时间 非对称双稳系统 乘性色噪声 加性白噪声}     

色关联乘性和加性色噪声驱动的多稳态系统的稳态特性

研究了色关联乘性和加性色噪声作用下的三稳态系统的稳态问题.首先利用一致有色噪声近似方法,推导出稳态概率密度函数的表达式,然后分析了乘性噪声和加性噪声的强度以及关联性对稳态概率密度函数的影响,研究结果表明:加性噪声强度、加性噪声和乘性噪声的关联强度和关联时间可以诱导系统的非平衡相变.     

色关联噪声驱动下非线性神经元模型的相干共振

研究了关联的加性离子通道噪声和乘性突触噪声共同作用下非线性积分发放神经元模型中的相干共振现象.运用绝热近似理论和统一色噪声近似方法,得到了神经元首次点火概率分布和神经元放电峰峰间隔的变差系数的近似表达式.研究表明,首次点火概率分布和变差系数是突触噪声强度、离子通道噪声强度、乘性色噪声自相关时间和噪声关联强度的函数,适当的噪声强度、噪声自相关时间和噪声关联强度可以减小神经元发放峰峰间隔的变差系数,使系统的相干性达到最大值,从而引起神经元出现相干共振现象.同时讨论了离子通道噪声强度、突触噪声强度、乘性色噪声自相关时间和噪声关联强度对系统相干共振的影响.     

Unstable state dynamics: Treatment of colored noise

The diffusion of a particle set near an unstable point in a bistable potential is considered. The scaling theory of fluctuations proposed originally for onedimensional systems driven by Gaussian white noise is extended to arbitrary dimensions. The merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension. It is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process. The two different methods to include asymptotic fluctuations-which are absent in the usual scaling solution-are shown to be equivalent. An alternate way of including asymptotic fluctuations is attempted by solving the associated Fokker-Planck equation using the Fer formula. The reason for the failure of this method is traced. After this, it is argued that the unified scaling theory should be applicable for treatment of colored noise as well, for the scaling assumption is independent of the statistical property of the driving noise. Explicit Monte Carlo simulation of a model onedimensional system driven by exponentially correlated Gaussian noise is performed and compared with the scaling solution to bolster this point. The agreement is very good.     

双色噪声驱动非对称双稳系统平均第一穿越时间研究

利用统一色噪声近似理论,研究乘性色噪声和加性色噪声驱动的非对称双稳系统中,势阱的非对称性和噪声对系统两个方向的平均第一穿越时间T₊（x_s1→x_s2）和T_-（x_s2→x_s1）的影响（x_s1和x_s2是双稳系统的两个稳定点）.数值结果表明：T₊（x_s1→x_s2）随乘性噪声的自关联时间τ₁以及加性噪声的自关联时间τ₂的增大而减小.T_-（x_s2→x_s1）随乘性噪声的自关联时间τ₁以及加性噪声的自关联时间τ₂的增大而增大.在曲线（T₊（x_s1→x_s2）,λ）和（T_-（x_s2→x_s1）,λ）上都存在单峰.T₊（x_s1→x_s2）随非对称系数r的增大而增大,T_-（x_s2→x_s1）随非对称系数r的增大而减小. 关键词： 统一色噪声近似 平均第一穿越时间 加性色噪声 乘性色噪声     

On stochastic complex beam-beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam-beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam-beam interaction model with random forcing [Physica A 336 (2004) 347-360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker-Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε.Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich-Khasminskii theorem for the same model under a broad-band excitation in our previous work.     

The study on a stochastic system with non-Gaussian noise and Gaussian colored noise

In this paper, a stochastic system with correlation between non-Gaussian noise and Gaussian colored noise is investigated. We carry out the functional methods to derive the approximate Fokker-Planck equation, and the expressions of stationary probability density function and mean first-passage time are presented. Also we explore the effects of correlation between non-Gaussian and Gaussian noise for the mean first-passage time.     

Escape driven by strongly correlated noise

We consider the colored-noise-driven archetypal bistability dynamics of the Ginzburg-Landau type. The focus is on the stationary behavior and the problem of escape from metastable states. The deterministic flow of the underlyingtwo-variable Fokker-Planck process is studied as a function of the noise correlation time . As a main result we find that the separatrix exhibits a cusp at asymptotically large noise color. The stationary probability is evaluated approximately (unified colored noise approximation) and compared with numerical exact results. The stationary probability forms the key input in the evaluation of the rate of escape. At very strong noise color, the escape path closely follows a nodal line, passing through the corresponding stable node. The asymptotic result for the escape rate at large is compared with exact calculations for the lowest, nonvanishing eigenvalue.     

Effects of colored noise and noise delay on a calcium oscillation system

As a calcium oscillations system is in steady state, the effects of colored noise and noise delay on the system is investigated using stochastic simulation methods. The results indicate that: (1) the colored noise can induce coherence bi-resonance phenomenon. (2) there exist three peaks in the R–_τ0$R\text{–}{\tau }_0$ (R$R$ is the reciprocal coefficient of variance, and _τ0${\tau }_0$ is the self-correlation time of the colored noise) curves. For the same noise intensity Q=1$Q=1$, the Gaussian colored noise can induce calcium spikes but the white noise cannot do this. (3) the delay time can improve noise induced spikes regularity as _τ0${\tau }_0$ is small, and R$R$ has a significant minimum with increasing τ$\tau$ as _τ0${\tau }_0$ is large. (4) large values of ζ$\zeta$ reduce noise induced spikes regularity.     

色噪声驱动的双奇异随机系统的熵流与熵产生

讨论一类 (非平衡约束下) 高斯色噪声驱动的双奇异随机系统对应的Fokker-Planck方程，结合Shannon信息熵定义给出此类系统的熵流、熵产生随时间演化的精确表达式.分析(非平衡约束下)所引入的奇异性强度参数、噪声相关时间与耗散参数的相互作用以及对熵流、熵产生的显著影响. 关键词： 信息熵 熵流与熵产生 双奇异随机系统 高斯色噪声     

色关联的乘性和加性色噪声激励下分段非线性模型的随机共振

研究了色关联的乘性和加性高斯色噪声与外周期信号共同激励下,分段非线性系统的随机共振.利用一致有色噪声近似和两态模型理论,推导出系统稳态概率密度和信噪比的解析表达式,分析了色噪声和周期信号对随机共振的影响,发现系统中存在传统的随机共振和真实的随机共振现象.同时,系统的信噪比曲线的峰值随着噪声互关联系数和互关联时间的增加而减小.     

Time evolution of information entropy for a stochastic system with double singularities driven by quasimonochromatic noise

<正>This paper deals with the time evolution of information entropy for a stochastic system with double singularities driven by quasimonochromatic noise.The dimension of the Fokker-Planck equation is reduced by the linear transformation. The exact expression of the time dependence of information entropy is obtained based on the definition of Shannon’s information entropy.The relationships between the properties of dissipative parameters,system singularity strength parameter,quasimonochromatic noise,and their effects on information entropy are discussed.     

色关联的色噪声驱动的分段非线性模型的平均首次穿越时间

研究了色关联的乘性高斯色噪声和加性高斯色噪声驱动的分段非线性系统中, 噪声强度和相关时间对平均首次穿越时间的影响. 利用一致有色噪声近似方法和最速下降方法, 推导出系统平均首次穿越时间的表达式. 研究结果表明: 系统的平均首次穿越时间随着乘性噪声的增加会出现单峰结构, 即“共振”现象, 峰值会随着加性噪声强度和噪声之间关联强度的增加而减小. 而平均首次穿越时间作为加性噪声的函数呈单调曲线, 说明乘性噪声和加性噪声对平均首次穿越时间的影响不同. 此外, 乘性和加性噪声关联时间以及互关联时间在正关联时和负关联时 对系统平均首次穿越时间的影响是不同的. 关键词： 色噪声 分段非线性系统 平均首次穿越时间     

色噪声激励下的FHN神经元系统

研究了色噪声激励下的FHN系统的相变问题和平均首次穿越时间,并推导了系统的定态概率密度函数和平均首次穿越时间表达式.结果表明,参数α,τ和β可以诱导相变,且存在锁定现象,各个参数对平均首次穿越时间都有很大影响,但影响效果有很大不同. 关键词： FHN系统 色噪声 平均首次穿越时间 定态概率密度     

色噪声驱动的双奇异随机系统随时间演化的熵变化率上界

通过变换的方法讨论了一类高斯色噪声驱动的双奇异随机系统所对应的Fokker-Planck方程, 并结合Shannon信息熵的定义及Schwartz不等式原理给出了经变换后该系统随时间演化的熵变化率上界的精确表达式. 分析了奇异性强度参数、噪声相关时间与耗散参数对熵变化率上界的显著影响.