Large deviations for the stochastic derivative Ginzburg-Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg-Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg-Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.     

Numerical solution of differential equations with colored noise

Using the general theory of numerical integration of stochastic differential equations, a constructive approach to numerical methods for a system with colored noise is proposed. Efficient methods up to the 5/2 strong order and up to the third weak order, including Runge-Kutta and implicit schemes, are presented. The algorithms are tested on the Kubo oscillator.     

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.     

Power-law distributions in random multiplicative processes with non-Gaussian colored multipliers

One class of universal mechanisms that generate power-law probability distributions is that of random multiplicative processes. In this paper, we consider a multiplicative Langevin equation driven by non-Gaussian colored multipliers. We analytically derive a formula that relates the power-law exponent to the statistics of the multipliers and numerically confirm its validity using multiplicative noise generated by chaotic dynamical systems and by a two-valued Markov process. We also investigate the relationship between our treatment and the large deviation analysis of time series, and demonstrate the appearance of log-periodic fluctuations superimposed on the power-law distribution due to the non-Gaussian nature of the multipliers.     

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.     

A stochastic return map for stochastic differential equations

A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O()] additive Gaussian colored noise. The construction is valid provided the correlation time isO() orO(1). The effective noise in the return map has nonzeroO( ²) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.     

倍增色噪声激励的非线性随机过程的精确数值模拟

本提出了一种求解含倍增有色噪声源项的非线性随机微分方程的精确和快速的数值算法,讨论了解对时间步长和轨道数选取的稳定性情况。     

A new method of analysis of the effect of weak colored noise in nonlinear dynamical systems

A systematic method for obtaining the asymptotic behavior of a dynamical system forced by colored noise in the limit of small intensity is developed. It is based on the search of WKB solutions to the Fokker-Planck equation for the joint probability density of the system and noise, in which the perturbation expansion is continued to the first correction beyond the Hamilton-Jacobi limit. The method can be applied to noise with correlation time of order unity. It is illustrated on the normal form of a pitchfork bifurcation, where it is pointed out that additive noise can induce a shift of the most probable value. This prediction is confirmed by numerical simulation of the stochastic differential equations.     

噪声对均质形核过程影响的晶体相场法研究

采用晶体相场模型,模拟了有色噪声诱发的均质形核过程. 结果表明,噪声强度在一定范围内对系统平衡热力学参量、形核势垒、临界晶核尺寸影响甚微; 而对形核孕育时间影响较大,形核孕育时间随噪声强度的增大呈指数减小趋势. 进一步分析表明,噪声对形核孕育时间的影响主要是由于噪声的引入改变了原子动力学系数.     

Structure changes induced by external multiplicative noise in the electrohydrodynamic instability of nematic liquid crystals

We study the influence of external multiplicative noise on the electrohydrodynamic instability (EHD) in nematic liquid crystals. It turns out that the correlation time _n and the intensityQ of the noise are the crucial parameters to control the system. Different types of noise lead to minor quantitative changes when compared to Gaussian white noise, leaving the qualitative aspects unchanged. With increasing noise intensity the threshold for the onset of the first instability changes drastically. We observe that the curvature arising when the threshold of the various instabilities is plotted as a function of the noise intensity changes as one is going, e.g., from the onset of Williams domains (WD) to the onset of the grid pattern (GP). This result reflects the transition in the flow structure from two-dimensional (WD) to three-dimensional (GP, DSM) flow patterns. As the intensity of the noise is increased further, the onset of the first instability becomes more complex. The measurement of the nonlinear onset time shows a strong dependence on the noise intensityQ, which is linear for WD and GP well above onset. The linear onset time shows an unexpected dependence on the noise intensity close to the onset of the first instability. For sufficiently long correlation times of the noise, a destabilization by noise is obtained.     

Impact of colored cross-correlated noises on stochastic resonance and mean extinction rate for a metapopulation system with a multiplicative periodic signal

In this paper, we aim to explore the mean extinction rate and the phenomena of the stochastic resonance (SR) for a metapopulation system induced by a multiplicative periodic signal, colored cross-correlated multiplicative and additive Gaussian noises. By use of the fast descent method and the adiabatic approximation theory for the signal-to-noise ratio, we obtain the expression of the signal-to-noise ratio (SNR). Numerical results indicate that the various SR phenomena occur in the metapopulation system due to the variation of the noise terms and the correlation time. Specifically, the noise correlation always plays a critical role in motivating the SR phenomenon, while the multiplicative noise exerts the inhibition effect on the SR. Interestingly, the weak additive noise can stimulate the resonant peak of the SNR, while the further increase of the noise intensity will lead to the reduction of the SR effect. On the other hand, the noise correlation time τ plays antipodal roles in motivating the SR phenomenon under different circumstances. With regard to the mean extinction rate of the population from the boom state to the extinction one, by performing the numerical calculations, it is found that the additive noise always accelerate the extinction of the population, while the correlation noise will slow down the decline for the population. The role that the noise correlation time plays in the population extinction depends on the values that λ takes.     

Hypersensitivity to small signals in a stochastic system with multiplicative colored noise

Recently we discovered the phenomenon of hypersensitivity to small time-dependent signals in a simple stochastic system, the Kramers oscillator with multiplicative white noise. In the present work we study, theoretically and experimentally with analog simulations, an influence of noise correlation time on hypersensitivity in a nonlinear oscillator with piecewise-linear current-voltage characteristic and multiplicative colored dichotomous noise. We found that the region of hypersensitive behavior is defined by universal scaling index, whereas the specifics of a particular system reveals itself only in the dependence of the above index on system parameters. The dependence of gain factor on noise correlation time is of bell-shaped (resonant) type. Received 27 April 2000 and Received in final form 2 November 2000     

色高斯噪声驱动双稳系统的多重随机共振研究

研究了由色关联乘性和加性色噪声作用下的双稳系统的随机共振问题,在绝热近似条件下得到了信噪比的表达式.通过分析所得的初始条件为 x(0)=x₊ 时的信噪比,发现了单随机共振和多重随机共振现象;分析了噪声强度、噪声关联时间和关联强度对系统信噪比的影响. 关键词： 多重随机共振 信噪比 双稳模型 色关联色噪声     

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.     

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     

Time-dependent Ginzburg-Landau equation in a car-following model considering the driver’s physical delay

In this paper, an extended car-following model considering the delay of the driver’s response in sensing headway is proposed to describe the traffic jam. It is shown that the stability region decreases when the driver’s physical delay in sensing headway increases. The phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. By applying the reductive perturbation method, we get the time-dependent Ginzburg-Landau (TDGL) equation from the car-following model to describe the transition and critical phenomenon in traffic flow. We show the connection between the TDGL equation and the mKdV equation describing the traffic jam.