A universal instability of many-dimensional oscillator systems

The purpose of this review article is to demonstrate via a few simple models the mechanism for a very general, universal instability - the Arnold diffusion—which occurs in the oscillating systems having more than two degrees of freedom. A peculiar feature of this instability results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations. The instability takes place generally for very special initial conditions (inside the so-called stochastic layers) which are, however, everywhere dense in the phase space of the systsm.The basic and simplest one of the models considered is that of a pendulum under an external periodic perturbation. This model represents the behavior of nonlinear oscillations near a resonance, including the phenomenon of the stochastic instability within the stochastic layer of resonance. All models are treated both analytically and numerically. Some general regulations concerning the stochastic instability are presented, including a general, semi-quantitative method-the overlap criterion—to estimate the conditions for this stochastic instability as well as its main characteristics.     

永磁同步风力发电机随机分岔现象的全局分析

永磁同步风力发电机在运行过程中不可避免地会受到风能的随机干扰,本文建立了在输入机械转矩存在随机干扰情况下永磁同步风力发电机的数学模型,采用胞映射方法分析了随机干扰强度变化时系统全局结构的演化行为,并通过数值模拟对理论分析进行验证.研究结果表明,随着随机干扰强度的增大,系统中会出现随机内部激变和随机边界激变,即由于随机吸引子与其吸引域内的随机鞍发生碰撞而产生的随机分岔现象和由于随机吸引子与其吸引域边界发生碰撞而产生的随机分岔现象.研究结果揭示了随机干扰对永磁同步风力发电机运行性能影响的机理,为永磁同步风力发电系统的运行和设计提供了理论依据.     

Modulated noisy biological dynamics: Three examples

Three examples of noisy biological dynamics modulated by a periodic signal are discussed. A minimal neuron model driven by stochastic noise and small periodic force show a firing statistic comparable with stochastic resonance as demonstrated in bistable systems. Similar results are obtained from responses to periodic vibrotactile stimulation on higher-order neuronal units of the somatosensory pathway. Finally, results from a bistable visual perception task exhibiting stochastic resonance are reported.     

Higher-order implicit strong numerical schemes for stochastic differential equations

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero. The regions of absolute stability of these implicit and related explicit methods are also examined.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.     

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.     

Noise-induced synchronous stochastic oscillations in small scale cultured heart-cell networks

This paper reports that the synchronous integer multiple oscillations of heart-cell networks or clusters are observed in the biology experiment.The behaviour of the integer multiple rhythm is a transition between super-and subthreshold oscillations,the stochastic mechanism of the transition is identified.The similar synchronized oscillations are theoretically reproduced in the stochastic network composed of heterogeneous cells whose behaviours are chosen as excitable or oscillatory states near a Hopf bifurcation point.The parameter regions of coupling strength and noise density that the complex oscillatory rhythms can be simulated are identified.The results show that the rhythm results from a simple stochastic alternating process between super-and sub-threshold oscillations.Studies on single heart cells forming these clusters reveal excitable or oscillatory state nearby a Hopf bifurcation point underpinning the stochastic alternation.In discussion,the results are related to some abnormal heartbeat rhythms such as the sinus arrest.     

Monte Carlo techniques for real-time quantum dynamics

The stochastic-gauge representation is a method of mapping the equation of motion for the quantum mechanical density operator onto a set of equivalent stochastic differential equations. One of the stochastic variables is termed the “weight”, and its magnitude is related to the importance of the stochastic trajectory. We investigate the use of Monte Carlo algorithms to improve the sampling of the weighted trajectories and thus reduce sampling error in a simulation of quantum dynamics. The method can be applied to calculations in real time, as well as imaginary time for which Monte Carlo algorithms are more-commonly used. The Monte-Carlo algorithms are applicable when the weight is guaranteed to be real, and we demonstrate how to ensure this is the case. Examples are given for the anharmonic oscillator, where large improvements over stochastic sampling are observed.     

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.     

用随机方法解河渠非恒定流

随机水力学是水力学研究的一个新发展。本文应用确定性模型与随机模型相结合的方法,对明渠非恒定渐变流的计算进行了研究。采用Saint Venant方程组为系统模型,但对系统的输入如明渠的糙率、过水断面、水力半径等,则采用适当的随机模型,产生的输出(流量、水位随时间和位置的变化)是随机性的、本文以柘溪水电站下游河道作为实例,用计算机进行了计算,并与实测结果作了印证。     

Effect of signal modulating noise in bistable stochastic dynamical systems

The effect of signal modulating noise in bistable stochastic dynamical systems is studied. The concept of instantaneous steady state is proposed for bistable dynamical systems. By making a dynamical analysis of bistable stochastic systems, we find that global and local effect of signal modulating noise as well as stochastic resonance can occur in bistable dynamical systems on which both a weak sinusoidal signal and noise are forced. The effect is demonstrated by numerical simulation.     

双稳系统的随机能量共振和作功效率

分析了处于双稳系统中的布朗粒子与外界的周期性外力和热随机力的功、热交互作用,建立了基于Langevin方程的随机能量平衡方程.围绕着受周期力、随机力和阻尼力共同作用的Langevin方程,采用动力学和非平衡热力学相结合的方法,从以"力"为立足点转到以"能量"为研究核心,深入分析了布朗粒子沿单一轨线运动时系统与环境之间的能量交换和作功效率,揭示了双稳系统的随机能量共振现象. 关键词： 双稳系统 随机能量共振 作功效率     

动态随机最短路径算法研究

静态最短路径问题已经得到很好解决, 然而现实中的网络大多具有动态性和随机性. 网络弧和节点的状态及耗费不仅具有不确定性且相互关联, 弧和节点的耗费都服从一定的概率分布, 因此把最短路径问题看作是一个动态随机优化问题更具有一般性. 文中分析了网络弧和节点的动态随机特性及其相互关系, 定义了动态随机最短路径; 给出了动态随机最短路径优化数学模型, 提出了一种动态随机最短路径遗传算法; 针对网络的拓扑特性设计了高效合理的遗传算子. 实验结果表明, 文中提出的模型和算法能有效地解决动态随机最短路径问题, 可以运用到交通、通信等网络的网络流随机优化问题中.     

具有固有频率涨落的记忆阻尼线性系统的随机共振

研究了在内噪声、外噪声(固有频率涨落噪声)及周期激励信号共同作用下具有指数型记忆阻尼的广义Langevin方程的共振行为.首先将其转化为等价的三维马尔可夫线性系统,再利用Shapiro-Loginov公式和Laplace变换导出系统响应一阶矩和稳态响应振幅的解析表达式.研究发现,当系统参数满足Routh-Hurwitz稳定条件时,稳态响应振幅随周期激励信号频率、记忆阻尼及外噪声参数的变化存在"真正"随机共振、传统随机共振和广义随机共振,且随机共振随着系统记忆时间的增加而减弱.数值模拟计算结果表明系统响应功率谱与理论结果相符.     

Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

STOCHASTIC DYNAMIC SENSITIVITY OF UNCERTAIN STRUCTURES SUBJECTED TO RANDOM EARTHQUAKE LOADING

The present study involves computation of stochastic sensitivity of structures with uncertain structural parameters subjected to random earthquake loading. The formulations are provided in frequency domain. A strong earthquake-induced ground motion is considered as a random process defined by respective power spectral density function. The uncertain structural parameters are modelled as homogeneous Gaussian stochastic field and discretized by the local averaging method. The discretized stochastic field is simulated by the Cholesky decomposition of respective co-variance matrix. By expanding the dynamic stiffness matrix about its reference value, the advantage of Neumann Expansion technique is explored within the framework of Monte Carlo simulation, to compute responses as well as sensitivity of response quantities. This approach involves only a single decomposition of the dynamic stiffness matrix for the entire simulated structure and the facility that several stochastic fields can be tackled simultaneously are basic advantages of the Neumann Expansion. The proposed algorithm is explained by an example problem.     

Stochastic stability of a fractional viscoelastic column under bounded noise excitation

The stability of a viscoelastic column under the excitation of stochastic axial compressive load is investigated in this paper. The material of the column is modeled using a fractional Kelvin–Voigt constitutive relation, which leads to that the equation of motion is governed by a stochastic fractional equation with parametric excitation. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The method of stochastic averaging is used to approximate the responses of the original dynamical system by a new set of averaged variables which are diffusive Markov vector. An eigenvalue problem is formulated from the averaged equations, from which the moment Lyapunov exponent is determined for the column system with small damping and weak excitation. The effects of various parameters on the stochastic stability and significant parametric resonance are discussed and confirmed by simulation results.     

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.