Asymptotic stability of a stochastic delay equation

The investigation of a nonlinear stochastic delay equation with structural tool and regenerative time delays is presented. The conditions of Hopf bifurcation are computed in order to describe the regions of stability and instability. Explicit expressions characterizing the influence of nonlinear and stochastic perturbations, valid in the first order centre manifold approximation, are derived. In addition to this, we describe the underlying mathematical ideas of the centre manifold reduction of delay differential equations to ordinary differential equations for fixed time delays.     

Transfer operators for contractive Markov systems and stochastic stability of the invariant measure

We consider contractive Markov systems, a generalization due to Werner of iterated function systems that contract on average. We study a transfer operator for such systems and determine an appropriate function space on which this operator acts quasi-compactly and has good spectral perturbation properties. We discuss how these results apply to the stochastic stability of the invariant measure.     

随机时滞反应扩散神经网络的指数稳定性

本文是讨论了一类随机变时滞反应扩散Hopfield神经网络的矩指数稳定性。利用平均Lya- punov泛函方法,随机偏微分方程的Ito公式和时滞微分不等式技巧,证明了随机变时滞反应扩散Hopfield神经网络的平衡解的矩指数稳定性,给出了这类随机变时滞神经网络平衡解矩指数稳定性的实用代数判据,得到了一些新结果。最后,给出了主要定理的一个应用实例。     

Modelling and estimating the reliability of stochastic dynamical systems with Markovian switching

In this paper, a general framework for the modelling of physical phenomena with stochastic dynamical systems switched by jump Markov processes is given. A methodology of the associated estimation procedures is provided. A particular attention is paid to the estimation of the underlying jump process, which is not observable.As an application, a stochastic model is proposed for the fatigue crack growth problem. The estimation of the model parameters is made on a real crack growth data set. We are thus able to simulate some crack growth paths which are used for reliability analysis through Monte Carlo techniques.     

Solutions of the FPK equation for time-delayed dynamical systems with the continuous time approximation method

This paper presents a method of finite-dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay and numerical solutions of probability density functions of the systems. Solutions of probability density functions of time-delayed systems are rare in the literature. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems and to compute the probability density functions efficiently. The solutions of the FPK equation for a linear time-delayed stochastic system are presented. The effects of different spectral differentiation schemes for the FDMP method on the probability density functions are compared.     

变时滞Cohen-Grossberg随机神经网络的均方指数稳定性

本文利用随机积分的It（^o）公式,时滞微分不等式及随机时滞神经网络的特性讨论变时滞CohenGrossberg随机神经网络的均方指数稳定性.     

一类具指数函数系数的非线性复差分方程

众所周知,在复微分方程和复差分方程领域中,Malmquist型方程是比Painlev′e方程和Riccati方程形式更一般的非线性方程.在本文中,我们运用Nevanlinna理论的差分模拟结果和微分域理论对一类具指数函数系数的Malmquist型复差分方程进行了研究.当上述Malmquist型复差分方程的有限级超越亚纯解具有较少的零点和极点时,我们得到其增长性和指数函数ez的增长性一致.该结果是对复微分Malmquist定理和复差分Malmquist定理的推广和补充.     

一类具多比例时滞递归神经网络的全局指数稳定性

比例时滞是一种不同于常时滞、有界变时滞和分布时滞的时变无界时滞.具比例时滞的系统在物理学、生物学和控制理论等领域发挥着重要的作用,但目前具比例时滞神经网络的动力学行为研究相对较少.本文对一类具多比例时滞递归神经网络的全局指数稳定性进行研究.首先,利用非线性变换将一类具多比例时滞的递归神经网络等价变换成一类变系数常时滞的递归神经网络,然后利用M-矩阵理论和同胚映射定理,以及时滞微分不等式技巧,得到了该系统平衡点的存在性、唯一性及其全局指数稳定性的时滞无关的充分条件,该条件依赖于神经网络的连接权值矩阵和神经元的激励函数.最后数值实验结果验证所得结论的正确性和与以往文献相比较低的保守性.     

时滞微分方程特征值的近似求解方法

通过将时滞微分方程变换为积分方程后并进行离散,得到相应的差分方程,从理论上建立了差分方程的系统矩阵的特征值与原线性时滞微分系统的特征值之间的关系,从而得到了一种求解时滞微分方程特征值的新方法。作为算例,计算了时滞的Logistic方程的前10阶特征值。误差分析表明,提出的近似方法不但简单,并且有很高的精度。     

具多变时滞中立型奇异微分系统的稳定性(英文)

本文讨论了中立型奇异泛函微分系统的稳定性问题.利用V泛函方法和差分算子的稳定性获得具变时滞中立型奇异微分系统的渐近稳定性判据.所得结果被描述为矩阵等式或者矩阵不等式,在计算上是可行和有效的,并给出例子说明了所得结果.     

Nonlinear magnetoquasistatic interface problem in a permanent-magnet machine with stochastic partial differential equation constraints

This study discusses an application of the stochastic collocation method for the solution of a nonlinear magnetoquasistatic interface problem that is constrained by a partial differential equation with random input data. Special attention is given to finding the robust design of a rotor and a stator of an electrical machine for the reduction of electromagnetic losses and a cogging torque under material uncertainties. For this reason, variations of material properties are modelled using the polynomial chaos expansion technique. To develop the domain derivative of a robust cost functional, the velocity method is applied with an adjoint technique. The utility and efficiency of the proposed method has been successfully demonstrated by applying it to the design of a permanent-magnet synchronous machine used in electric vehicles.     

一类非线性时滞双曲型偏微分方程的振动性

研究一类非线性时滞双曲型偏泛函微分方程解的振动性，利用微分不等式方法和广义Riccati变换，获得了该类方程在第一类边值条件下振动的新的充分条件，所得结果通过实例加以阐明．     

多滞后时变区间Lurie控制系统的指数稳定性

引进了多滞后时变区间Lurie控制系统是指数稳定的概念,用矩阵测度和时滞微分不等式研究了多滞后时变区间Lurie控制系统x(t)=N[P(t),Q(t)]x(t) ∑i=1mN[Pi(t),Qi(t)]x(t-Ti) bf(σ（t)) σ（t)=c^Tx(t) 的指数稳定性,得到了该系统指数稳定的一些判据,推广和改进了前人的结果,并给出了应用的例子。     

Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems

A new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non‐symmetric matrix is proposed. The algorithm extends the deterministic inverse power method using a spectral discretization approach. The convergence and accuracy of the algorithm is studied for both symmetric and non‐symmetric matrices. The method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors. Copyright © 2006 John Wiley & Sons, Ltd.     

A stochastic approach to the closure of accessible sets of control systems with application on homogeneous spaces

Given a control system on a compact manifold M, we study conditions for the foliation defined by the accessible sets to be dense in M. For this, we relate the control system to a stochastic differential equation and, by the support theorem, we give a characterization of the density in terms of the infinitesimal generator of the diffusion and its invariant measures.     

一类具阻尼项的三阶非线性中立型泛函微分方程的振动性和渐近性

研究一类具阻尼项的三阶非线性中立型连续分布时滞泛函微分方程解的振动性和渐近性,利用广义Riccati变换、积分平均技巧和一些分析技巧,对α≠β的情况,建立了一组保证该方程每一解振动或者收敛到零的充分条件,所得结果改进和推广了最新文献中的一些熟知的准则.     

Kovacic's algorithm and its application to some families of special functions

We apply the Kovacic algorithm to some families of special functions, mainly the hypergeometric one and that of Heun, in order to discuss the existence of closed-form solutions. We begin by giving a slightly modified version of the Kovacic algorithm and a sketch proof.     

Ruin Probability in a Generalised Risk Process under Rates of Interest with Homogenous Markov Chains

This article explores recursive and integral equations for ruin probabilities of generalised risk processes, under rates of interest with homogenous Markov chain claims and homogenous Markov chain premiums. We assume that claim and premium take a countable number of non-negative values. Generalised Lundberg inequalities for the ruin probabilities of these processes are derived via a recursive technique. Recursive equations for finite time ruin probabilities and an integral equation for the ultimate ruin probability are presented, from which corresponding probability inequalities and upper bounds are obtained. An illustrative numerical example is discussed.     

New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation

In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems subjected to general, external, stochastic excitation is considered. The starting point of our approach is a Hopf-type equation, governing the evolution of the joint, response–excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response–excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker–Planck–Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker–Planck–Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach.

A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper.