Criteria for Strong and Weak Random Attractors

The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors.      

H 2-regularity random attractors of stochastic non-Newtonian fluids with multiplicative noise

In this paper, the authors study the long time behavior of solutions to stochastic non-Newtonian fluids in a two-dimensional bounded domain, and prove the existence of H2-regularity random attractor.     

Hausdorff Dimension of Invariant Sets for Random Dynamical Systems

Suppose X() is a compact random set, invariant with respect to a continuously differentiable random dynamical system (RDS) on a separable Hilbert space. It is shown that the Hausdorff dimension dim _H (X()) is an invariant random variable, and it is bounded by d, provided the RDS contracts d-dimensional volumes exponentially fast. Both exponential decrease of d-volumes as well as the approximation of the RDS by its linearization are assumed to hold uniformly in . The results are applied to reaction diffusion equations with additive noise and to two-dimensional Navier–Stokes equations with bounded real noise.     

Asymptotic behaviors of solutions for dissipative quantum Zakharov equations

The dissipative quantum Zakharov equations are mainly studied. The existence and uniqueness of the solutions for the dissipative quantum Zakharov equations are proved by the standard Galerkin approximation method on the basis of a priori estimate. Meanwhile, the asymptotic behavior of solutions and the global attractor which is constructed in the energy space equipped with the weak topology are also investigated.     

Uniform attractors for non-autonomous Klein-Gordon-Schrǒdinger lattice systems

The existence of a compact uniform attractor for a family of processes corresponding to the dissipative non-autonomous Klein-Gordon-Schrodinger lattice dynamical system is proved. An upper bound of the Kolmogorov entropy of the compact uniform attractor is obtained, and an upper semicontinuity of the compact uniform attractor is established.     

Uniform attractors for non-autonomous Klein-Gordon-Schr/Sdinger lattice systems

The existence of a compact uniform attractor for a family of processes corre- sponding to the dissipative non-autonomous Klein-Gordon-SchrSdinger lattice dynamical system is proved. An upper bound of the Kolmogorov entropy of the compact uniform attractor is obtained, and an upper semicontinuity of the compact uniform attractor is established.     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

On the Determination of Nonlinear Response Distributions for Oscillators with Combined Harmonic and Random Excitation

The analysis of two different nonlinear systems, both subject to an excitation that comprises a harmonic and a random component, is presented in this paper. Both systems are known to exhibit different coexisting attractors for a purely harmonic forcing. The random part causes a disturbance of the response of these systems: Even though a dominating effect of the attractors for the deterministic case is still visible, the random disturbance also leads to occasional jumps between the areas surrounding the different attractors. To access the likelihood of the system being found in a specific state, probability density functions are approximated numerically by means of a localized statistical linearization.     

具有随机拉索刚度的预应力钢桁架随机内力摄动分析

本文在建立拉索式预应力钢桁架基本计算理论的基础上，考虑到拉索刚度的随机性，推导了拉索式平面预应力钢桁架随机内力摄动分析的一般公式及敏度分析式，并给出了算例。为进一步进行预应力钢桁架可靠性分析奠定了基础。     

Global attractor of 2D autonomous g-Navier-Stokes equations

In this paper, the existence of global attractors for the 2D autonomous g-Navier-Stokes equations on multi-connected bounded domains is investigated under the general assumptions of boundaries. The estimation of the Hausdorff dimensions for global attractors is given.     

The fractal structure of attractor for the generalized Kuramoto-Sivashinsky equations

I.IntroductionSincethereexistspectralbarriersandspectralgapconditions,theexistenceofaninertialmanifoldformanynonlineardissipativeevolutionequationsisstillamystery.Recently,Edenetal[5]havediscoveredthatfornonlinearsemigroup,definedbynonlineardissipativeevolutionequationsincludingZDNavier-Stokesequations,thereexistsatinliefractaldimensionalinertialsetwhichmayberepresentedbyaunionoffractillsetsandattractor,ifitisLipschitzcontinuousandissqueezingonacompacti,ositiveinvariantset.Ontileotherhand,S…     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Bounds for the Effective Elastic Properties of Completely Random Planar Polycrystals

Bounds have been developed for the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are supposed to be uncorrelated. Explicit results for the aggregates of orthotropic crystals are demonstrated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Solutions for a system of nonlinear random integral and differential equations under weak topology

I.IntroductionTheexistenceandcomparisonresultsofsolutionsfornonlinearVolterraintegralequationsinstrongtopologyofBanachspaceshavebeenobtainedbyVaughnI"'"],LakshmikanthamIl3]andLakshmikantham-Leela114l.TheexistenceresultsofweaksolutionsfortheCauchyproblemof…     

λ-PDF AND GEGENBAUER POLYNOMIAL APPROXIMATION FOR DYNAMIC RESPONSE PROBLEMS OF RANDOM STRUCTURES

A bounded, mono-peak, and symmetrically distributed probability density function, called λ-PDF, together with the Gegenbauer polynomial approximation, is used in dynamic response problems of random structures. The λ-PDF can reasonably model a variety of random parameters in engineering random structures. The Gegenbauer polynomial approximation can be viewed as a new extension of the weighted residual method into the random space. Both of them can be easily used by scientists and engineers, and applied to a variety of response problems of random structures. The numerical example shows the effectiveness of the proposed method to study dynamic phenomena in random structures.     

平稳随机振动荷载识别的逆虚拟激励法（一）

本文将虚拟激励法作了逆向推广，从而用确定性方式求解了平稳随机振动的荷载谱识别问题。即已知部分响应的自谱与互谱，反求多点激励间的自谱与互谱。并通过计算机模拟，讨论了识别的精度及影响因素等问题。     

动力学系统精细算法的逼近机理与误差上界

发现以下三者的协同作用是实现精细算法高精度,高效率的内在机理和根本原因：１）指数矩阵ｅ＾Ｈｚ的Ｍａｃｌａｕｒｉｎ级数展开式绝对收敛;２）初始Ｍａｃｌａｕｒｉｎ级数展开式中有效展开项总数能够通过递推算法以指数方式扩展;３）新增有效展开项的系数能够通过递推算法以指数或拟指数方式逼近其真值。此外,本文还给出了精细算法的截项误差递推公式和相关的误差上界,发现随着保留项数Ｍ或递推阶数Ｎ的增大,精细算法的逼近     

非比例阻尼线性体系平稳随机地震响应计算的虚拟激励法

应用复振型分解方法,将非比例阻尼线性体系在地震作用下的动力方程求解问题转化为若干个广义复振子的求解与叠加问题。通过假定地震地面运动为一零均值的平稳随机激励,应用虚拟激励法原理,推导得到了广义复振子动力坐标的解析计算公式,进而得到了以复振型为基础的非比例阻尼线性体系随机地震响应计算的一般实数解析解答。算例证实了这种方法的可靠性及高效率。     

二阶随机参激系统的不变测度与Lyapunov指数

用统一的模型，研究了一类典型的二阶系统在宽带和窄带随机参数激励情形下，系统的不变测度与最大Ｌｙａｐｕｎｏｖ指数，由最大Ｌｙａｐｕｎｏｖ指数给出了系统几乎必然稳定的充分必要条件。     

桥梁结构移动平稳随机荷载识别新方法

移动的平稳随机荷载对简支梁的作用,相当于固定的调制函数已知的非平稳随机荷载对简支梁结构的作用.本文以此为基础,研究了移动平稳随机荷载的识别问题.首先基于虚拟激励法的思想,利用特征值分解及奇异值分解技术,由虚拟位移响应反演虚拟广义坐标,有效地避免了矩阵求逆,得到高精度的广义坐标谱;同时利用Wiener-Khintchine关系及Duhamel积分,由广义坐标谱值反演得到平稳随机激励谱密度.仿真算例表明,只要适当地选取参振振型及测点位置,本文方法可以有效地识别桥梁结构的移动平稳随机荷载.