Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

Blow-up of a Stable Stochastic Differential Equation

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete—in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.     

Random attractors for asymptotically upper semicompact multivalue random semiflows

The present paper studied the dynamics of some multivalued random semi- flow.The corresponding concept of random attractor for this case was introduced to study asymptotic behavior.The existence of random attractor of multivalued random semiflow was proved under the assumption of pullback asymptotically upper semicompact,and this random attractor is random compact and invariant.Furthermore,if the system has ergodicity,then this random attractor is the limit set of a deterministic bounded set.     

同源演变随机激励下的非平稳响应

在复模态分析与演变谱分析的基础上任意点的三维坐提出了时不变线性系统的在一般演变随机的激励下的时变均方响应的一个简便实用解法。并以简化模型为例，用本方法考察了变速车辆的路面响应问题。     

Extremal exponents of random dynamical systems do not vanish

A family of random diffeomorphisms on a manifoldM is said to be a random dynamical system or RDS if it has the so-called cocycle property. The multiplicative ergodic theorem assignsd (=dimM) Lyapunov exponents to every invariant measure of the system. Take the maximum of the leading exponents associated with the various invariant measures. The resulting number is said to be the maximal exponent of the system. The minimal exponent is defined in a similar fashion. It is shown that the minimal exponent of an RDS on a compact manifold is negative, provided not all invariant measures are determined by the future of. A similar statement relates the maximal exponent with the past of. We proceed by introducing Markov systems and Markov measures. This notion covers flows of stochastic differential equations as well as products of random diffeomorphisms in Markovian dependence, in particular, products of iid diffeomorphisms. Markov measures are characterized by the fact that they are functionals of the past. Consequently, if there exists a non-Markovian invariant measure, then the maximal exponent does not vanish. Typically, Markov systems do have non-Markovian invariant measures. Finally, for linear systems we recover results of Ledrappier. In particular, these results provide another proof of Furstenberg's theorem on the positivity of the leading exponent of a product of iid unimodular matrices.     

Stability and Capsizing of Ships in Random Sea – a Survey

This report is a survey of methods of stochastic and nonlinear dynamics in ship stability. After a brief introduction we describe the sea as a stationary random field. We then derive the general equations of motion of a ship from ‘first principles’, specializing to the case of the equations of motion for roll, heave and sway using strip theory from which eventually the ‘archetypal’ nonlinear random differential equation for the roll motion follows. This determines in particular how and where the stochasticity of the sea enters the equation. We then analyze simple nonlinear models of ship motion by means of the theory of random dynamical systems which amounts to studying invariant measures, Lyapunov exponents, random attractors and their (random) domain of attraction and to using stochastic bifurcation theory to describe qualitative changes.     

Stochastic stability of three-dimensional linear systems under parametric random action

The stability of a three-dimensional linear system, driven by a parametric random excitation is considered. The stability of the system is investigated using a combination of stochastic averaging method and a probabilistic approach. For this purpose, it is necessary to find the transient probability density of the components of the vector random process. Thus, the invariant measure of the system may be calculated from the stationary solutions of the associated Fokker–Planck equations. These solutions are obtained numerically, using the sweep method. As a comparison criterion, a digital simulation of It? equations has been carried out using the Monte-Carlo simulation (MCS) method. As an application, the example of instability of a thin-walled bar under the effect of parametric random action is considered     

Invariant manifolds for products of random diffeomorphisms

This paper is concerned with the construction of invariant families of submanifolds for products of random diffeomorphisms on a compact Riemannian manifold. These submanifolds can be obtained for almost arbitrary parameters disjoint from the Lyapunov spectrum of the resulting cocycle. Local measurable families are constructed and the globalization problem is discussed. We present a globalization result for generalized stable and unstable manifolds.     

Differential Invariants of a One-Parameter Group of Local Transformations: One Independent Variable

We propose a new procedure for the construction of an invariant differentiation operator of a one-parameter group of local transformations in the space of one independent variable and m dependent variables. We prove that, for a known universal invariant, a complete set of functionally independent differential invariants of any order of this group can be constructed by using one quadrature and differentiation. The relationship between first-order differential invariants and systems of Riccati-type equations is analyzed.     

考虑周长约束的圆柱薄壳初始几何缺陷随机场建模方法

利用随机场对圆柱薄壳结构的初始几何缺陷进行建模,并据此建立了一种用于含初始几何缺陷轴压圆柱薄壳屈曲分析的随机分析方法。首先,指出已有将圆柱薄壳初始几何缺陷表征为二维高斯随机场的方法会导致与实际不相符的初始几何缺陷,如圆柱周长显著增大或缩小的几何缺陷。其次,提出一种考虑周长不变约束的随机场建模方法,以剔除与实际不相符的随机几何缺陷。最后,基于所建立的初始几何缺陷随机场模型,利用非干涉多项式混沌展开法进行圆柱薄壳的随机屈曲分析,给出临界屈曲载荷的概率分布。数值试验结果表明,基于随机场理论的初始几何缺陷建模方法可有效刻画几何缺陷对结构承载能力的影响,而提出的约束随机场建模方法又能有效减小结果的分散性。     

Pesin’s Formula for Random Dynamical Systems on \mathbf {R}^d

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $\mathbf {R}^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $\mathbf {R}^d$ . Finally we will show that a broad class of stochastic flows on $\mathbf {R}^{d}$ of a Kunita type satisfies Pesin’s formula.     

Stochastic finite element method based on local averages of random fields

The stochastic finite element method (SFEM) based on the local, averages of random fields, which was proposed in [5], is now generalized to analyze the structures with several correlated random parameters. The covariance matrix of the local averages of a random vector field is derived. The SFEM based on the local averages of random vector fields is formulated. The numerical examples show that the generalized SFEM preserves the advantages of the original one, i. e., rapid convergence, good accuracy and insensitivity to the correlation structures of random parameters.Project supported by National Natural Science Foundation of China.     

On the local average of random field in stochastic finite element analysis

The distinguishing feature of stochastic finite element analysis is that it involves the discretization of the parameter space of random fields of material properties, the geometry of structure and / or the loads. It is shown in earlier investigations that a reasonable procedure of discretization is to take the local averages of the random fields on each element. In the present paper the formulae for the covariance of the local averages of a homogeneous random vector field on rectangular elements are generalized by relaxing the condition. For an inhomogeneous random field and /or non-rectangular elements, a procedure of using Gaussian quadrature to evaluate the means and covariances of the local averages is proposed. Thus, the stochastic finite element method (SFEM) based on the local averages of random fields is adapted to a structure with irregular shape and / or inhomogeneous random fields. The effects of the mesh geometry, the ratio of element size to the correlation scale as well as the number of Gaussian quadrature points on the convergence of SFEM are discussed. It is found that even better results could be obtained by utilizing appropriate Gaussian quadrature instead of exact local average.Project supported by National Natural Science Foundation of China.     

Inertial Manifolds and Forms for Stochastically Perturbed Retarded Semilinear Parabolic Equations

We construct inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. These inertial manifolds are finite-dimensional invariant surfaces, which attract exponentially all trajectories. We study the corresponding inertial forms, i.e., the restriction of the stochastic equation to the inertial manifold. These inertial forms are finite-dimensional Ito equations and they completely describe the long-time dynamics of the system under consideration. The existence of inertial manifolds and the properties of inertial forms allow us to show that under mild additional conditions the system has a global (random) attractor in the sense of the theory of random dynamical systems.     

Statistical theory of radiative transfer in layered media

An equation for the probability density of the wave intensity which takes into account absorption, is obtained with a help of the invariant imbedding method. The limiting case when the medium occupies a half-space, is considered. The field intensity is found for the case of a source inside the medium. The conditions of applicability of the linear theory or radiative transfer are obtained. Numerical solutions of the equations corresponding to the statistical theory of radiative transfer in a layered medium with random inhomogeneities are discussed.     

Topological visualisation of focal structures in free shear flows

This paper describes a method for identifying and visualising the three-dimensional geometry of focal (vortex) structures in complex flows. The method is based primarily on the classification of the local topology as it is identified from the values of the velocity gradient tensor invariants. The identification of the local topology is reference frame invariant. Therefore, focal (vortex) structures can be unambiguously identified in these flows. A novel flow visualisation method is introduced whereby focal structures are rendered using a solid model view of the local topology. This new approach is applied to the identification of focal structures in three-dimensional plane mixing layer and plane wake flows.     

Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws

General results about maps of convex sets in ? ⁿ are proved. We outline their extensions to an infinite-dimensional context. Such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps. Here, we explore applications only in the finite-dimensional context. More specifically, we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite-difference schemes applied to systems of conservation laws in several space variables. In particular, we establish a final characterization of the invariant regions under the Lax-Friedrichs scheme and also give sufficient conditions for the local invariance. Further, we give sufficient conditions for the global and local invariance of regions under flux-splitting finite-difference schemes. An example of the multi-dimensional Euler equations for non-isentropic gas dynamics is discussed.     

Characterization of Measures Satisfying the Pesin Entropy Formula for Random Dynamical Systems

In this paper we first give a formulation of SRB (Sinai–Ruelle–Bowen) property for invariant measures of stationary random dynamical systems and then prove that this property is sufficient and necessary for a formula of Pesin's type relating entropy and Lyapunov exponents of such dynamical systems. This result is a random version of the main result in Part I of Ledrappier and Young's celebrated paper [11].