Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 132–142, February 2008.     

Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation

Exact stationary response solutions of nonlinear dynamic systems

Summary Exact stationary solutions are constructed for nonlinear dynamic systems subjected to stochastic excitation. The results are then applied to both classical and nonclassical stochastic vibration systems.     

Sufficient practical-stability conditions for nonlinear impulsive systems

A new approach to solving the practical-stability problem for nonlinear impulsive systems is described.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 131–134, October 2004.     

基于纤维随机分布的单向复合材料拉伸破坏过程模拟

单向复合材料的极限拉伸强度受纤维强度变化和纤维随机分布的影响。本文提出了一种合理的纤维随机分布假设,并以此建立了剪滞数值模型。利用蒙特卡罗模拟方法结合现有模型分析了单向复合材料的拉伸失效过程,得到极限拉伸强度。与文献模型相比较的结果表明,本文的模型结果更理想。     

On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.     

Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations

The stability of stationary solutions of nonlocal reaction-diffusion equations on a bounded intervalJ of the real line with homogeneous Dirichlet boundary conditions is studied. It is shown that it is possible to have stable stationary solutions which change sign once onJ in the case of constant diffusion when the reaction term does not depend explicitly on the space variable. The problem of the possible types of stable solutions that may exist is considered. It is also shown that Matano's result on the lap-number is still true in the case of nonlocal problems.On leave from IST, Lisbon, Portugal.     

A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Analytical and numerical studies of multi-degree-of-freedom(MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge.This paper presents a highly-efficient method for determining the stationary probability density functions(PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov(FPK) equation when constructi...     

Exact solutions for stationary responses of several classes of nonlinear systems to parametric and/or external white noise excitations

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external white noise excitations are constructed by using Fokker-Planck-Kolmogorov equation approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence of nonconservative forces on the first integrals of the corresponding conservative systems and are called generalized-energy-dependent (G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and as an example, the equivalent stochastic systems for the second order G.E.D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may be found by searching the equivalent G.E.D. systems.Project Supported by The National Natural Science Foundation of China. Accepted by XVIIth International Congress of Theoretical and Applied Mechanics.     

Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium

The nonlinear stability of the conduction-diffusion solution of a fluid mixture heated and salted from below (and of a homogeneous fluid heated from below) and saturating a porous medium is studied with the Lyapunov direct method. Both Darcy and Brinkman models have been used. The porous medium is bounded by two horizontal parallel planes and is rotating about a vertical axis. Necessary and sufficient conditions of unconditional stability are proved (i.e., the critical linear and nonlinear stability Rayleigh numbers coincide). Our results generalize those given by Straughan [21] for a homogeneous fluid in the Darcy regime. In the case of a mixture two stabilizing effects act: that of the rotation and of the concentration of the solute. Received March 05, 2002 / Published online June 4, 2002 RID="a" ID="a" e-mail: lombardo@dmi.unict.it RID="b" ID="b" e-mail: mulone@dmi.unict.it Communicated by Brian Straugham, Durham     

Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems

We define a sequence of uniform Lyapunov exponents in the setting of Banach spaces and prove that the Hausdorff dimension of global attractors is bounded from above by the Lyapunov dimension of the tangent map. This result generalizes the papers by Douady and Oesterlé (1980) and Ledrappier (1981) in finite dimension and Constantinet al. (1985) for Hilbert spaces.     

Exact solutions of some general nonlinear wave equations in elasticity

A similarity analysis of a nonlinear wave equation in elasticity is studied; in particular, one with anharmonic corrections. The symmetry transformation give rise to exact solutions via the method of invariants. In some cases, graphical figure of the solutions are presented. Furthermore, we consider some cases wherein the velocities of the longitudinal and transversal plane waves are variable. Finally, a brief discussion on how a symmetry analysis on a perturbation of the elasticity equation can be pursued.     

基于蒙特卡罗随机有限元方法的随机多孔介质内流体自然对流不确定性研究

为分析孔隙率不确定性对多孔介质方腔内自然对流换热的影响,发展了一种基于KL（Karhunen-Loeve展开）-蒙特卡罗随机有限元算法的随机多孔介质内自然对流不确定性分析数理模型及有限元数值模拟程序框架。通过K-L展开及基于拉丁抽样法生成多孔介质孔隙率随机实现,并耦合多孔介质自然对流有限元程序,进行随机多孔介质内自然对流传热数值模拟,得出了多孔介质内流场与温度场平均值与标准偏差,并分析了孔隙率不确定性条件下Da数对Nu数的影响。结果表明,孔隙率不确定性对多孔介质方腔内自然对流有重要影响。随机多孔介质内流场及温度场与确定性条件下的流场及温度场存在一定偏差,Nu数标准偏差随着Da的增大先增大后减小。     

Monte Carlo simulation in the stochastic analysis of non-linear systems under external stationary Poisson white noise input

A method for the evaluation of the probability density function (p.d.f.) of the response process of non-linear systems under external stationary Poisson white noise excitation is presented. The method takes advantage of the great accuracy of the Monte Carlo simulation (MCS) in evaluating the first two moments of the response process by considering just few samples. The quasi-moment neglect closure is used to close the infinite hierarchy of the moment differential equations of the response process. Moreover, in order to determine the higher order statistical moments of the response, the second-order probabilistic information given by MCS in conjunction with the quasi-moment neglect closure leads to a set of linear differential equations. The quasi-moments up to a given order are used as partial probabilistic information on the response process in order to find the p.d.f. by means of the C-type Gram-Charlier series expansion.     

Bifurcation and stability analysis for a non-smooth friction oscillator

Summary Friction-induced self-sustained oscillations, also known as stick-slip vibrations, occur in mechanical systems as well as in everyday life. On the basis of a one-dimensional map, the bifurcation behaviour including unstable branches is investigated for a friction oscillator with simultaneous self-and external excitation. The chosen way of mapping also allows a simple determination of Lyapunov exponents.Dedicated to Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. h.c. mult. Erwin Stein on the occasion of his 65th birthday.     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…     

各向异性势问题充要的随机边界积分方程

本文针对各向异性势问题提出了一类充分必要的随机边界积分方程。数值计算结果表明在退化尺度附近，充要的随机边界积分方程较习用的随机边界积分方程有较大的优越性。     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Construction of the stationary probability density for a family of SDOF strongly non-linear stochastic second-order dynamical systems

In this paper, a new procedure is proposed to construct the stationary probability density for a family of the single-degree-of-freedom (SDOF) strongly non-linear stochastic second-order dynamical systems subjected to parametric and/or external Gaussian white noises. First of all, the Fokker-Planck-Kolmogorov (FPK) equation associated with the original Itô stochastic differential equation is replaced by the equivalent FPK equation by adding arbitrary anti-symmetric diffusion coefficient. Then, a family of invariant measures depending on the arbitrary anti-symmetric diffusion coefficient and another arbitrary function is constructed by vanishing the probability flows in two directions. Finally, the drift vector associated with a family of Itô stochastic differential equations is deduced by giving, a priori, these two arbitrary functions. It is shown that the known invariant measures dependent on energy are only the special cases of invariant measures presented in this paper, while some other classes of invariant measures are independent of energy. Thus, the invariant measures constructed in this paper are those belonging to the most general class of the SDOF strongly non-linear stochastic second-order dynamical systems so far.