Bifurcation and Chaos Analysis of Stochastic Duffing System Under Harmonic Excitations

The Chebyshev polynomial approximation is applied to the dynamic response problem of a stochastic Duffing system with bounded random parameters, subject to harmonic excitations. The stochastic Duffing system is first reduced into an equivalent deterministic non-linear one for substitution. Then basic non-linear phenomena, such as stochastic saddle-node bifurcation, stochastic symmetry-breaking bifurcation, stochastic period-doubling bifurcation, coexistence of different kinds of steady-state stochastic responses, and stochastic chaos, are studied by numerical simulations. The main feature of stochastic chaos is explored. The suggested method provides a new approach to stochastic dynamic response problems of some dissipative stochastic systems with polynomial non-linearity.     

Global synchronization of stochastic delayed complex networks

This paper is concerned with the delay-dependent synchronization criterion for stochastic complex networks with time delays. Firstly, expectations of stochastic cross terms containing the It? integral are investigated by utilizing stochastic analysis techniques. In fact, in order to obtain less conservative delay-dependent conditions for stochastic delay systems including stochastic complex (or neural) networks with time delays, how to deal with expectations of these stochastic cross terms is an important problem, and expectations of these stochastic terms were not dealt with properly in many existing results. Then, based on the investigation of expectations of stochastic cross terms, this paper proposes a novel delay-dependent synchronization criterion for stochastic delayed complex networks. In the derivation process, the mathematical development avoids bounding stochastic cross terms. Thus, the method leads to a simple criterion and shows less conservatism. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.     

随机结构在随机激励下的二阶摄动随机势能泛函及其应用

利用小参数摄动法,建立了随机结构在随机激励下的二阶振动随机势能泛函。并由此推导了二阶摄动随机变原理,作为应用,建立了随机有限元的计算列式。     

Analysis of period-doubling bifurcation in double-well stochastic Duffing system via Laguerre polynomial approximation

The Laguerre polynomial approximation method is applied to study the stochastic period-doubling bifurcation of a double-well stochastic Duffing system with a random parameter of exponential probability density function subjected to a harmonic excitation. First, the stochastic Duffing system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then nonlinear dynamical behavior about stochastic period-doubling bifurcation can be fully explored. Numerical simulations show that similar to the conventional period-doubling phenomenon in the deterministic Duffing system, stochastic period-doubling bifurcation may also occur in the stochastic Duffing system, but with its own stochastic modifications. Also, unlike the deterministic case, in the stochastic case the intensity of the random parameter should also be taken as a new bifurcation parameter in addition to the conventional bifurcation parameters, i.e. the amplitude and the frequency of harmonic excitation.     

二维弹性随机边界元与可靠性分析

本文利用对随机变量求偏导数的方法，推导了二维弹性随机边界积分方程及其相应的公式，考虑了随机边界条件、材料性能参数随机量以及几何尺寸随机因素等。本文还将所发展的方法用于结构静强度的可靠性分析。算例分析表明，本文数值结果与ＭｏｎｔｅＣａｒｌｏ模拟和理论解相比是十分满意的。     

A theoretical analysis on stochastic behaviour of short cracks and fatigue damage of metals

In this paper, the closed form of solution to the stochastic differential equation for a fatigue crack evolution system is derived, and the relationship between metal fatigue damage and crack stochastic behaviour is investigated. It is found that the damage extent of metals is independent of crack stochastic behaviour if the stochastic deviation of the crack growth rate is directly proportional to its mean value. The evolution of stochastic deviation of metal fatigue damage in the stage close to the transition point between short and long crack regimes is also discussed.     

Nonlinear stochastic dynamics: A survey of recent developments

This paper provides an overview of significant advances in nonlinear stochastic dynamics during the past two decades, including random response, stochastic stability, stochastic bifurcation, first passage problem and nonlinear stochastic control. Topics for future research are also suggested. The project supported by the National Natural Science Foundation of China (19972059)     

Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise

A global analysis of stochastic bifurcation in a special kind of Duffing system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell flow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. Through numerical simulations the evolution of the Poincare cell maps of the random flow against the variation of noise intensity is explored systematically. Our study reveals that as a powerful tool for global analysis, the generalized cell mapping method using digraph is applicable not only to deterministic bifurcation, but also to stochastic bifurcation as well. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.     

弯曲梁随机刚度有限元法

弯曲梁的随机材料参数和随机几何参数可结合为一个随机参数-弯曲刚度。以位移的均值和协方差函数的变分原理为基础,根据基本方程推导出具有随机刚度的静定弯曲梁的有限元法。这种方法不再采用摄动技术,可直接建立位移的均值和协方差函数。     

随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

Integral Representations of Increments of Stochastic Processes

A formulation is presented in which the increment of a stochastic process is represented as an integral of the derivative of the process. It is shown that this representation is an alternative to the more common approach of writing equations for the differentials of stochastic processes. A possible advantage of the integral formulation is that its reliance on derivatives, rather than differentials, makes the operations of stochastic calculus more closely resemble those of ordinary deterministic calculus. This similarity to well-known mathematics may serve to make stochastic calculus accessible to a broader audience than in the past. The integral formulation is herein shown to be compatible with the Itô differential rule for non-Gaussian processes and is used to describe the increment of the nonstationary response of a system governed by a vector stochastic equation with parametric delta-correlated excitation.     

Stochastic boundary element method for reliability analysis of plate and beams composite structures

I.IntroductionTheengineeringstructuresareoftells,'1>:'rectcdtotheactionofthestochasticloadingthatvarieswiththetime,forexample,theengineeringstructuresactedonbytheearthquake,theoceanstructuresactedonbydynamicpressure,andthevehiclesofthetransportationinflue…     

随机有限元方法与结构可靠性

简要介绍和综合论述了随机结构的有限元分析及可靠性分析方法的研究及发展情况．包括随机结构的离散化、随机有限元控制方程的求解、随机有限元可靠性方法及随机结构的可靠性分析等．讨论了随机结构分析中存在的一些问题．概要论述了可能的发展方向．     

多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.     

Method of stochastic normal forms

—The method of normal forms, originally developed for deterministic non-linear dynamical systems, is extended to include stochastic excitations, with the objective of obtaining an optimal reduction of dimensionality of the system while retaining its essential dynamic characteristics. Similar to the deterministic case, the crucial step in the normal-form computation is to find the so-called resonant terms which cannot be eliminated through a non-linear change of variables. Subsequent to the reduction of dimensionality, the associated stochastic normal form is obtained using a Markovian approximation. It is shown that the second order stochastic terms must be retained, in order to capture the stochastic contributions of the stable modes to the drift terms of the critical modes. Furthermore, for a specific class of non-linear systems, the results obtained from the stochastic normal form analysis are the same as those obtained from an extended stochastic averaging procedure. Thus, for this particular class, the two methods are equivalent.     

Stabilization of stochastic cycles and chaos suppression for nonlinear discrete-time systems

In this paper we consider a nonlinear discrete-time control system with regular and chaotic dynamics forced by stochastic disturbances. The problem addressed is the design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states for the corresponding stochastic system. To solve this problem, we propose a new method based on the stochastic sensitivity function technique. This function approximates a dispersion of random states distributed around deterministic cycle. Explicit formulas for the intercoupling between stochastic sensitivity function and considered system parameters are worked out. The problem of the design of the required stochastic sensitivity function for cycles by feedback regulators is solved. Coefficients of the feedback regulator are constructed and corresponding attainability sets are described. The effectiveness of the proposed approach is demonstrated on the stochastic Verhulst model. It is shown that constructed regulators provide a low level of sensitivity and suppress chaotic oscillations.     

拟哈密顿系统非线性随机最优控制

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果, 包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略, 即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度为目标的控制, 为将它们应用于工程实际而作的部分可观测系统最优控制、有界控制、时滞控制、半主动控制、极小极大控制的进一步研究, 以及综合考虑这些实际问题的非线性随机最优控制的综合策略, 非线性随机最优控制在滞迟系统、分数维系统等中的若干应用, 介绍与这些研究有关的背景, 并指出今后有待进一步研究的问题.     

A new approach to almost-sure asymptotic stability of stochastic systems of higher dimension

The almost sure asymptotic stability of higher-dimensional linear stochastic systems and of a special class of nonlinear stochastic systems with homogeneous drift and diffusion coefficients of order one is studied. Based on the well-known Khasminskii's theorem, a new approach for obtaining the regions of almost sure asymptotic stability and instability without evaluating the stationary probability density of the diffusion process defined on unit hypersphere is proposed. Two examples of two and three degree-of-freedom linear stochastic systems are given to illustrate the application and effectiveness of the proposed approach combined with stochastic averaging.     

单层悬索体系在风中的随机稳定性研究

本文建立了单层悬索体系在含有较低紊流成分的自然风中的随机稳定方程，应用Ｓｃｈｕｓｓ随机稳定理论分析，并建立可靠性评判准则，实例分析指出，由于微小紊流成分的随机摄动，致系统产生不稳定运动。     

随机激励的耗散的Hamilton系统理论的研究进展

近几年中,利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等,提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法,并在此基础上提出了等效非线性系统法,提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法,并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制,从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．