Hopf bifurcation in general Brusselator system with diffusion

The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established. Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results.     

Diffusion-driven instability and Hopf bifurcation in Brusselator system

The Hopfbifurcation for the Brusselator ordinary-differential-equation （ODE） model and the corresponding partial-differential-equation （PDE） model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.     

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...     

Determination of stress intensity factor with direct stress approach using finite element analysis

In this article,a direct stress approach based on finite element analysis to determine the stress intensity fac-tor is improved.Firstly,by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack,we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity fac-tor,which is nearly equal to the stress intensity factor given by the asymptotic solution.Secondly,the crack problem is solved numerically by the finite element method.Depending on the modeling capability of the software,we designed an adaptive mesh model to simulate the stress singularity.Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corre-sponding crack problem.Therefore,the stress intensity factor may be calculated from the stress distribution in the appro-priate interval,with a high accuracy.     

Hopf Bifurcation of the Generalized Lorenz Canonical Form

Hopf bifurcation of a unified chaotic system – the generalized Lorenz canonical form (GLCF) – is investigated. Based on rigorous mathematical analysis and symbolic computations, some conditions for stability and direction of the periodic obits from the Hopf bifurcation are derived.     

An Energy Analysis of the Local Dynamics of a Delayed Oscillator Near a Hopf Bifurcation

Hopf bifurcation exists commonly in time-delay systems. The local dynamics of delayed systems near a Hopf bifurcation is usually investigated by using the center manifold reduction that involves a great deal of tedious symbolic and numerical computation. In this paper, the delayed oscillator of concern is considered as a system slightly perturbed from an undamped oscillator, then as a combination of the averaging technique and the method of Lyapunov's function, the energy analysis concludes that the local dynamics near the Hopf bifurcation can be justified by the averaged power function of the oscillator. The computation is very simple but gives considerable accurate prediction of the local dynamics. As an illustrative example, the local dynamics of a delayed Lienard oscillator is investigated via the present method.     

Computation of a two-dimensional stress intensity factor by the boundary element method

Summary This paper presents a boundary element formulation for elastostatic problems. The formulation is expressed in terms of the matrix notation, so that it is easily applicable to an available system of matrix structural analysis. A computer program developed is used to calculate the stress intensity factor K _I for some example problems in plane elasticity. Comparison is made between the boundary element calculations and other solutions, whereby the effectiveness of the boundary element method is demonstrated.

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Berechnung eines zweidimensionalen Spannungsintensitätsfaktors mit der Methode der Bandelemente

Übersicht Dieser Aufsatz bietet eine Formulierung elastostatischer Probleme durch die Methode der Randelemente an. Die Formulierung benutzt eine Darstellung in Matrizenbezeichnung, so daß sie auf ein verfügbares System für Matrizenstrukturberechnung einfach anwendbar ist. Ein ausgebautes Computerprogramm wird auf die Berechnung des Spannungsintensitätsfaktors K _i für einige Beispiele in der ebenen Elastizität angewendet. Die mittels der Randelemente erhaltenen Ergebnisse werden mit anderen Lösungen verglichen. Dadurch zeigt sich die Wirksamkeit der Methode der Randelemente.

     

Stability and slightly supercritical oscillatory regimes of natural convection in a 8:1 cavity: solution of the benchmark problem by a global Galerkin method

The global Galerkin method is applied to the benchmark problem that considers an oscillatory regime of convection of air in a tall two‐dimensional rectangular cavity. The three most unstable modes of the linearized system of the Boussinesq equations are studied. The converged values of the critical Rayleigh numbers together with the corresponding oscillation frequencies are calculated for each mode. The oscillatory flow regimes corresponding to each of the three modes are approximated asymptotically. No direct time integration is applied. Good agreement with the previously published results obtained by solution of the time‐dependent Boussinesq equations is reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings subject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functions. The procedure is applied to predict the response of Duffing–van der Pol oscillator under both external and parametric excitations of wide-band stationary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of the procedure are analyzed. The proposed procedure is also applied to study the asymptotic stability in probability and stochastic Hopf bifurcation of Duffing–van der Pol oscillator under parametric excitations of wide-band stationary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining the asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stability analysis using linearized equation is correct only if the linear stiffness term does not vanish.     

多自由度转子-轴承系统周期运动分岔及稳定性研究

建立考虑诸多因素的转子-轴承系统多自由度模型,将与Newmark结合的打靶法应用到多自由度转子-轴承系统的周期稳定性分析中。着重研究了转子-轴承系统失稳转速随系统偏心量、轴承间隙、润滑油动力粘度以及轴承长径比的变化规律,研究结果表明:提高系统偏心量、减小轴承间隙、增大润滑油动力粘度以及选择适当的轴承长径比均能提高转子-轴承系统的失稳转速;对于不同的参数值,系统表现出不同的分岔规律,系统发生半速涡动时表现为倍周期分岔或拟周期分岔,发生油膜振荡时则表现为拟周期分岔。     

Equivalence of the notch stress intensity factor,tip opening displacement and energy release rate for a sharp V-notch

For an infinite elastic plane with a sharp V-notch under the action of symmetrically loading at infinity, the length of crack initiation ahead of the V-notch’s tip is estimated according to a modified Griffith approach. Applying a new conservation integral to the perfectly plastic strip (Dugdale model) ahead of the V-notch’s tip, the relationship between notch stress intensity factor (NSIF) and notch tip opening displacement (NTOD) is presented. Also, the relationship between NSIF and perfectly plastic strip size (PPSS) is found. Since there are three fracture parameters (NSIF, NTOD, and PPSS) with changeable notch opening angle in two basic relationships, it is necessary to select one critical parameter with changeable notch opening angle or two independent critical parameters, respectively. With the help of a characteristic length, it is found by this new conservation integral that the NSIF, NTOD and energy release rate are equivalent in the case of small-scale yielding. Especially, the characteristic length possesses clear physical meaning and, for example, depends on both the critical NSIF and SIF or both the NTOD and CTOD, respectively, in which SIF and CTOD are from the tip of a crack degenerated from the sharp V-notch. The dependence of NSIF on NTOD and PPSS is presented according to the equivalence, and the critical NSIF depending on the critical NTOD with a notch opening angle is also predicted.     

一类生物流体力学连续系统的分岔研究

连续动力系统的非线性动力学研究,由于其应用的广泛性与问题的复杂性,近年来越来越受到重视。本文对一类生物流体力学中的连续系统-动脉局部狭窄时血液流动的分岔特性进行了研究,采用有限差分方法,将由偏微分方程组描述的边境动力系统约化为由常微分方程组描述的高维离散动力系统。求得了离散动力系统的平衡解并分析其稳定性,同时讨论了流场中变量空间分布的变化情况。求得了离散动力系统的前三个Lyapunov指数,以此作为系统是否发生混沌的判别条件。     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

Identification of imperfections in thin plates based on the modified potential energy principle

A procedure to identify the imperfection in thin plates is proposed in this paper. The modified potential energy principle, which serves as the theoretical basis of the identification procedure, is improved to allow for the experimental measurements in static tests. Several typical examples are studied to illustrate the effectiveness of the procedure.     

基于能量原理的双稳态各向同性柱壳力学特性研究

双稳态结构可以在伸展和卷拢两种状态下均保持稳定,作为一种新型可展开空间结构,其在航空航天等领域具有广阔的应用前景。选取各向同性材料作为研究对象,并假定该材料为理想弹塑性材料。建立柱壳的力学计算模型,推导结构变形时应变能的解析表达式,基于能量原理导出柱壳在两个稳定状态下的卷曲半径。结果表明,基于全量理论和增量理论的模型均能反映结构的双稳态特性,但后者能更准确地预测两个稳态对应的卷曲半径的大小。对于文中给定的柱壳,全量理论导出的卷曲半径R3x相对于增量理论结果误差为5.2%~11.6%,R5y相对误差为29.8%~41.3%。另外,应用有限元软件ABAQUS对各向同性柱壳结构的双稳态过程进行数值模拟,数值结果与增量理论结果的相对误差为0.9%~13.2%,二者吻合较好。     

裂纹面受荷载作用的应力强度因子的计算

基于比例边界有限元法计算了裂纹面有荷载作用情况下裂纹尖端的应力强度因子,给出了有限介质裂纹面作用荷载的比例边界有限元方程的基本求解过程.对于随径向坐标任意变化的一类面荷载的积分能够显式计算,不需要引入额外的近似;并将计算结果与解析解和数值结果进行对比,结果表明比例边界有限元法在计算裂纹面作用荷载时的应力强度因子是有效且精确的.此外,该方法可方便地处理各向异性材料裂纹问题,本文给出了正交各向异性矩形盘裂纹面受均布荷载情况的应力强度因子.