Classification of parametrically constrained bifurcations

If the constraint boundary relates to a bifurcation parameter, a bifurcation is said to be parametrically constrained. Relying upon some substitution, a parametrically constrained bifurcation is transformed to an unconstrained bifurcation about new variables. A general form of transition sets of the parametrically constrained bifurcation is derived. The result indicates that only the constrained bifurcation set is influenced by parametric constraints, while other transition sets are the same as those of the corresponding nonparametrically constrained bifurcation. Taking parametrically constrained pitchfork bifurcation problems as examples, effects of parametric constraints on bifurcation classification are discussed.     

黏弹性圆柱形壳动力学高余维分岔、普适开折问题

讨论两端受到谐波激励的黏弹性圆柱形壳的非线性动力学行为,利用奇异性理论,研究了分岔方程的普适开折问题,严格证明了它是一个高余维分岔问题。余维数为5（含有一个模参数）,给出了它的所有可能的普适开折形式。在分岔参数满足某些条件时得到该分岔问题的转迁集及分岔图,展示了一些新的动力学行为,改进和完善了奇异性分析方法。     

Some dynamical behavior of the Stuart-Landau equation with a periodic excitation

The lock-in periodic solutions of the Stuart-Landau equation with a periodic excitation are studied. Using singularity theory, the bifurcation behavior of these solutions with respect to the excitation amplitude and frequency are investigated in detail, respectively. The results show that the universal unfolding with respect to the excitation amplitude possesses codimension 3. The transition sets in unfolding parameter plane and the bifurcation diagrams are plotted under some conditions. Additionally, it has also been proved that the bifurcation problem with respect to frequence possesses infinite codimension. Therefore the dynamical bifurcation behavior is very complex in this case. Some new dynamical phenomena are presented, which are the supplement of the results obtained by Sun Liang et al.     

Bifurcation on synchronous full annular rub of rigid-rotor elastic-support system

An aero-engine rotor system is simplified as an unsymmetrical-rigid-rotor with nonlinear-elastic-support based on its characteristics. Governing equations of the rubbing system, obtained from the Lagrange equation, are solved by the averaging method to find the bifurcation equations. Then, according to the two-dimensional constraint bifurcation theory, transition sets and bifurcation diagrams of the system with and without rubbing are given to study the influence of system eccentricity and damping on the bifurcation behaviors, respectively. Finally, according to the Lyapunov stability theory, the stability region of the steady-state rubbing solution, the boundary of static bifurcation, and the Hopf bifurcation are determined to discuss the influence of system parameters on the evolution of system motion. The results may provide some references for the designer in aero rotor systems.     

ROBUST CONTROL OF PERIODIC BIFURCATION SOLUTIONS

The topological bifurcation diagrams and the coefficients of bifurcation equation were obtained by C-L method. According to obtained bifurcation diagrams and combining control theory, the method of robust control of periodic bifurcation was presented, which differs from generic methods of bifurcation control. It can make the existing motion pattern into the goal motion pattern. Because the method does not make strict requirement about parametric values of the controller, it is convenient to design and make it. Numerical simulations verify validity of the method.     

近哈密顿系统的Hopf分岔

简化了Wiggins提出的关于近哈密顿系统的Hopf分岔条件,并结合硬弹簧Duffing系统,研究了该类系统的Hopf分岔行为,并用数值积分的方法验证了结果的正确性。     

矩形液池热毛细对流转捩途径研究

主要研究矩形液池热毛细对流的分岔转捩. 通过测量流体内部温度振荡情况, 详细研究了热毛细对流的转捩过程和转捩途径. 实验发现, 矩形液池热毛细对流的转捩过程依次经历了定常、规则振荡、不规则振荡的阶段. 对于不同普朗特数的硅油在不同长高比情况下, 通向混沌的途径不同. 在转捩过程中, 随着温差的增加, 普朗特数在16 (1cSt) 以下和普朗特数为25 (1.5cSt)、长高比为26 的硅油热毛细对流主要以准周期分岔的转捩方式为主;而普朗特数为25 以上的则以倍周期分岔的转捩方式为主;两种分岔有时还会伴随有切分岔形式的出现.实验中还观察到了表面波动和对流涡胞振荡等现象.      

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are Used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.     

时变小扰动Hamilton系统的Hopf分岔

运用Melnikov方法研究了时变小扰动Ｈamilton系统周期轨道发生Ｈopf分岔的条件,并将这些条件应用到一类三维时变小扰动非自治系统,数值结果验证了本文理论的正确性,进一步数值积分表明,所研究的系统还存在复杂而有规律的环面分岔行为。     

非线性Mathieu方程1／2亚谐分叉解的实验研究

本文对一类Mathieu方程的1/2亚谐分叉特性进行了实验研究,得出了在整个参数平面上具有不同拓朴结构分叉图的实验曲线,研究了确定非线性系统衰减参数的方法。并对各种特定的物理系统,可能出现的不同拓朴结构的分叉图和所具有的不同参数区域进行了讨论。     

BIFURCATION IN A TWO-DIMENSIONAL NEURAL NETWORK MODEL WITH DELAY

IntroductionForunderstandingthedynamicsofneuralnetworks ,thepropertiesofstabilityandbifurcationinasimplifiednon_self_connectionneuralnetwork u1(t) =-μ1u1(t) aF(u2 (t-τ2 ) ) , u2 (t) =-μ2 u2 (t) bG(u1(t-τ1) ) ( 1 )hasbeenstudied .Forexample ,inRef.[1 ]ChenandWustudiedtheexistenceoftheslowlyoscillatingperiodicsolutionbyusingthemethodofdiscreteLiapunovfunction .InRef.[2 ]thesumoftimedelaysτ=τ1 τ2 beingregardedasabifurcationparameter,theexistenceoflocalHopfbifurcationandthepropertiesof…     

Dynamic bifurcation of the <Emphasis Type="Italic">n</Emphasis>-dimensional complex Swift-Hohenberg equation

This paper is concerned with the bifurcation of a complex Swift-Hohenberg equation. The attractor bifurcation of the complex Swift-Hohenberg equation on a one- dimensional domain （0, L） is investigated. It is shown that the n-dimensional complex Swift-Hohenberg equation bifurcates from the trivial solution to an attractor under the Dirichlet boundary condition on a general domain and under a periodic boundary condition when the bifurcation parameter crosses some critical values. The stability property of the bifurcation attractor is analyzed.     

Pattern formation in a nonequilibrium phase transition for a generalized Burgers–Fisher equation

A nonequilibrium phase transition of a generalized Burgers–Fisher equation describing biological pattern formation with a periodic boundary condition is examined. In the presence of a weak external force, some approximate bifurcation solutions near a critical point and new spatially periodic patterns are obtained by using the perturbation method in an infinite-dimensional space. The result shows that the external force delays the bifurcation.     

用微分求积方法计算二维薄板在超音速流中的非线性颤振

采用了一种微分求积方法将二维薄板在超音速气流作用下的非线性动力学方程离散为常微分方程,并用Runge-Kutta数值方法进行了计算.为验证微分求积方法的结果,与伽辽金方法计算结果进行了比较,取得了一致的结果.微分求积法的计算结果用分叉图、相平面、时域曲线以及功率谱进行了描述,结果表明在特定的参数区间存在混沌运动,而通向混沌的道路是经过一系列周期倍化分叉产生的.     

Bifurcation in two-dimensional neural network model with delay

A kind of 2-dimensional neural network model with delay is considered. By analyzing the distribution of the roots of the characteristic equation associated with the model, a bifurcation diagram was drawn in an appropriate parameter plane. It is found that a line is a pitchfork bifurcation curve. Further more, the stability of each fixed point and existence of Hopf bifurcation were obtained. Finally, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions were determined by using the normal form method and centre manifold theory. Foundation item: the National Natural Science, Foundation of China (19831030) Biography: WEI Jun-jie, Professor, Doctor, E-mail: weijj@hit.edu.cn     

The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

The stability of thin viscous sheets has been studied so far in the special case where the base flow possesses a direction of invariance: the linear stability is then governed by an ordinary differential equation. We propose a mathematical formulation and a numerical method of solution that are applicable to the linear stability analysis of viscous sheets possessing no particular symmetry. The linear stability problem is formulated as a non-Hermitian eigenvalue problem in a 2D domain and is solved numerically using the finite-element method. Specifically, we consider the case of a viscous sheet in an open flow, which falls in a bath of fluid; the sheet is mildly stretched by gravity and the flow can become unstable by ‘curtain’ modes. The growth rates of these modes are calculated as a function of the fluid parameters and of the geometry, and a phase diagram is obtained. A transition is reported between a buckling mode (static bifurcation) and an oscillatory mode (Hopf bifurcation). The effect of surface tension is discussed.     

STABILITY AND LOCAL BIFURCATION IN A SIMPLY-SUPPORTED BEAM CARRYING A MOVING MASS

The stability and local bifurcation of a simply-supported flexible beam(Bernoulli- Euler type)carrying a moving mass and subjected to harmonic axial excitation are investigated. In the theoretical analysis,the partial differential equation of motion with the fifth-order nonlinear term is solved using the method of multiple scales(a perturbation technique).The stability and local bifurcation of the beam are analyzed for 1/2 sub harmonic resonance.The results show that some of the parameters,especially the velocity of moving mass and external excitation,affect the local bifurcation significantly.Therefore,these parameters play important roles in the system stability.     

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.     

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support. The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration. The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method. The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter,nonlinear spring stiffness. Based on this, the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness. The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.     

Fundamental resonance and bifurcation of large generator end winding when its clamping plates are loose

Introduction Theendwindingunderelectromagneticforcewillvibratewhenthegeneratorisrunning. Becauseofeffectofmanyfactors,suchasmanufacturetechniqueandpracticalrunningcondition, thevibrationmayresultinabrasion,fatigueandfractureoftheendwindinginsulation,even makecoilshortcircuited.Thisisoneofthedifficultiesthatpuzzledpeopleinengineeringfield.Forexample,someelectricpowerstationshavehadaccidentssuchasendwindinginsulation frettedwhichresultedinshortcircuitbecauseclampingplatesareloose;Clipofdown_lea…