INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

ＩＮＶＡＲＩＡＮＴＭＡＮＩＦＯＬＤＳＡＮＤＴＨＥＩＲＳＴＡＢＩＬＩＴＹＩＮＡＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＭＥＡＳＵＲＥ－ＰＲＥＳＥＲＶＩＮＧＭＡＰＰＩＮＧＳＹＳＴＥＭＳＬｉｕＪｉｅ（刘杰）（ＲｅｃｅｉｖｅｄＤｅｃ．１．１９９４;Ｃｏｍｍｕｎｉｃ...     

INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

ＩＮＶＡＲＩＡＮＴＭＡＮＩＦＯＬＤＳＡＮＤＴＨＥＩＲＳＴＡＢＩＬＩＴＹＩＮＡＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＭＥＡＳＵＲＥ－ＰＲＥＳＥＲＶＩＮＧＭＡＰＰＩＮＧＳＹＳＴＥＭＳＬｉｕｊｉｅ（刘杰）（ＲｅｃｅｉｖｅｄＤｅｃ．１．１９９４；Ｃｏｍｍｕｎｉｃ...     

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods. The project supported by the National Natural Science Foundation of China     

Controlling hyperchaos in planar systems by adjusting parameters

IntroductionIn 1 990 ,Ott,GrebogiandYorke (OGY)introducedtheconceptofcontrolofchaosandgaveamethodforcontrollingchaos,knownastheOGYmethod[1].AfterOGY ,manyothermethodsforcontrollingchaoshavebeendeveloped[2 - 8].Otherrelatedtopics,suchasnoiseeffectsandoptimizationofthecontrol,havealsobeeninvestigated[9- 12 ].TheOGYmethodconsidersthemapξn+ 1=Fp(ξn) ,ξ∈R2 ,wherep∈Risaparameter,whichhasasaddleatacertainvalueoftheparameter.Thereforethereareaone_dimensionalstablemanifoldandaone_dimension…     

弹性动力学反问题的非线性及其迭代反演

分析了弹性动力学反问题的非线性及在迭代反演过程中表现出的复杂非线性现象。迭代反演的结果依赖于反演系统参数和迭代初值,而系统参数对应的Mandelbrot集和迭代初值对应的Julia集都是复杂的分形集。随反演系统状态参数的变化,完全确定性的反演系统却可能产生一系列无规则的,不可预测的迭代输出序列。反演迭代过程中出现的分形和混沌现象反映了表面简单的反演迭代后隐藏的复杂性,正是这种复杂性给迭代系统参数的合理选择带来困难,进而使反演迭代不总能给出满意的结果。     

Bifurcation analysis for a modified Jeffcott rotor with bearing clearances

A HB (Harmonic Balance)/AFT (Alternating Frequency/Time) technique is developed to obtain synchronous and subsynchronous whirling motions of a horizontal Jeffcott rotor with bearing clearances. The method utilizes an explicit Jacobian form for the iterative process which guarantees convergence at all parameter values. The method is shown to constitute a robust and accurate numerical scheme for the analysis of two dimensional nonlinear rotor problems. The stability analysis of the steady-state motions is obtained using perturbed equations about the periodic motions. The Floquet multipliers of the associated Monodromy matrix are determined using a new discrete HB/AFT method. Flip bifurcation boundaries were obtained which facilitated detection of possible rotor chaotic (irregular) motion as parameters of the system are changed. Quasi-periodic motion is also shown to occur as a result of a secondary Hopf bifurcation due to increase of the destabilizing cross-coupling stiffness coefficients in the rotor model.     

Bifurcation and Chaos in the Duffing Oscillator with a PID Controller

We discuss in this paper the bifurcation, stability and chaos of the non-linear Duffing oscillator with a PID controller. Hopf bifurcation can occur and we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system and so is fractal basin boundary for soft spring system. The Melnikov method is used to obtain the criterion of fractal basin boundary.     

On Controlling Chaos in Cournot-Games with Two and Three Competitors

Controlling chaotic economic systems is an important problem. The Ott, Grebogi and Yorke (1990) method is a way to control chaos gradually. However, in order to apply this method the control parameter has to be close to a prescribed value. To achieve this, the targeting method of Shinbrot Ott, Grebogi and Yorke can be applied first to speed up convergence. In this paper the Ott et al method is modified so that the system is guided gradually out of the chaotic region and in the stability zone. The control parameter does not have to be in a prescribed region. The method is applied to the Puu formulation of Cournot dynamic duopoly and oligopoly models.     

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 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…     

惯性/高度表组合导航系统卡尔曼滤波算法的浑沌及其控制

本文研究了组合导航系统中应用卡尔曼滤波器进行信息最优估计时的浑沌现象和浑沌 控制,并以惯性系统和高度定位组合导航系统为例,研究了当载体受周期激励时,由于采样频 率选择不当,卡尔曼滤波算法将导致浑沌结果,从而影响系统导航精度。本文还提出了利用变 步长的参数调整法来抑制这种浑沌。     

NONLINEAR NUMERICAL ANALYSIS OF A FLEXIBLE ROTOR EQUIPPED WITH SQUEEZE COUPLE STRESS FLUID FILM JOURNAL BEARINGS

This study performs a dynamic analysis of a rotor supported by two squeeze couple stress fluid film journal bearings with nonlinear suspension.The numerical results show that the stability of the system varies with the non-dimensional speed ratios and the dimensionless parameter l*.It is found that the system is more stable with higher dimensionless parameter l*. Thus it can conclude that the rotor-bearing system lubricated with the couple stress fluid is more stable than that with the conventional Newtonian fluid.The modeling results thus obtained by using the method proposed in this paper can be used to predict the stability of the rotor-bearing system and the undesirable behavior of the rotor and bearing center can be avoided.     

Events Maps in a Stick-Slip System

This paper describes a one-dimensional map generated by a two degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction. The iterated map allows a much simpler representation and a better understanding of some dynamic features of the system. Some applications of the map are illustrated and its behaviour is simulated by means of an analytically defined one-dimensional map. A method of reconstructing one-dimensional maps from experimental data from the system is introduced. The method uses cubic splines to approximate the iterated mappings. From a sequence of such time series the parameter dependent bifurcation behaviour is analysed by interpolating between the defined mappings. Similarities and differences between the bifurcation behaviour of the exact iterated mapping and the reconstructed mapping are discussed.     

Van der Pol type self-excited micro-cantilever probe of atomic force microscopy

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.     

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

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Effect of Lateral Linear Stiffness on Nonlinear Characteristics of Hunting Motion of a Railway Wheelset

Railway wheelset experiences the problem of hunting above a critical speed, which is a kind of self-excited oscillation. At the critical speed, it is known that the system undergoes a subcritical Hopf bifurcation. Therefore, for clarifying the nonlinear characteristics of hunting it is very important to detect, for example, the nonlinear forces in the wheelset due to the creep forces acting between the wheels and rails, and the nonlinear component of the resorting forces by the suspensions. However, it is impossible to determine each force quantitatively. In the present paper, it is first shown, by using the center manifold theory and the method of normal form, that the nonlinear characteristics of the bifurcation in a wheelset model with two degrees of freedom are governed by a single parameter, hence each nonlinear force need not be detected when examining the nonlinear characteristics. Also, a method of determining the governing parameter from experimentally observed radiuses of the unstable limit cycle is proposed. Next, we experimentally investigate the variation of the parameter due to the presence of linear spring suspensions in the lateral direction and discuss the variation of the nonlinear characteristics of the hunting motion, which depends on the lateral stiffness. As a result, the improvement of the stability of the wheelset against the disturbance by the linear spring suspensions is clarified.     

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     

Invariant Manifolds for Random Dynamical Systems with Slow and Fast Variables

We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modeling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.      

Application of a non-linear local analysis method for the problem of mixed convection instability

We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra_s, the stationary solution is a pitchfork bifurcation point of the system.     

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。