Calculation of Coefficients of Simplest Normal Forms of Hopf and Generalized Hopf Bifurcations

Consider the following dynamical system x.=f(x),x∈Rn(1)wherexis the state vector andfis analytic, assumingthatx=0 is an equilibrium of system (1), i.e.f(0) =0. Further, suppose that the Jacobian matrixevaluated on the equilibriumx=0 presents a purelyimag…     

Simplest normal forms of generalized Neimark-Sacker bifurcation

The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.     

WAMS-based monitoring and control of Hopf bifurcations in multi-machine power systems

A method is proposed to monitor and control Hopf bifurcations in multi-machine power systems using the information from wide area measurement systems （WAMSs）. The power method （PM） is adopted to compute the pair of conjugate eigenvalues with the algebraically largest real part and the corresponding eigenvectors of the Jacobian matrix of a power system. The distance between the current equilibrium point and the Hopf bifurcation set can be monitored dynamically by computing the pair of conjugate eigenvalues. When the current equilibrium point is close to the Hopf bifurcation set, the approximate normal vector to the Hopf bifurcation set is computed and used as a direction to regulate control parameters to avoid a Hopf bifurcation in the power system described by differential algebraic equations （DAEs）. The validity of the proposed method is demonstrated by regulating the reactive power loads in a 14-bus power system.     

混流式水轮机调节系统非线性建模与分叉分析

基于Matlab仿真软件,考虑弹性水击对水轮机调节系统的影响,建立弹性水击水轮机微分代数模型,并与调速器和发电机共同构成水轮机调节系统非线性模型。在此基础上运用非线性动力学分叉理论对PID调速器参数进行Hopf分叉行为研究,得出PID参数三维分叉曲面、二维分叉曲线及稳定域,并通过对时域图、相轨迹图等的综合分析,得到取不同调速器参数时系统的非线性动力学特性,为机组PID参数选择提供理论依据。最后通过系统仿真得出,在小扰动下线性模型和非线性模型均能适用,在大扰动下非线性模型能更真实地反映机组动态特性。     

三维食物链的非线性振荡

利用中心流型定理和Lyapunov函数,证明了一个三维食物链的Hopf分支的存在性,从而可知在三维食物链系统中也存在非线性振荡。本文的方法显然也适合其它三维非线性系统。     

一类具有对称性的幂零向量场的高阶规范形

规范形是研究非线性向量场的动分岔问题强有力的工具,它包含了原系统在平衡点附近的所有动力学特性.对于一类具有γ对称的线性部分Jacobian矩阵为幂零矩阵的非线性向量场,在Ushiki规范形理论的基础上,利用无穷小形变的方法,得到了一维和二维的幂零向量场的具有γ对称的三阶、五阶规范形,并推导和证明了具有γ对称的且1-节退化的向量场的k阶规范形.     

非解析径向轴承支承转子系统非线性动力特性分析

Nomenclaturex,y,x ·, y·-displacement and velocity of rotor-x,-y,x-·,-y·-displacement and velocity of rotor , di mension less¨x,¨y-acceleration of rotor¨-x,¨-y-acceleration of rotor ,di mensionlessω-rotating speed of rotor-ω-rotating speed of rotor ,di mensionlessx,y,z-Cartesian coordinates-x,-y,-z-Cartesian coordinates ,di mensionless2m-mass of rotorg-acceleration of gravityG-weight of rotor ,di mensionlesse=e2x e2y-mass eccentricity of rotorex,ey-mass eccentricity of rotor in thexan…     

一个三维对称系统的二次Hopf分叉控制

详细分析一个三维对称系统——Langford系统的动态分叉特性,设计非线性状态反馈控制器实现对该系统二次Hopf分叉控制,推导出控制增益与分叉参数之间的解析关系式。利用数值模拟绘制控制前后系统的动力学运动轨迹,说明控制效果,验证理论解析解的正确性。     

Stochastic Bifurcation of Rectangular Thin Plate Vibration System Subjected to Axial Inplane Gaussian White Noise Excitation

A stochastic nonlinear dynamical model is proposed to describe the vibration of rectangular thin plate under axial inplane excitation considering the influence of random environment factors. Firstly, the model is simplified by applying the stochastic averaging method of quasi-nonintegrable Hamilton system. Secondly, the methods of Lyapunov exponent and boundary classification associated with diffusion process are utilized to analyze the stochastic stability of the trivial solution of the system. Thirdly, the stochastic Hopf bifurcation of the vibration model is explored according to the qualitative changes in stationary probability density of system response, showing that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simple way on the potential applications of stochastic stability and bifurcation analysis.     

Non-linear 2-DOF model and centre manifold theory to study limit cycle oscillations caused by drum-brake judder

This paper presents the research on the laws of systematic-parameter dependent variation in the vibration amplitude of drum-brake limit cycle oscillations (LCO). We established a two-degree non-linear dynamic model to describe the low-frequency vibration of the drum brake, applied the centre manifold theory to simplify the system, and obtained the LCO amplitude by calculating the normal form of the simplified system at the Hopf bifurcation point. It is indicated that when the friction coefficient is smaller than the friction coefficient at the bifurcation point, the amplitude decreases; whereas with a friction coefficient larger than the friction coefficient of bifurcation point, LCO occurs. The results suggest that it is applicable to suppress the LCO amplitude by changing systematic parameters, and thus improve the safety and ride comfort when applying brake. These findings can be applied to guiding the design of drum brakes.     

一类时滞竞争捕食系统模型的Hopf分支

研究了一类两捕食者一食饵的时滞竞争系统模型的Hopf分支。以食饵的成熟时滞为分支参数,通过分析系统模型相应特征方程根的分布,得到系统模型局部渐近稳定和产生局部Hopf分支的充分条件。最后,利用仿真实例对所得结果进行了验证。     

一类时滞两食饵一捕食者系统的周期解

对带有时滞的两食饵一捕食者系统进行研究.讨论了非负平衡点的性质,运用Hopf分岔方法,以时滞τ为参数给出了系统经历Hopf分岔的条件,得到了构成系统的三物种具有周期循环生物现象的结论.     

有限时滞Lienard方程Hopf分支的数值逼近

研究了一类有限时滞的Lienard方程Hopf分支的数值逼近问题．首先，以滞量为参数，应用Cooke和JHale的方法，得到Hopf分支存在的条件；然后，利用欧拉方法将得到的差分方程表示为映射，利用离散动力系统的分支理论，给出了差分方程Hopf分支存在的奈件和连续系统与其数值逼近间的关系．证明了当该系统在r=ro产生Hopf分支时，其数值逼近也在相应的参数rh处具有Hopf分支，并且忙rb=ro＋o（h）．     

一类具有非线性发生率的时滞病毒模型Hopf分支

研究了一类具有Beddington-DeAngelis非线性发生率的时滞病毒动力学模型。以时滞为参数,研究了模型的Hopf分支,得到时滞病毒模型的局部稳定性和产生局部Hopf分支的充分条件。最后,给出仿真实例,验证了理论分析结果的正确性。     

一个二维离散系统的分岔分析

应用动力系统的局部分支理论,研究一个二维离散动力系统当参数变化时产生的复杂动力学性质。我们应用中心流形定理和分岔理论证明了这个二维离散动力系统存在叉型分岔、倍周期分岔和Hopf分岔。     

一类食饵具有阶段结构的时滞捕食系统Hopf分支

以捕食者的妊娠时滞为参数,研究了一类食饵具有阶段结构和Beddington-DeAngelis功能性反应函数的时滞捕食系统。通过分析特征方程根的分布,讨论了系统正平衡点的局部稳定性和Hopf分支的存在性,并利用仿真实例证明了理论分析的正确性。     

一个混沌系统的霍普夫分叉分析

讨论了一个混沌系统的霍普夫分叉情况,并通过计算系统的第一李雅普诺夫系数判断了其分叉的方向,对相应的动力系统行为也作了简要的分析.     

Henon系统动力学行为的MATLAB仿真研究

利用MATLAB对Henon系统的动力学行为的演变进行仿真与分析.通过对时域图、相图、功率谱和分岔图的分析,表明Henon系统可通过阵发性途径走向混沌,其间歇性与霍夫分岔和倍周期分岔密切相关.     

用Excel绘制常微分方程的分支图

用Excel2003绘制了常微分方程Hopf分支,Poincare分支,多重闭轨的分支以及同宿轨线的分支结构图。     

非线性双圆盘转子-密封系统的数值分析

研究目的：求斛双圆盘转了-密封系统的非线性振动特性和运动响应 创新要点：采用有限元法（FEM）和拉格朗目方程求解双圆盘转子-密封系统,进而为研究多级转了系统的非线性振动问题提供有效方法。研究方法：基于有限元法（FEM）和拉格朗目方程。同时利用四阶龙格-库塔法求解系统动特性运动响应情况,利用风岔图、时间历程图、轴心轨迹图、庞加莱映射和幅值谱等分析图研究双圆盘转子-密封系统的非线性振动特性。重要结论：随着转速的增大,双圆盘转子-密封系统呈现丰富的非线性运动形式,包括周期性运动、多周期运动,准周期运动以及混沌运动。在右端圆盘不平衡质量小于34kg、密封间隙范围为0.376mm-0.54mm、密封长度大于0.13m或者密封压差高于0.104MPa的情况下均有利于提高双圆盘转子-密封系统的稳定性。