IMPROVEMENT OF MATRIX REPRESENTATION METHOD IN NORMAL FORM THEORY

In this paper, based on the invariant subspace theory and adjoint operator concept of linear operator, a new matrix representation method is proposed to calculate the normal forms of n order general nonlinear dynamic systems. In the method, there is no need to determine the structure of the class of normal forms in advance. Because the subspace is not related to the dimensions of the system and the order of the normal forms directly, it is determined only by a given vector field. So the normal forms with high orders and dimensions can be calculated by the method without difficulties. In this paper, is used the method for selecting the minimal subspace and solving homological equations in the subspace, the examples show that the method is very effective.

一种求解非线性振动系统渐近解的新方法——计算向量场Normal Form系数的简单方法

利用非线整变换,本文推导出了一种只需通过简单代数运算即可算出Hopf分叉Normal Form系数的简单方法,用这种方法求解非线性振动问题,只需把原方程变换成本文讨论的典则形式的一阶微分方程组,然后进行简单的代数运算即可得到原非线性振动方程的解,这种方法简单方便容易掌握。

基于Normal Form方法的UPFC多个功能控制器间交互影响分析

统一潮流控制器(UPFC)的重要特点之一就是它具有的多控制功能,包括潮流控制、电压控制和稳定性控制等等。然而时域仿真研究表明统一潮流控制器的多个控制器之间可能存在着较强的交互影响。本文研究了规范型(Normal Form)方法用于分析UPFC控制器间的交互影响问题的可能性,并引入一非线性交互指标定量地指示出控制器间的交互强弱程度。所提方法应用到新英格兰10机39节点系统,并通过时域仿真验证了所提规范型理论分析方法的有效性。

基于Normal Form理论的电力系统3阶解析解

在对电力系统动态方程进行泰勒级数展开的基础上,运用normal form理论推导出其3阶解析解。分别对单机无穷大系统和2区域4机系统进行了仿真,通过在时域和频域定义的两个近似度指标对比分析线性解析解,2阶解析解及3阶解析解在分析电力系统受扰后的动态特性精确性。并且应用prony算法对仿真结果进行定量分析,检测到了数值积分解中高阶复合模态的存在。从而,验证了推导的3阶解析解对受扰后系统进行分析的有效性。

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

The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.

基于Normal Form方法的电机系统分岔控制

研究了无刷直流电机系统等效非线性数学模型,分析了系统的Hopf分岔行为,并设计了washout滤波器,对系统产生的分岔行为进行控制。引入直接Normal Form(规范型)计算方法,求出系统的Hopf分岔规范式。通过规范式系数讨论控制参数的选择原则以及对Hopf分岔类型及极限环幅值的影响。理论和仿真结果表明:在施加控制器之前,系统发生亚临界Hopf分岔,极限环不稳定,系统进入混沌状态;在施加控制器之后,当受控系统规范式系数实部小于零时,系统发生超临界Hopf分岔,原系统不稳定极限环被控制为稳定极限环,从而抑制了混沌的产生,保证了电机运行性能的稳定性。

Computation of the Simplest Normal Forms for Resonant Double Hopf Bifurcations System Based on Lie Transform

The simplest normal form of resonant double Hopf bifurcation was studied based on Lie operator. The coefficients of the simplest normal forms of resonant double Hopf bifurcation and the nonlinear transformations in terms of the original system coefficients were given explicitly. The nonlinear transformations were used for reducing the lower- and higher-order normal forms, and the rank of system matrix was used to determine the coefficient of normal form which could be reduced. These make the gained normal form simpler than the traditional one. A general program was compiled with Mathematica. This program can compute the simplest normal form of resonant double Hopf bifurcation and the non-resonant form up to the 7th order.

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.

双频内共振系统的 Normal Form及其简化

讨论双频内共振系统的 Normal Form及其降维问题.利用发展的 Normal Form直接方法,导出了任意双频内共振系统 Normal Form的一般形式.指出 Poincare共振项分为内共振项和非内共振性两类,并定义了内共振项的阶.提出了一种普遍适用的降维变换,并证明了该变换可将任意双频内共振系统的 Normal Form方程降到3维.应用举例表明,该变换不仅适用于半单问题,也适于非半单问题（即强:1:1内共振系统）.

TO SOLVE MATRIX EQUATION sum (A^iXBi=c) BY THE SMITH NORMAL FORM

By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.

ADJOINT OPERATOR METHOD AND NORMAL FORMS OF HIGHER ORDER FOR NONLINEAR DYNAMICAL SYSTEM

Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z(2)-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.

COMPLEX INNER PRODUCT AVERAGING METHOD FOR CALCULATING NORMAL FORM OF ODE

This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.

一种基于NORMAL FORM变换的时间序列轨迹预测方法

随着基于GPS的功角测量技术在电力系统中的广泛应用,在线轨迹的预测成为了可能,其对于暂态稳定紧急控制具有重要意义。本文基于Normal Form分析,提出了一种复幂指数时间序列的预测方法;针对轨迹预测特点和各种预测模型对观测数据窗长短的要求不同,提出了一种变结构的思想。所提预测方法是基于Prony非线性拟合,Prony方法及其最近的研究显示了其在线应用的可能。轨迹预测过程和结果使用中国电科院8机36母线系统进行分析,其时域仿真结果验证了所提方法的有效性和可行性。

NORMAL FORM OF THE NONSEMI-SIMPLE BIFURCATION PROBLEM

On the basis of the method proposed in [1], the paper gives the method for finding the normal form of nonsemi-simple bifurcation problems. As an example, it analyses the normal form of a general nonlinear dynamical system with the nonsemi-simple double zero eigenvalues, and gives out the expression for the coefficients in the normal form by using those in the original system.

Local integrable differential systems and their normal forms

In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems. Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity. Finally we pose some open problems on this subject.

用Normal Form直接法研究壁板的热颤振与控制

建立了二维受热壁板在超音速气流中的颤振方程。运用分岔理论求得了系统的Hopf分岔点,应用Nor-mal Form直接法计算得到系统Hopf分岔Normal Form系数。引入wash-out滤波器技术对壁板热颤振进行了主动控制,延迟系统Hopf分岔的产生而不改变分岔类型。最后采用数值模拟验证了理论分析。

Using Normal Form of Matrices over Finite Fields to Construct Cartesian Authentication Codes

In this paper, one construction of Cartesian authentication codes from the normal form of matrices over finite fields are presented and its size parameters are computed. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the P I and P S , which denote the largest probabilities of a successful impersonation attack and of a successful substitution attack respectively, of these codes are also computed.

一类随机Normal-Form的研究与应用

利用计算非线性随机动力系统的随机Normal Form方法,研究了一类带噪声的vanderPol Duffing振动系统¨y+μ(y2±1) y-(α+σξt(ω))y+βy3=0的随机分岔问题.结合中心流形定理,借助计算机代数系统Mathematica编写了计算随机Normal Form的程序,得到了中心流形上的Normal Form,实现了系统的降维,从而为计算系统的最大Lyapunov指数并进而分析随机分岔提供了基础.

非半单分叉的Normal Form计算

在文［１］基础上，提出一种仅知道派生线性系统零实部特征值时求解非线性系统非半单分叉ＮｏｒｍａｌＦｏｒｍ的方法．通过适当的分类，将要求解的线性代数方程组分为若干相互独立的方程组．将所求系数向量按字典序列排列后，各独立方程组的系数矩阵是上三角矩阵．在非共振情形，各系数向量可按反字典序列递推求出．在共振情形，根据文中的二个定理，巧妙地由一简单的常数矩阵的最大秩子矩阵，定位其系数矩阵的满秩子矩阵，解决了这类方程组的降维简化．通过消元法，把简化后的方程化成类似于半单分叉ＮｏｒｍａｌＦｏｒｍ求解过程中方程的形式，其解法也类似．该方法非常易于在计算机代数软件平台上程序化．

Normal Forms of Symplectic Matrices

In this paper,we prove that for every symplectic matrix M possessing eigenvalues on the unit circle,there exists a symplectic matrix P such that P-1 MP is a symplectic matrix of the normal forms defined in this paper.