将参数视为状态变量，在不截断的情况下，研究了非共振含参双Hopf分叉系统的最简规范形。在采用非线性恒同变换时引入了变时间尺度函数及变参数尺度函数两个变换函数，借助于计算机代数语言Mathematica，推导出最一般情况下含一个参数的非共振双Hopf分叉系统的最简规范形的前五阶系数的表达式，并根据其中的规律推导出该系统高阶最简规范形的通式。

主要在传统规范形的基础上，研究了非共振双Hopf分叉系统的最简规范形。通过对矩阵理论和近恒同变换的应用，详细分析了当n=3和5时，双Hopf分叉系统的最科规范形，得出当n≥5时，传统规范形可以进一步科化，得到系统的最科规范形。最后根据分析和计算的结果，在计算机语言Mathematica的帮助下，发现在非共振双Hopf分叉系统的n（n>5）阶最简规范形议程中，只存在一项k（5<k≤n，k为奇数）阶齐次单项式。

提出了锥形模短芯棒钢管拉拔的非线性动力学模型，利用非线性动力系统理论研究了钢管拉拔过程中抖纹产生的机理分析了拉拔系统工艺参数对霍普夫分岔临界点即钢管产生抖纹的临界点的影响，以及造成芯棒产生稳定极限环振荡的原因，发现钢管内表面的干摩擦力是芯棒产生自激振荡的根本原因，当拉拔速度到达临界值时即诱发芯棒的振动；由于芯棒振动的幅值与芯杆长度成正比，使高道次拉拔产生大幅振动采用新型空心芯棒向管坯内表面补充润滑剂，改善内表面的润滑状态，可从根本上避免钢管在拉拔过程中产生抖纹。

在定常空气动力的作用下，对含立方非线性刚度的二元机翼颤振系统的极限环颤振进行了研究。应用中心流形理论将原四维系统降为二维，采用后继函数法对分岔点类别进行了定性的分析，从而确定平衡点的性质，并应用范式理论对分岔点处中心流形约化方程进行化简，进而研究了系统参数对极限环颤振的稳定性以及幅值的影响。

efficient method for calculating the normal form and the associated non-linear transformations for the semi-simple case without central manifold reduction is given in this paper. The one-step transformation concept is adopted for easy programming. This method can be used to calculate high order normal forms of high-dimensional ordinary differential equations of non-linear oscillators. Aprog ram in MATHEMATICA language is designed to perform the calculation. Three examples are given in order to verify the method and to show the efficiency of the program.

An efficient method to calculate the normal form and the associated nonlinear transfolmations for the semi-simple case is given in this paper. The one step transforma-tion concept is adopted to make the approach very easy to be programmed. An intelligent judgement is used to simplify the tedious calculation. This method can be used to calculate high order normal form (without limitation, up to the capacity of the computer) of high dimensional (until dimension 9) ordinary differential equations of the nonlinear oscillators. A program in Mathematica language is designed to perform the calculation. Six examples are given in order to verify the method and to show the efficiency of the program.

A general four-dimensional normal form of a double Hopf bifurcation is considered. As a particular case, the normal form of a forced (non-autonomous) non-linear oscillator having two frequencies, namely the linear natural frequency and the excitation frequency, is studied in detail. When these two frequencies form two purely imaginary sub-blocks of order two in the real Jordan block, the system constitutes a double Hopf bifurcation. In this paper, the normal form of the double Hopf bifurcation is reduced when the two frequencies are not in resonance. In order to use the method of normal form, the non-autonomous problem is transformed into an autonomous one by a generalized co-ordinate transformation. The method of undetermined coeticients is used to "nd the double Hopf bifurcation normal form. The coeticients of similar monomials rather than similar powers of e are compared to get the normal form to various orders. The steady state periodic solutions and the bifurcation equations of the forced non-linear vibration system in the case of non-resonant are studied. A Mathematica program is designed to" nd the normal form. Three examples are given to use the Mathematica program and to compare them with the existing results.

Calculation of the higher order averaging equations of a non-linear oscillator is very tedious using the classical averaging method. This is also true for higher order normal forms. This paper presents an alternative method, which is a combination of the method of normal form and the classical averaging method. A simple and efficient program is given to calculate the higher order averaging equations by using the symbolic computer algebra system Mathematica. Furthermore, the program can be used to calculate the higher order coefficients of normal form. Four examples are given and compare well with the existing results

利用接近恒同的非线性变换，计算出了非共振双Hopf分叉系统规范形和系数。利用广义坐标变换，将非共振单自由度非线性强迫振动系统变换为双Hopf分叉系统，用规范形理论给出了一种计算该类系统定常解及分叉特性的方法。