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。