中文摘要：

反响的双 Hopf 分叉的最简单的正常形式基于谎言操作员被学习。反响的双 Hopfbifurcation 和以原来的系统，系数明确地被给的非线性的转变的最简单的正常形式的系数。非线性的转变被用于减少 lower-ordernormal 和 higher-ordernormal 形式，并且系统矩阵的等级被用来决定能被减少的正常形式的系数。这些使获得的正常形式比传统的简单。Ageneral 节目与 Mathematica 被编译。这个程序能计算反响的双 Hopf 分叉的最简单的正常形式和非反响的形式直到第 7 顺序。

英文摘要：

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.