一类含参分叉系统最简规范形系数的计算

Computation of the simplest normal form a bifurcation system with parameters

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Hop f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M athem atica,推导出最一般情况下含一个参数的非共振双Hop f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。

The simplest normal form of the nonresonant double Hopf bifurcation with parameters are obtained without truncating the original differential equation, i.e. the perturbation parameter of the system is treated as one-dimensional state variable. The computation of the simplest normal form is not only based on near-identity nonlinear transformations, but also on time and parameter rescaling. The expressions of the simplest normal forms from order two to order five for nonresonant double Hopf Bifurcation with parameters are obtained with the help of computer algebra language Mathematica from which the coefficients of the simplest normal form of this system up to any high order are deduced.