﻿ 具有幂零奇点三维系统的二次规范型及普适开折 Quadratic Normal Forms and Universal Unfoldings of Three-Dimensional Systems with Nilpotent Singular Points

Vol.05 No.04(2016), Article ID:18978,9 pages
10.12677/AAM.2016.54073

Quadratic Normal Forms and Universal Unfoldings of Three-Dimensional Systems with Nilpotent Singular Points

Dandan Xie1,2, Nana Zhang1,2

1School of Mathematics and Statistics, Shandong Normal University, Jinan Shandong

2School of Science, Linyi University, Linyi Shandong

Received: Oct. 26th, 2016; accepted: Nov. 8th, 2016; published: Nov. 18th, 2016

ABSTRACT

In this paper, for three-dimensional dynamic systems, we discuss and analyze three classes of equations which have nilpotent singular points. The normal forms of quadratic polynomials are obtained by normal formal theory. Then, the universal unfoldings are obtained by using coordinate translation.

Keywords:Three-Dimensional System, Nilpotent Singular Point, Quadratic Polynomial, Normal Form, Universal Unfolding

1山东师范大学数学与统计学院，山东 济南

2临沂大学理学院，山东 临沂

1. 引言

2. 规范型的基本理论

， (2.1)

， (2.2)

， (2.3)

。 (2.4)

1) 如果，则存在，使。这时(2.4)中不再出现二次项。

2) 如果，则对适当的，因此可按中的基底

， (2.5)

。 (2.6)

3. 具有幂零奇点三维系统的二次规范型及普适开折

， (3.1)

3.1. 情况一

。 (3.2)

。 (3.3)

3.2. 情况二

， (3.4)

。 (3.5)

3.3. 情况三

，(3.6)

。 (3.7)

Quadratic Normal Forms and Universal Unfoldings of Three-Dimensional Systems with Nilpotent Singular Points[J]. 应用数学进展, 2016, 05(04): 630-638. http://dx.doi.org/10.12677/AAM.2016.54073

1. 1. 罗定军, 张祥, 董梅芳. 动力系统的定性与分支理论[M]. 北京: 科学出版社, 2001.

2. 2. Wiggins, S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd Edition, Springer-Verlag, New York.

3. 3. 张锦炎, 冯贝叶. 常微分方程几何理论与分支问题[M]. 北京: 北京大学出版社, 2000.

4. 4. 傅希林, 范进军. 非线性微分方程[M]. 北京: 科学出版社, 2011.