一类带全局中心的三次系统的定性分析

Qualitative analysis of a class of cubic systems with global center

研究了一类E13系统的有限远奇点和无穷远奇点的定性性态,原点O(0,0)为全局中心时的所有可能的全局结构,以及系统产生Hopf分支的充分条件.运用定性理论中的Poincare形式级数法和Hopf分支理论研究了系统的Hopf分支,利用Frmmer方法和Briot-Bouquet变换研究无穷远奇点的性态,利用Poincare变换研究了原点O(0,0)为全局中心时的所有可能的全局结构.得到几个关于有限远奇点和无穷远奇点的定性性态的定理,得到关于原点O(0,0)为全局中心的充要条件,给出了原点O(0,0)为全局中心时的所有可能的全局结构图,得到系统产生Hopf分支的一个定理.对系统的定性分析得到了完整的结论.

The qualitative behavior of all the singular points at finity and infinity of a class of E3^1 system are studied, all the possible global structures of a class of E3^1 system are obtained when O（0,0） is a global center, and a sufficient condition for the system to generate the Hopf bifurcation is given. By using the Poincare series method and Hopf bifurcation theory of the qualitative theory, the Hopf bifurcation of the system is studied. By using the Frommer method and Briot-Bouquet transformation, the qualitative behavior of the singular pionts at infinity are studied. When O（0,0） is a global center, all the possible global structures of the system are obtained by using the transformation. Several theorems on the qualitative behavior of the finity and infinite singular points of the system are obtained. A sufficient and necessary of the conditon is provided for O（0,0） is a global center. The graphs of all the possible global structures system bifurcation is are given when O（0,0） is a global center. One theorem on the system to generate the Hopf obtained . The intact conclusion on the qualitative analysis of the system are obtained.