一类五次多项式系统的奇点量与极限环分支  

Singular Point Quantities and Bifurcations of Limit Cycles for a Class of Quintic Polynomial Systems

作　　者：黄文韬[1] 刘一戎[2] 唐清干[1] 

机构地区：[1]桂林电子工业学院计算科学与数学系,桂林541004 [2]浙江师范大学数学系,金华321004

出　　处：《数学物理学报（A辑）》2005年第5期744-752,共9页Acta Mathematica Scientia

基　　金：国家自然科学基金(10361003);广西教育厅科学基金(D200356)资助

摘　　要：该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题.该系统的原点是高次奇点,赤道环上没有实奇点.首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量,然后分别讨论了系统原点、无穷远点中心判据.给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例.这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.In this article, the bifurcations of limit cycles at the degenerate critical point and at infinity for a class of quintic polynomial system are investigated. In the system, the origin is degenerate critical point and the equator contains no real critical point. Firstly, algebraic recursive formulas for computing singular point quantities of the origin and infinity are derived respectively. The first five singular point quantities at the origin and first four singular point quantities at infinity for the system are given in order to get conditions of center and investigate bifurcations of limit cycles. At last, the authors construct a quintic system which allows the appearance of five limit cycles in the neighborhood of the origin and two limit cycles around infinity. As far as the authors know, this is the first time that the problem of limit cycles bifurcated from a degenerate singular point and from infinity under the synchronous perturbed conditions is investigated.

关 键 词：高次奇点 无穷远点 焦点量 奇点量 极限环分支 

分 类 号：O175.12[理学—数学]