一类三次多项式系统的定性分析  

Qualitative Analysis for a Class of Cubic Polynomial Differential System

作　　者：蒋自国[1] 

机构地区：[1]阿坝师范高等专科学校数学与财经系,四川汶川623002

出　　处：《四川理工学院学报（自然科学版）》2013年第1期71-75,共5页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：四川省教育厅自然科学基金项目(12ZB168);阿坝师专重点科研课题项目(ASA11-27;ASA12-22)

摘　　要：研究一类具有二实不变直线的三次多项式微分系统x'=y(1-x2),y'=-x+δy+nx2+mxy+ly2+bxy2,分析了奇点的性态,并运用形式级数法对原点O进行了中心-焦点判定。利用旋转向量场的理论和Bendixson判据得出了系统不存在极限环的充分条件,利用Hopf分支问题的Liapunov第二方法得到了该系统极限环存在性和稳定性的若干充分条件。A class of cubic polynomial differential system with two real invariant line x＇=y（1-x^2）,y＇=-x＋δy＋nx^2＋mxy＋ly^2＋bxy^2 is studied. The character of the critical point is studied and center-focus to critical point O is judged by form series＇ method. Using the theory of a rotating vector field and Bendixson＇s theory, the sufficient conditions for no-existence of limit cycles is obtained. By using the second method of Liapunov for Hopf bifurcation problem, some sufficient conditions for the existence and stability of limit cycle of the system is obtained.

关 键 词：多项式系统 奇点 极限环 存在性 唯一性 

分 类 号：O175.12[理学—数学]