具有三对特殊方向的一类平面齐五次系统的全局拓扑结构

Global Topological Structure of a Kind of Plane Homogeneous Fifth System with Three Special Direction

研究了一类平面齐五次系统dxdxdt=a50x5+a41x4y+a32x3y2+a23x2y3+a14xy4+a05y5,dydt=b50x5+b41x4y+b32x3y2+b23x2y3+b14xy4+b05y5当其只有唯一的有限远奇点且具有三对特殊方向时的全局拓扑结构及系数条件.假设系统只有唯一的有限远奇点(0,0),不妨设b50=0,其特殊方向由示性方程G(θ)=0给出,引进poincare变换研究无穷远奇点,再根据定理中的系数条件,列出系统所有可能的无穷远奇点和特殊方向,并判断其类型,由此画出系统具有三对特殊方向时的全局相图.

This paper studies the global topological structure of a kind of plane homogeneous fifth system with three special direction {dx/dt=a50x^t＋a41x^4y＋a32x^3y^2＋a23x^2y^3＋a14xy^4＋a05y^5,;dy/dt=b50x^5＋b41x^4y＋b32x^3y^2＋b23x^2y^3＋b14xy^4＋b05y^5 Suppose the system has are only one finite singular point （0,0）, then we can assume bs0 =0, which special directions are determined by equation G（O）= O, introduce Poincare transformation to discuss infinite singular points, according to the coefficient conditions, list all possible infinite singular points and special directions, judging their type, drawing out all kinds of phase portraits when it has three special direction.