三次系统存在三叶玫瑰分界线环的充要条件  

A Necessary and Sufficient Condition of the Existence of Trifoliate Rose Separatrix Cycle in Cubic System

作　　者：沈伯骞[1] 谭欣欣[2] 

机构地区：[1]辽宁师范大学数学系,辽宁大连116029 [2]大连大学数学系,辽宁大连116622

出　　处：《辽宁师范大学学报（自然科学版）》1999年第3期177-181,共5页Journal of Liaoning Normal University:Natural Science Edition

摘　　要：证明了具有三叶玫瑰曲线解的三次系统的一般形状形为ｘ·＝ ｋ１（３ｘ＋ ｘ２－ ｙ２－ ４ｘ３－ ４ｘｙ２）＋ ｋ２（３ｘ２－ ３ｘ３－ ５ｘｙ２）＋ ｋ３（１８ｘｙ＋ ３２ｘ２ｙ）＋ ｋ４（６ｘｙ＋ ４ｘ２ｙ＋ ４ｙ３）ｙ·＝ ｋ１（３ｙ－ ２ｘｙ－ ４ｘ２ｙ－ ４ｙ３）＋ ｋ２（３ｘｙ－ ５ｘ２ｙ＋ ３ｙ３）＋ ｋ３（３ｘ２＋ ９ｙ２－ ８ｘ３＋ ２４ｘｙ２）＋ ｋ４（３ｘ２－ ３ｙ２－ ４ｘ３－ ４ｘｙ２）并证明了此系统存在三叶玫瑰分界线环的充要条件为ｋ１＝ ０ 与（ｋ３－ １１５ ｋ４）２＋ １１５ｋ２２＜ １６２２５ｋ２４ 同时成立．We proved that the general form of a quadratic system with trifoliate rose solution is as follows: x·=k 1(3x+x 2-y 2-4x 3-4xy 2)+k 2(3x 2-3x 3-5xy 2) +k 3(18xy-32x 2y)+k 4(6xy+4x 2y+4y 3) y·=k 1(3y-2xy-4x 2y-4y 3)+k 2(3xy-5x 2y+3y 3) +k 3(3x 2+9y 2-8x 3+24xy 2)+k 4(3x 2-3y 2-4x 3-4xy 2) we also proved that the necessary and sufficient conditions of existence of the trifoliate rose separatrix cycle in this system is that both (k 3-115k 4) 2+115k 2 2<16225k 2 4 and k 1=0 hold simaltaneously.

关 键 词：三次系统 分界线环 三叶玫瑰曲线 充要条件 

分 类 号：O175.12[理学—数学]