某一类平面上E_3动力系统的全局结构

本文证明了Ｅ３系统ｄｘ／ｄｔ＝Ｐ３（ｘ，ｙ）ｄｙ／ｄｔ＝ｘ有的定曲线Ｆ（ｘ，ｙ）＝ｘ～２＋ｙ～２－１＝０为特解当且仅当 它具有形式： ｄｘ／ｄｔ＝ａ１ｘ＋ａ２ｙ－ａ１ｘ～３－（１＋ａ２）ｘ～２ｙ－ａ１ｘｙ～２－（１＋ａ２）ｙ３，ｄｙ／ｄｔ＝ｘ，进而讨论了系统的所有 有限远奇点和无限远奇点的性质，并且利川Ｄｕｌａｃ函数证明了ｘ～２＋ｙ～２＝１是它唯一可能的极限环。 最后我们得到由图（１）－（１０）说明的所有可能的全局结构．

THE GLOBAL STRUCTURE OF A CERTAIN CLASS OF PLANAR DYNAMICAL SYSTEM E_3

we investigated the global structure of dx/dt= P(x, y )dy/dt= x(1)with a parti- cular solution F (x, y)=x~2+y~2-1=0, and obtained that system (1) has an algebraic curve(2)as its solutain, if it can be written in the form of dx/dt=a1x+ a2y-a1x~3-(1+a2)x~2y-a1xy~2-(1+a2)y~3,dy/dt=x(3). Furtherly, we analysed all the finite and infinite singular points of syst- em(3), and showed. by means of Dulac's function, that system(3)has no limit cycles except curve(2). Then, the possible globle structures of the system could be obtained, as shown in figures(1)-(10).