以三次曲线xy^2=ax^3+cx+d为解的三次系统

The Cubic System With Solution xy^2=ax^3+cx+d

本文讨论了以三次曲线xy^2=ax^3+cx+d为解的三次系统,给出了这类系统的一般形式,我们证明了当xy^2=ax^3+cx+d没有闭分支时,以其为解的三次系统不存在极限环;当xy^2=ax^3+cx+d存在闭分支时,以其为解的三次系统可以以该闭分支为极限环,同时我们也给出了闭分支为唯一极限环和不存在极限环的充分条件。

In this paper, we discuss the system with the solution xy^2=ax^3+cx+d,give out the general of this system, prove that this system possesses no limit cycles when xy^2=ax^3+cx+d has no closed branch. We also prove that the closed branch of the curve may be the limit cycle of this system, give out the sufficient conditions of non-existence of limit cycles and sufficient condition so fclos branch being a unique limit cycle.