一类三次系统含单奇点的极限环

A class cubical system of limit cycle of include singular point

证明三次系统x.=y-εy3,y.=x(1-x2)+(α-x2)y,ε>0,当0<1-α<<1时,在区域y<1ε内含单奇点的极限环的存在性与唯一性.根据Hopf分支定理,证明了当0<1-α<<1时,存在含单奇点的极限环,再由唯一性定理证明了当0<1-α<<1时,含单奇点的极限环的唯一性.

For cubic system x=y-εy^3,y=x（1-x^2）＋（α-x^2）y,the existence and uniqueness of limit cycles is proved in ｜y｜〈1/√ε,when 0〈1-α〈〈1 and wher ε〉0.According to Hopf bifuration theorem, the existence of limit cycles is proved, when 0 〈 1 -α 〈〈 1. And by the uniqueness theorem, the uniqueness of limit cycles is proved, when 0 〈 1 - α〈〈 1.