一类三次系统极限环的存在唯一性

The Existence and Uniqueness of Limit Cycle for a Class of Cubic System

本文讨论了三次系统x=-y-x·y+a·y^2+a·x·y^2,y=x-δ·y+a·x^2+β·x^3-a·y^3的包围原点的极限环。得到了如下的结果:1) 若 a·δ≥0,在全平面上无极限环。2) 若 a·δ≤-3,不存在包围原点的极限环。3) 若-3<a·δ<0,|δ|(?)1,存在包围原点的极限环。4) 若-3<a·δ<0,α~2-3β≤0,至多存在一个包围原点的极限环。

The aim of this paper is to discuss the.limit cycle for the cubic system:x=-y-xy+ay^2+axy^2 y=x-δ·y+αx^2+βx^3-ay^3 (1) We obtain fotlowing results 1) If aδ≥0,(1) possesses no limit cycle.2) If aδ≤-3,(1) possesses no limit cycle surrounding the origin.3) If 0>aδ>-3,|δ|<<1,(1) possesses limit cycle surrounding the origin.4) If 0>aδ>-3,α~2-3β≤0,(1) possesses at most one limit cycle surrounding the origin.