一类具有细鞍点的二次系统的同缩轨道的存在性和它产生的极限环

Existence of the Homoclinic Loop for a Quadratic Systems and Limit Cycle Appearing Near It

本文给出二次系统存在过一个细鞍点的同缩轨道(即过一个细鞍点的分界线环,记为S_N^((1)))的参数条件,并证明微扰参数在S_N^((1))的内侧邻域至少产生二个极限环。

The quadratic systems with a weak saddle and a focus (or center) maybe changed intox= -y-mx + 1x2 +mxy+y2 =P(x, y),(1)y = x(l + ax + by) = Q(x,y), 1+b>0, |m|<2.Without loss of genearlly, we assume a≥0. In this paper, we obtain that if systems ( 1 ) satisfies the following conditions(A)m(b+21)<0 (B)0<a<(1+1) (C)1≤1 or a-41(l+b)<0Then, there exists m*, such thatthe system ( 1 ) has a SN(1) where the SN(1) passes through the weak saddle and surroundes the singular point 0 as m = m*. Then, we consider the systemx= -y- (m+λ)x + lx2 +mxy + y2, .y = x(l+ax + by), (l+b)>0, |m|<2.we obtain that if the system (l)λ satisfies conditons (A), (B) and (D) 1<min {1, l+b}, Then the system (1)λ=0,m-m* has a SN(1)(homoclinic loop), From the bifurcation point(λ, m) = (0, m*), exists a perturbation having at least 2 limit cycles appearing near the SN(1).