On The Stability of Double Homoclinic Loops
Consider a planar system of the form(x)=f(x,y)(y)=g(x,y) (1)in which f,g ∈C∞ and satisfy f(-x,-y) =-f(x,y), g(-x,-y)=-g(x,y). Suppose that system (1.1) has a double-homoclinic loop L with a saddle point O at the origin. We can denote it by L=L1∪L2∪O, where Li(i=1,2) are homoclinic loops. We will suppose L is isolated, that is , there is no periodic orbit in a neighborhood of L.As we know, if there exists a neighborhood U of L such that ω(A)=L(α(A) = L) for any points A in the intersection of U and the exterior of L, then L is said to be out stable(unstable). Ifω(A) = L1(α(A)= L1) for any points A in the intersection of U and the interior of L1, then L1 is said to be inner stable (unstable).It is well known that for (1), if C1=(fx + gy)(0)＜0(＞0) then L is out stable(unstable) while L1 and L2 are inner stable(unstable). Thus, in the following we will suppose C1 = 0. Also, for definiteness, we assume that L is oriented clockwise. For any natural number n≥2, there exists a C∞ coordinate change (x y) → T(x,y) = (y1 y2) with T(-x, -y)= -T(x,y) such that system (1) locally becomes the C∞ system(y)1=∑ni=1aiy1(y1y2)i-1+y1(y1y2)nR1=Y1(y1,y2),(y)2=∑ni=1biy2(y1y2)i-1+y2(y1y2)nR2=Y2(y1,y2), (2)where b1 = -a1＜0, R1,R2 ∈C∞ and Yi(-y1,-y2) =-Yi(y1, y2),i=1,2. From (2), we havedy2/dy1=y2/y1-1+n∑i=2Ci(y1y2)i-1+(y1y2)nR, (3)where R ∈ C∞, and Ci are constants. It is easy to see that if ai + bi = 0 for i = 2,…,k,k ＜ n, then Ci = 0 for i = 2,…,k, and Ck+1 = (ak+1 + bk+1)/a1.If (e)(f,g)/(e)(x,y)(0,0)=(0 a1 a1 0),then we have the formulae of C2 given byC2=1/2a1fxxx-fxyy+gxxy-gyyy+1/a1(fxy(fyy-fxx)+gxy(gyy-gxx)-fxxgxx+fyygyy)|(0,0)
Maoan Han Yuhai Wu
Department of Mathematics, Shanghai Jiao Tong University Shanghai 200030, P . R . China
国际会议
上海
英文
50-51
2003-11-09（万方平台首次上网日期，不代表论文的发表时间）