一类含有同宿轨的系统的Melnikov函数的零点

该文考虑了一类具有同宿轨的三次多项式系统对应的Melnikov函数的零点问题. 这一Melnikov函数可写为Abel积分线性组合的形式. 在推导出的Abel积分的Picard-Fuch方程与相关性质的基础上, 作者得到了Melnikov函数至多只有一个零点, 这表明围绕一个平衡点至多有一个极限环分岔出. 进一步作者还给出了分岔图, 即给出了围绕一个平衡点的Melnikov函数有一个零点的充分必要条件.

Zeros of the Melnikov function of a system with homoclinic orbits

The paper deals with the zeros of Melnikov function about one cubic system with homoclinic orbits. This Melnikov function can be rewritten as a linear combination of Abelian integrals. A Picard-Fuchs equation of Abelian integrals is derived and an induced equation is obtained to determine the properties of Abelian integrals. According to these properties, the Melnikov function has at most one zero point which means there is at most one limit cycle surrounding one singularity. Furthermore the sufficient and necessary condition for that the Melnikov function has exact one zero point is given.