多项式微分系统的极限环分支

LIMIT CYCLE BIFURCATIONS IN PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS

与Ｈｉｌｂｅｒｔ第十六问题相关联，本文讨论了平面多项式微分系统的极限环分支，将其分为四种类型，其中前两类与相平面上的某些奇点相关联，后两类则在相平面的一定区域内不与任何奇点相关联．文中的主要结论是：至今为止对一些具体多项式微分系统，特别是二、三次系统研究中所得到的极限环都与前两类极限环分支，即奇点分支与奇闭轨分支相关联，由此观点出发，在具体系统的研究中。

Associated with the Hilbert’s 16 th problem, the limit cycle bifurcations for planar polynomial systems are discussed, and are classified into four different types, among which the first two types are related to certain critical points on the phase plane, and the last two are not related to any critical point in a certain region of the phase plane. It is concluded in this paper that the limit cycles obtained for many concrete systems so far, in particular, for quadratic and cubic systems all can be considered as bifurcated from the bifurcation of the first two kinds, i.e., the bifurcations of critical points and singular closed orbit. From this point of view, it can be avioded from the bifurcations of the last two types which are more complicated to be studied.