一类四次多项式系统原点的中心条件与极限环分支

Conditions of the origin to be a center and bifurcation of limit cycles for a quartic polynomial system

讨论一类四次多项式微分系统的中心条件与极限环分支问题。通过对该实系统所对应的伴随复系统奇点量的计算,得到系统的原点成为中心的必要条件,并对它的充分性进行严格的证明。从奇点量导出焦点量,得到了原点成为8阶细焦点的条件,最后证明该系统从在原点邻域有8个小振幅极限环。这是首次得到四次系统在细焦点可分支出8个极限环。

The bifurcation of limit cycles and conditions of origin to be a center for a biquadratic polynomial system is investigated. By the computation of the singular point values for the concomitant complex system of the real sys- tem, the necessary conditions of origin of system to be a center is obtained, and the sufficiency for the conditions is strictly proven. The focal values are derived from of the singular points, and the conditions that the origin to be an 8 order weak focal is obtained. Finally, it is proved that this system has small amplitude limit cycles in the neigh- borhood of the origin. This is the first time that an example of a biquadratic system with eight limit cycles bifurcated from a weak focal is given.