一类七次系统三次幂零奇点的焦点判定与极限环分支

FOCUS DETERMINATION AND LIMIT CYCLE BIFURCATION FOR A KIND OF THE THREE-ORDER NILPOTENT SINGULAR POINT IN THE SEVEN-ORDER SYSTEM

对一类原点为三次幂零奇点的七次微分系统,利用已经计算出的该七次微分系统原点的前10个拟Lyapunov常数,经过分析推导,从而得出原点成为10阶细焦点的充要条件,在此条件的基础上对系统作微小扰动,进而得出在原点充分小的邻域内恰有10个包围原点的极限环的结论。

For a kind of the seven-order differential system, whose origin is the three-order nilpotent singular point, this paper makes use of the first ten quasi-Lyapunov constants from the calculated to probe into the sufficient conditions by which the origin becomes the fine focus of the ten-order. Based on the sufficient conditions, when the system is slightly perturbed, ten limit cycles are derived, which are in the neighborhood of the origin.