一类七次多项式系统高次奇点的极限环分支与拟等时中心

摘要

本文研究了一类七次多项式系统高次奇点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换和复变换将系统的高次奇点化为复域中的初等原点,然后求出了新系统在原点的前45个奇点量,从而导出了高次奇点为中心和最高阶细焦点的条件.在此基础上给出了七次系统在高次奇点分支出8个极限环的实例.最后通过一种新的算法求出高次奇点为中心时的周期常数,得到了高次奇点为拟等时中心的必要条件,并一一证明了这些条件的充分性.%Investigated in this paper are the center conditions, pseudo-isochronous center conditions and bifurcation of limit cycles at higher-order singular points for a class of septic system. Firstly, the higher-order singular point is transformed into the origin by a homeomorphic transformation and a complex transformation. Then the first 45 singular point quantities at the origin are calculated and the conditions for the higher-order singular point to be a center and the highest order fine focus are deduced as well. With these conclusions, a septic system which allows the appearance of 8 limit cycles in the neighborhood of higher-order singular points is constructed. Finally, a new algorithm is applied to find necessary conditions for pseudo-isochronous centers, and then the sufficiency of these conditions is proved.