一类有十一个赤道环的七次多项式系统

A Class of Seven Degree Polynomial System With Eleven Limit Cycles at the Equator

研究了一类七次系统无穷远点的中心条件与赤道极限环分支问题.通过将实系统转化为复系统研究,给出了计算无穷远点奇点量的递推公式,并在计算机上用Mathematica推导出该系统无穷远点前13个奇点量,进一步导出了无穷远点成为中心的条件和十三阶细焦点的条件,在此基础上首次得到了七次系统无穷远点分支出11个极限环的一个实例.

In this paper, center conditions and bifurcation of limit cycles from the equator in a class of polynomial system of degree seven are studied. By converting real planar system into complex system, the recursion formula for the computation of singular point quantities of the infinity are given, and, with computer algebraic system Mathematica, the first 13 singular point quantities of the infinity are deduced. At the same time, the conditions for the infinity to be a center and 13 degree fine focus are derived respectively. A system of degree seven that bifurcates 11 limit cycles from the infinity is constructed at the first time.