一类具有13个参数的7次系统的极限环分支

Bifurcation of Limit Cycles in a Septic System with 13 Parameters

研究了一类具有幂零奇点的7次多项式微分系统的极限环分支与中心问题.借助于数学软件MATHEMATICA,推导出系统在原点的前14个拟Lyapunov常数,从而得到了系统的原点为中心的充要条件,证明了系统在3阶幂零奇点处可以分支出14个极限环,给出了7次李雅谱诺夫系统在3阶幂零奇点处的环性数的下界.

In this paper, the bifurcation of limit cycles and the center conditions of a class of septic polynomial differential systems with nilpotent singular points are investigated. With the help of the mathematical software MATHEMATICA, the first 14 quasi-Lyapunov constants at the origin of the system are deduced. As a result, necessary and sufficient conditions for the origin of the system to be a center are obtained. The result that there exist 14 limit cycles created from the three-order nilpotent singular point is also proved. Moreover, a lower bound of cyclicity of three-order nilpotent singular point for septic Lyapunov systems is given. s