一类在无穷远点分支出十个极限环的多项式微分系统

Polynomial Differential System with Ten Limit Cycles at Infinity

运用一种间接的方法研究了一类七次系统在无穷远点的中心条件和极限环分支问题.首先通过变换将原系统在无穷远点的极限环分支问题转化到在原点来研究,从而计算出该系统在原点的前98个奇点量,推导出原点成为中心和最高细焦点的条件,最后构造出在原点(即无穷远点)充分小的领域内分支出10个极限环的实例,首次证明了七次多项式系统在无穷远点能分支出10个极限环.

An indirect method is used to study bifurcations of limit cycles at infinity for a class of seventh-order polynomial differential system. First, the problem for bifurcations of limit cycles in the system at infinity is transformed into that at the origin. By the computation of fist 98 singular quantities, the conditions of the origin （ correspondingly, infinity） to be the highest degree fine focus are derived. Finally, the system that bifurcates nine limit cycles in the neighborhood of infinity is constructed, which is proved that ten limit cycles can bifurcated at infinity for a class of seven-order polynomial system firstly.