一类四次Kolmogorov系统的中心与极限环

The center and limit cycles of a quartic Kolmogorov system

为了研究一类四次Kolmogorov系统在正平衡点(1,1)处的中心和极限环分支问题,运用计算机代数软件Mathematica计算系统的前6阶奇点量,导出(1,1)成为系统中心的必要条件,并用分析方法和Gr9bner基方法证明了条件是充分的。证得系统在该平衡点处可分支出6个极限环。

The center and bifurcation of limit cycles at equilibrium point（1,1）for a class of quartic Kolmogorov are studied.By using computational algebra system Mathematica,the first six singular point values are computed and the necessary conditions of（1,1）to be a center are deduced.By the analytical method and Grobner basis method,the sufficiency of the conditions are proved.Finally,it is proved that the system can bifurcate 6 limit cycles in the equilibrium point.