一类三次系统的细中心与局部临界周期分支

Weak Centers and Local Bifurcation of Critical Periods for a Cubic System

研究了一类三次实自治系统细中心与局部临界周期分支问题.利用计算机代数系统Mathematica分别计算出了该实系统所对应的伴随复系统在三个中心条件下的复周期常数,得到原点为k(k=0,1,2,3)阶细中心的充分必要条件.得到了当原点为k(k=0,1,2,3)阶细中心时,该系统从原点能分支不多于k个局部临界周期分支.证明了该实系统恰有3个局部临界周期分支.

The weak centers and the local bifurcation of critical periods for a cubic real autonomous system were studied. Using the computer algebra system Mathematica, the complex period constants of the concomi- tant complex system for the real system were calculated respectively in the three central conditions. The suffi- cient and necessary conditions of kth-order （k = 0,1,2,3） weak center for the origin were obtained. There are no more than k local bifurcation of critical periods from the origin of the system when the origin is k th-or- der （k = 0,1,2,3） weak center. The conclusion is proved that the real system has exactly 3 local bifurcation of critical periods.