具有一对共轭复不变直线的三次系统的中心判定问题

Center Determination Problem for a Class of Cubic System with a Pair of Invariant Conjugate Imaginary Straight Lines

对于一类具有一对共轭复不变直线和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是其前五阶焦点量全为零.此中心条件是通过不变代数曲线构造积分因子或对称原理得以证明.

A class of cubic systems with a pair of invariant conjugate imaginary straight lines and a center-focus type singular point, is proved to have a center at the origin if and only if the first five focal values vanish. The presence of a center at the origin is proved by constructing integrating factor formed from invariant algebraic curves or by symmetry principle.