具有四点异宿环和同宿环共存的平面五次多项式系统

The Planar Quintic Polynomial Systems with a Four-point Heteroclinic Cycle and a Homoclinic Cycle Coexist

给出一平面五次多项式微分系统存在五次代数不变曲线的条件.经分析,获得系统在一定条件下同时存在一个四点异宿环和一个同宿环(它们内部均只含一个焦点).进一步根据旋转向量场理论研究了它们各自分支出极限环的条件.

The conditions of one family of planar quintic polynomial differential system which possesses a quintic algebraic invariant curve are presented. By analysis, the system has a heteroclinic cycle connecting four saddle points and a homoclinic cycle under certain conditions （their inner unique sigular point is a focus, respectively）. Then the rotated vector field is constructed to bifurcate cycles from the homoclinic and heteroclinic cycles, respectively.