一类三次系统的中心焦点判定及极限环的唯一性

The decision of center and focus and uniqueness oflimit cycles of a cubic system

研究三次系统dx/dt=-y+δx+a2xy+a5xy2,dy/dt=x(1+by2),给出系统的中心焦点判定,证明焦点量W0W1≥0时,O(0,0)外围不存在极限环;W0W1<0时,O(0,0)外围至多一个极限环.

To study the decision of center and focus and the uniqueness of limit cycles for a class of cubit system dx/dt=-y+δx+a2xy+a5xy2, dy/dt=x(1+by2) . It is proved that if focal quantities W0W1≥0, the system has no limit cycles;if W0W1<0, the system has at most one limit cycle.