Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems (Q1821246)

Some results of existence, nonexistence and uniqueness of limit cycles for a quadratic system \[ \dot x=y+P_ 2(x,y),\quad \dot y=-x+dy+Q_ 2(x,y), \] where \(P_ 2\) and \(Q_ 2\) are homogeneous polynomials of degree 2 are given. The basic hypothesis is that the function F(x,y)\(\cdot g(x,y)\) does not change the sign, where \[ F(x,y)=(x-dy)P_ 2(x,y)+yQ_ 2(x,y),g(x,y)=xQ_ 2(x,y)-yP_ 2(x,y). \] By using these results, the authors improve some known results of the quadratic system with a unique finite singular point and of the bounded quadratic system.