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

Abstract

Given a quadratic system (QS) with a focus or a center at the origin we write it in the form ẋ = y + P 2(x, y) , ẏ = − x + dy + Q 2(x, y) where P 2 and Q 2 are homogeneous polynomials of degree 2. If we define F( x, y) = ( x − dy) P 2( x, y) + yQ 2( x, y) and g( x, y) = xQ 2( x, y) − yP 2( x, y) we give results of existence, nonexistence, and uniqueness of limit cycles if F( x, y) g( x, y) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two finite singularities.