Uniqueness of Limit Cycles for Quadratic Vector Fields

Abstract

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x^{\prime }=a_{1}x-y-a_3{x}^2+(2a_2+a_5)xy+a_6{y}^2$, $y^{\prime }=x+a_{1}y+a_2{x}^2+(2a_3+a_4)xy-a_2{y}^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1,a_2,\dots ,a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.

24 pages