On the Number of Limit Cycles of Planar Quadratic Vector Fields with a Perturbed Center

We investigate the number of limit cycles of a planar quadratic vector field with a perturbed center-like singular point. An upper bound is obtained on the number of δ-good limit cycles of such a vector field (Theorem 1). Here δ is a parameter characterizing the limit cycles: it shows how far those cycles are from the singular points of the vector field and from the infinite points. The bound also includes another parameter, κ, characterizing the vector field. More precisely, κ gives an estimate on the distance from the vector field to the set consisting of quadratic vector fields with a line of singular points. Earlier, Ilyashenko and Llibre found a bound on the number of δ-good limit cycles of those vector fields which are sufficiently far from the fields with a center-like singular point. Theorem 1 and that bound complement each other and yield a new bound on the number of δ-good limit cycles of a quadratic vector field, regardless of its distance to the vector fields with a center-like singular point (Theorem 2). 1. Quadratic vector fields and their limit cycles A planar quadratic vector field is a vector field given by a system of differential equations of the form