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 $\delta$-good limit cycles of such a vector field (Theorem 1). Here $\delta$ 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, $\kappa$, characterizing the vector field. More precisely, $\kappa$ 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 $\delta$-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 $\delta$-good limit cycles of a quadratic vector field, regardless of its distance to the vector fields with a center-like singular point (Theorem 2).