Transversal conics and the existence of limit cycles

Abstract

This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincar\'e--Bendixson regions by using transversal conics. We present several examples of known systems in the literature showing different features about limit cycles: hyperbolicity, Hopf bifurcation, sky-blue bifurcation, rotated vector fields, \ldots for which the obtained Poincar\'e--Bendixson region allows to locate the limit cycles. Our method gives bounds for the bifurcation values of parametrical families of planar vector fields and intervals of existence of limit cycles.