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 Poincaré--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 Poincaré--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.

28 pages; 20 figures