Abstract

In this talk we deal with a planar system of type $x'=-yG(x,y), y'=xG(x,y), '=d/dt,$ where the set of critical points ${G(x,y)=0}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. The aim of this work is to study the maximum number of limit cycles that can bifurcate from the associated period annulus when a polynomial perturbation of degree $n$ is considered. This bound is given in terms of the number $K$ of critical straight lines and the degree $n$ of the perturbation. It is based on the explicit computation of the Abelian integrals governing the bifurcation and on a new result concerning estimates on the number of zeroes of a particular family of real functions. Previous results for $K le 4$ from other authors will be briefly revisited.

Joint work with Armengol Gasull and Joan Torregrosa (Dept. Matemàtiques UAB)