Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x'=-yF(x, y), y'=xF(x, y), where the set {F(x, y)=0} consists of k different isolated points, is defined by k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n. Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1, the provided upper bound is reached. Finally, the case k=2 is also treated. © 2013 Elsevier Masson SAS.