Pseudo-Abelian integrals along Darboux cycles

The aim of this seminar is to present some new results about the analytic properties of pseudo-Abelian integrals. These integrals naturally appear in the study of phase portrait of plane polynomial vector field. They represent the linearization of Poincare return map. Thus, zeros of these integrals correspond to limit cycles bifurcating from the center. More precisely, we consider a polynomial perturbation of an integrable, non-Hamiltonian system with first integral of Darboux type. The linear part (in perturbation parameter) of the Poincare return map is given by pseudo-Abelian integral. This integral satisfies certain variation relation. Using this relation one can prove that the number of zeros of pseudo-Abelian integral is locally uniformly bounded under generic hypothesis.