Essential perturbations of polynomial vector fields with a period annulus

Abstract

Chicone-Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial diferential systems of arbitrary degree for which the expressions of its Poincaré-Liapunov constants are known, we give the structure of its k-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the diferential systems.