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 differential systems of arbitrary degree for which the expressions of its Poincare--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 differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.