On the finite cyclicity of open period annuli

Abstract

Let $\Pi$ be an open, relatively compact period annulus of real analytic vector field $X_0$ on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from $\Pi$ under a given multiparameter analytic deformation $X_{\lambda }$ of $X_0$ is finite provided that $X_0$ is either a Hamiltonian or generic Darbouxian vector field