Abstract

We give an upper bound for the number of zeros of an Abelian integral. This integral controls the number of limit cycles that bifurcate, by a polynomial perturbation of arbitrary degree $n$, from the periodic orbits of the integrable system $(1+x)dH=0$, where $H$ is the quasi-homogeneous Hamiltonian $H(x,y)=x2k/(2k)+y2/2$. The tools used in our proofs are the Argument Principle applied to a suitable complex extension of the Abelian integral and some techniques in real analysis.

This is a joint work with Armengol Gasull and Changjian Liu.