On the limit cycles bifurcating from the periodic orbits of a Hamiltonian system

Abstract

This paper concerns the weak 16th Hilbert problem and considers the Hamiltonian center ẋ=−y2n−1, ẏ=x2n−1, and we perturb it by all polynomials of degree 2n−1 for n=2,3,4,5,6,7,8. We prove that the maximum number of limit cycles that can bifurcate from the periodic orbits of this center for n=2,3,4,5,6,7,8, under the mentioned perturbations and using the averaging theory of first order, is 1,4,3,2,5,6,7, respectively.