Bifurcation of limit cycles from some uniform isochronous centers

© 2015 Watam Press. This article concerns with the weak 16'th Hilbert problem. More precisely, we consider the uniform isochronous centers (equation presented) for n = 2, 3, 4, and we perturb them by all homogeneous polynomial of degree 2, 3, 4, respectively. Using averaging theory of first order we prove that the maximum number N(n) of limit cycles that can bifurcate from the periodic orbits of the centers for n = 2, 3, under the mentioned perturbations, is 2. We prove that N(4) ≤ 2, but there is numerical evidence that N(4) = 2. Finally we conjecture that using averaging theory of first order N(n) = 2 for all n 1. Some computations have been made with the help of an algebraic manipulator as mathematica.