Bifurcation of limit cycles from a quintic center via the second order averaging method

This paper is concerned with the bifurcation of limit cycles from a quintic system with one center. By using the averaging theory, we show that under any small quintic homogeneous perturbations, up to order 1 in ε, at most three limit cycles bifurcate from periodic orbits of the considered system, and this upper bound can be reached. Up to order 2 in ε, at most seven limit cycles emerge from periodic orbits of the unperturbed one.