Limit Cycles Bifurcating from the Period Annulus of Quasi-Homogeneous Centers

We provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous center, which can be obtained using the Abelian integral method of first order. We show that these bounds are the best possible using the Abelian integral method of first order. We note that these centers are in general non-Hamiltonian. As a consequence of our study we provide the biggest known number of limit cycles surrounding a unique singular point in terms of the degree n of the system for arbitrary large n.