Bifurcation of Limit Cycles from a Polynomial Degenerate Center

Using Melnikov functions at any order, we provide upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the degenerate center (x) over dot = -y((x(2) + y(2))/2)(m) and (y) over dot = x((x(2) + y(2))/2)(m) with m >= 1, when we perturb it inside the whole class of polynomial vector fields of degree n. The positive integers m and n are arbitrary. As far as we know there is only one paper that provide a similar result working with Melnikov functions at any order and perturbing the linear center (x) over dot = -y, (y) over dot = x.