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 ẋ = −y((x + y)/2) and ẏ = x((x + y)/2) 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 ẋ = −y, ẏ = x. ∗The authors are partially supported by a MCYT/FEDER grant number MTM2008-00694 and by a CIRIT grant number 2009SGR 381 †The author is partially supported by a MCYT/FEDER grant number MTM2008-03437 and by a CIRIT grant number 2009SGR 410