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 The number of limit cycles from a quartic center by the Melnikov function of any order Xia Liu Keywords:Melnikov functions, bifurcation, limit cycles, generators. Abstract: In this paper, we consider cubic perturbations of the integral system $(1+x)^3dH$, where $H=(x^2+y^2)/2.$ Our main results show that the first four Melnikov functions associated to the perturbed system yield at most five limit cycles.