On the number of limit cycles of a perturbed cubic polynomial differential center

For ε sufficiently small, we consider the planar polynomial differential system x ̇ = − y ( y 2 + A x 2 + B x + C ) + ε P ( x , y ) , y ̇ = x ( y 2 + A x 2 + B x + C ) + ε Q ( x , y ) , where P ( x , y ) and Q ( x , y ) are polynomials of degree n . By using the averaging method of first order, we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system.