Linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system

Perturbing the cubic polynomial differential systems ẋ=−y(a1x+a0)(b1y+b0), ẏ=x(a1x+a0)(b1y+b0) having a center at the origin inside the class of all polynomial differential systems of degree nn, we obtain using the averaging theory of second order that at most 17n+1517n+15 limit cycles can bifurcate from the periodic orbits of the center.