Limit cycles for a generalization of polynomial Liénard differential systems

We study the number of limit cycles of the polynomial differential systems of the formx˙=y-f1(x)y,y˙=-x-g2(x)-f2(x)y,where f1(x) = εf11(x) + ε2f12(x) + ε3f13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f2(x) = εf21(x) + ε2f22(x) + ε3f23(x) where f1i, f2i and g2i have degree l, n and m respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when f1(x) = 0 we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that this differential system can have bifurcating from the periodic orbits of the linear center x˙=y,y˙=-x using the averaging theory of third order.