On the number of limit cycles for a generalization of liénard polynomial differential systems

We study the number of limit cycles of the polynomial differential systems of the form $$\dot x = y - g-1(x), \quad \dot y = -x-g-2(x)-f(x) y,$$ where g1(x) = εg11(x) + ε2g12(x) + ε3g13(x), g2(x) = εg21(x) + ε2g22(x) + ε3g23(x) and f(x) = εf1(x) + ε2 f2(x) + ε3 f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center = y, = -x using the averaging theory of third order. © 2013 World Scientific Publishing Company.