Affiliation:  ^{[1]} Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trabalhador Sao Carlense 400, BR13566590 Sao Carlos, SP  Brazil
^{[2]} Univ Autonoma Barcelona, Dept Matemat, E08193 Barcelona, Catalonia  Spain
^{[3]} Univ Estadual Campinas, Dept Matemat, Rua Sergio Buarque Holanda 651, BR13083859 Campinas, SP  Brazil
Total Affiliations: 3

We develop the averaging theory at any order for computing the periodic solutions of periodic discontinuous piecewise differential system of the form dr/d theta = r' = [ F+(theta, r, epsilon) if 0 <= theta <= alpha, F(theta, r, epsilon) if alpha <= theta <= 2 pi, where F+/(theta, r, epsilon) = Sigma(k)(i=1) epsilon(i) Fi(+/) (theta, r) + epsilon(k+1) R+/(theta, r, epsilon) with theta is an element of S1 and r is an element of D, where D is an open interval of R+, and epsilon is a small real parameter. Applying this theory, we provide lower bounds for the maximum number of limit cycles that bifurcate from the origin of quartic polynomial differential systems of the form <(x)over dot> = y + xp(x, y), <(y)over dot> = x + yp(x, y), with p(x, y) a polynomial of degree 3 without constant term, when they are perturbed, either inside the class of all continuous quartic polynomial differential systems, or inside the class of all discontinuous piecewise quartic polynomial differential systems with two zones separated by the straight line y = 0. (AU) 