We develop the averaging theory at any order for computing the periodic solutions of discontinuous piecewise differential system of the form r = F + (θ, r, ε) if 0 ≤ θ ≤ α, F − (θ, r, ε) if α ≤ θ ≤ 2π, where F ± (θ, r, ε) = k i=1 ε i F ± i (θ, r) + ε k+1 R ± (θ, r, ε) with θ ∈ S 1 and r ∈ D, where D is an open interval of R + , and 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 formẋ = −y + xp(x, y),ẏ = 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.

Abstract. Let p be a uniform isochronous cubic polynomial center. We study the maximum number of small or medium limit cycles that bifurcate from p or from the periodic solutions surrounding p respectively, when they are perturbed, either inside the class of all continuous cubic polynomial differential systems, or inside the class of all discontinuous differential systems formed by two cubic differential systems separated by the straight line y = 0.In the case of continuous perturbations using the averaging theory of order 6 we show that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 3, and this number can be reached. For a subfamily of these systems using the averaging theory of first order we prove that at most 3 medium limit cycles can bifurcate from the periodic solutions surrounding p, and this number can be reached.In the case of discontinuous perturbations using the averaging theory of order 6 we prove that the maximum number of small limit cycles that can appear in a Hopf bifurcation at p is 5, and this number can be reached. For a subfamily of these systems using the averaging method of first order we show that the maximum number of medium limit cycles that can bifurcate from the periodic solutions surrounding p is 7, and this number can be reached.We also provide all the first integrals and the phase portraits in the Poincaré disc for the uniform isochronous cubic centers.

Abstract. In this paper we classify the global phase portraits in the Poincaré disc of all quartic polynomial differential systems with a uniform isochronous center at the origin.

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