Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers

Let pp be a uniform isochronous cubic polynomial center. We study the maximum number of small or medium limit cycles that bifurcate from pp or from the periodic solutions surrounding pp 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=0y=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 pp 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 pp, 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 pp 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 pp 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.