Simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli

We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system (x',y')= (-y((x-1)² + y²),x((x-1)² + y²) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously. This problem is an generalization of PerTor2014 to the piecewise systems class. When the polynomial perturbation has degree n, we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree than the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. But, when the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three type of configurations: (9,3), (6,6) and (4,8). In the non-piecewise scenario only 5 limit cycles were found