Birth of limit cycles for a class of continuous and discontinuous differential systems in (d + 2)-dimension

The orbits of the reversible differential system , , , with and , are periodic with the exception of the equilibrium points (0, 0, z1,… , zd). We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system , , , using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree n, and second inside the class of all discontinuous piecewise polynomial differential systems of degree n with two pieces, one in y > 0 and the other in y < 0. In the first case, this maximum number is nd(n − 1)/2, and in the second, it is nd + 1.313237250COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPESFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP88881.030454/2013-012012/18780–0; 2012/23591–1; 2013/21078–