In the first part of this talk we present a joint work with Armengol Gasull (UAB, Spain). The results assure the existence of many periodic solutions for a second order cubic periodic equation. The tools used are the Malkin bifurcation theorem and the Fonda-Sabatini-Zanolin theorem. The latter assures the existence of periodic solutions for systems with a Hamiltonian structure (in dimension 2), without a bifurcation function or a nondegeneracy condition. In the second part of the talk we present a generalization of the nondegeneracy condition that appears in the Malkin bifurcation theorem.