In this work we study the orbital stability/instability in the energy space of a specific family of periodic wave solutions of the general -model for all . This family of periodic solutions are orbiting around the origin in the corresponding phase portrait and, in the standing case, are related (in a proper sense) with the aperiodic Kink solution that connect the states with . In the traveling case, we prove the orbital instability in the whole energy space for all , while in the standing case we prove that, under some additional parity assumptions, these solutions are orbitally stable for all . Furthermore, as a by-product of our analysis, we are able to extend the main result in (de Loreno and Natali 2020 arXiv:2006.01305) (given for a different family of equations) to traveling wave solutions in the whole space, for all .