A special class of periodic orbits of parameter dependent families of truncated resonant normal forms is constructed. Their existence is shown for arbitrarily large periods. Explicit analytical criteria are derived for constructing the complete basin of attraction of an invariant circle for normal forms for which the eigenvalue of the linearized mapping is not a multiple root of unity. Persistence of these basins is shown for small perturbations of the parameters. Hence, the existence of bounded orbits for all times established, generically.