Using an analytical approach, we derive an
explicit formula for the subharmonic Mel'nikov potential ${\rm
L}^{^{{\p}/{\q}}}$ for perturbations of twist maps. Our method
based on the integrability of map and the variational approach of
twist map. If ${\rm L}^{^{{\p}/{\q}}}$ is non--constant the
perturbed twist map is non--integrable and all the resonant curves
are destroyed for $\abs{\varepsilon}\ll 1$. We also apply our
result to show the existence of such subharmonic bifurcations for
a mapping representing localized oscillatory solutions of a
discrete NLS equation with conservative and dissipative
perturbations.