  The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearity
is used to study the persistence and decay of localized patterns in the
presence of time-periodic parametric forcing. A novel resonance phenomenon
between the forcing period and the time required to nucleate one wavelength of
the pattern outside the pinning region is identified. The resonance generates
distinct regions in parameter space characterized by the net number of
wavelengths gained or lost in one forcing cycle. These regions are well
described by an asymptotic theory based on the wavelength
nucleation/annihilation time near the boundaries of the pinning region. The
resulting theory leads to predictions that are qualitatively correct and, in
some cases, provide quantitative agreement with numerical simulations.
