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The generalized Swift--Hohenberg equation with a quadratic-cubic nonlinearityis used to study the persistence and decay of localized patterns in thepresence of time-periodic parametric forcing. A novel resonance phenomenonbetween the forcing period and the time required to nucleate one wavelength ofthe pattern outside the pinning region is identified. The resonance generatesdistinct regions in parameter space characterized by the net number ofwavelengths gained or lost in one forcing cycle. These regions are welldescribed by an asymptotic theory based on the wavelengthnucleation/annihilation time near the boundaries of the pinning region. Theresulting theory leads to predictions that are qualitatively correct and, insome cases, provide quantitative agreement with numerical simulations.

A new resonance mechanism in the Swift--Hohenberg equation with time-periodic forcing