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Localized States in Driven Dissipative Systems with Time-Periodic Modulation


The generalized Swift--Hohenberg equation is used to study the persistence and decay of localized patterns in the presence of time-periodic parametric forcing in one and two dimensions. A localized state that was stable with constant forcing may begin to breathe under periodic forcing, i.e. grow for part of the forcing cycle via nucleation of new wavelengths of the pattern followed by wavelength annihilation during another part of the cycle. The breathing dynamics occur as the forcing parameter exits the region of stability of the localized pattern on either side and the fronts that define the edges of the state temporarily depin. The parameters of the forcing determine if there will be net growth, a balance, or net decay on average.

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 resonances generate distinct regions in parameter space characterized by the net number of wavelengths gained or lost in one forcing cycle. Canard trajectories, in which the localized state follows an unstable solution branch for some amount of time before quickly jumping to a stable one, appear near the transitions between each region. In one dimension, the partitioning of the parameter space is well described by an asymptotic theory based on the wavelength nucleation/annihilation time near the boundaries of the region of stability. This theory leads to predictions that are qualitatively correct and, in some cases, provide quantitative agreement with numerical simulations.

The underlying resonance mechanism is a more general phenomenon and is also studied in the context of coupled oscillator systems with a periodically modulated Adler equation as a simple model. A strikingly similar partitioning of the parameter space is observed, with the resonances occurring this time between the period of the frequency modulation and the time for the generation of a phase slip.

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