We study localized states in the Swift–Hohenberg equation when time-periodic parametric forcing is introduced:
$$u_t= r u-\left(1+\partial_{x}^2\right)^2u+bu^2-u^3,\qquad r=r_0+\rho\sin 2\pi t/T,\qquad b=1.8.$$
The presence of a time-dependent forcing introduces a new characteristic time which creates a series of resonances with the depinning time of the fronts bounding the localized pattern.
The resulting dynamics partition the parameter space by the net number of wavelengths of the pattern gained or lost during the course of a forcing cycle.
Asymptotic analysis can provide insight into and, in some cases, quantitative predictions about the sweet-spot and pinching structure that is generated.

For more details, see

SIAM Journal on Applied Dynamical Systems 2015 14:2, 860–892
Author Institutional Affiliation | University of California, Berkeley; Imperial College, London |

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Keywords | Swift–Hohenberg Equation, Parametric Forcing, Localized Structures |