# Spatial forcing of pattern-forming systems that lack inversion symmetry Academic Article

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### abstract

• The entrainment of periodic patterns to spatially periodic parametric forcing is studied. Using a weak nonlinear analysis of a simple pattern formation model we study the resonant responses of one-dimensional systems that lack inversion symmetry. Focusing on the first three $n:1$ resonances, in which the system adjusts its wavenumber to one $n\mathrm{th}$ of the forcing wavenumber, we delineate commonalities and differences among the resonances. Surprisingly, we find that all resonances show multiplicity of stable phase states, including the $1:1$ resonance. The phase states in the $2:1$ and $3:1$ resonances, however, differ from those in the $1:1$ resonance in remaining symmetric even when the inversion symmetry is broken. This is because of the existence of a discrete translation symmetry in the forced system. As a consequence, the $2:1$ and $3:1$ resonances show stationary phase fronts and patterns, whereas phase fronts within the $1:1$ resonance are propagating and phase patterns are transients. In addition, we find substantial differences between the $2:1$ resonance and the other two resonances. While the pattern forming instability in the $2:1$ resonance is supercritical, in the $1:1$ and $3:1$ resonances it is subcritical, and while the inversion asymmetry extends the ranges of resonant solutions in the $1:1$ and $3:1$ resonances, it has no effect on the $2:1$ resonance range. We conclude by discussing a few open questions.

### publication date

• August 8, 2014