Spatial forcing of pattern-forming systems that lack inversion symmetry.

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 nth 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.