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We study the excitation of spatial patterns by resonant, multi-frequencyforcing in systems undergoing a Hopf bifurcation to spatially homogeneousoscillations. Using weakly nonlinear analysis we show that for small amplitudesonly stripe or hexagon patterns are linearly stable, whereas square patternsand patterns involving more than three modes are unstable. In the case ofhexagon patterns up- and down-hexagons can be simultaneously stable. Thethird-order, weakly nonlinear analysis predicts stable square patterns andsuper-hexagons for larger amplitudes. Direct simulations show, however, that inthis regime the third-order weakly nonlinear analysis is insufficient, andthese patterns are, in fact unstable.

Pattern Selection in the Complex Ginzburg-Landau Equation with Multi-Resonant Forcing