Motivated by the rich variety of complex patterns observed on the surface of fluid layers that are vibrated at multiple frequencies, we investigate the effect of such resonant forcing on systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. We use an extension of the complex Ginzburg-Landau equation (CGLE) that systematically captures weak forcing functions with a spectrum consisting of frequencies close to the 1:1-, 2:1-, and 3:1-resonance. We first examine the case where the multi-resonant forcing is unmodulated in time. Our third-order, weakly nonlinear analysis shows that for small amplitudes only stripe patterns or hexagons (up and down) are linearly stable; for larger amplitudes rectangles and super-hexagons (or super-triangles) may become stable. The larger-amplitude super-hexagons arise in a transcritical bifurcation because of the quadratic interaction introduced by the 3:1-forcing, and are linearly stable only on the upper branch. Numerical simulations show, however, that in the latter regime the third-order analysis is insufficient: super-hexagons are unstable. Instead large-amplitude hexagons can arise and be bistable with the weakly nonlinear hexagons.

By slowly modulating the amplitude of the 2:1-forcing component we render the bifurcation to subharmonic patterns subcritical despite the quadratic interaction introduced by the 3:1-forcing, so there is always a range in which the weakly nonlinear analysis with cubic truncation is valid. Our weakly nonlinear analysis shows that quite generally the forcing function can be tuned such that resonant triad interactions with weakly damped modes stabilize subharmonic patterns comprised of four or five Fourier modes, similar to quasi-patterns exhibiting elements with 4-fold and 5-fold rotational symmetry, respectively. Using direct simulations of the extended CGLE we confirm our weakly nonlinear analysis. In simulations domains of different complex patterns compete with each other on a slow time scale. As expected from energy arguments, with increasing strength of the triad interaction the more complex patterns eventually win out against the simpler patterns. We characterize these ordering dynamics using the spectral entropy of the patterns. For system parameters reported for experiments on the oscillatory Belousov-Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns.