Complex Patterns in Oscillatory Systems

Complex Patterns in Oscillatory Systems Jessica Maral Conway 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.