2009 PIMS-CSC Seminar - 02

Superlattice patterns and quasipatterns, while well-studied in waves on the surface of vertically vibrated viscous fluids (Faraday waves), have found little attention in forced oscillatory systems. We study such patterns, comprised of 4 or more Fourier modes at different orientation, by applying multi-frequency forcing to systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. For weak forcing composed of 3 frequencies near the 1:2- and 1:3-resonance, such systems can be described by a suitably extended complex Ginzburg-Landau equation with time periodic coefficients. Using Floquet theory and weakly nonlinear analysis we obtain the amplitude equations for simple patterns (comprised of 1, 2, or 3 modes) and superlattice patterns. By judicious choice of the forcing function we stabilize these patterns via spatiotemporal resonance and find stable subharmonic 4- and 5-mode patterns. We confirm our analysis through numerical simulation.