Superlattice Patterns in the Complex Ginzburg-Landau Equation with Multi-Resonant Forcing

Abstract

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 that systematically captures weak forcing functions with a spectrum consisting of frequencies close to the 1:1-, the 1:2-, and the 1:3-resonance. By slowly modulating the amplitude of the 1:2-forcing component we render the bifurcation to subharmonic patterns supercritical despite the quadratic interaction introduced by the 1:3-forcing. 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, which are similar to quasi-patterns with 4-fold and 5-fold rotational symmetry, respectively. Using direct simulations of the extended complex Ginzburg-Landau equation 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.

22 pages, 18 figures (38 total plots and pictures). For associated .mov files, see http://www.esam.northwestern.edu/~riecke/research/ModCGL/Movies

References in corpus (17)

• The World of the Complex Ginzburg-Landau Equation
• Phase Diagram of the Two-Dimensional Complex Ginzburg-Landau Equation
• Pattern formation in 2-frequency forced parametric waves
• Soft quasicrystals - Why are they stable?
• Resonance tongues and patterns in periodically forced reaction-diffusion systems
• Grain boundary pinning and glassy dynamics in stripe phases
• Ordering kinetics of stripe patterns
• Multi-frequency control of Faraday wave patterns
• Design of parametrically forced patterns and quasipatterns
• Broken symmetries and pattern formation in two-frequency forced Faraday waves
• Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves
• Convergence properties of the 8, 10 and 12 mode representations of quasipatterns
• Resonant triad dynamics in weakly damped Faraday waves with two-frequency forcing
• Quasipatterns in parametrically forced systems
• Sideband Instabilities and Defects of Quasipatterns
• Geometric Diagnostics of Complex Patterns: Spiral Defect Chaos
• Pattern Selection in the Complex Ginzburg-Landau Equation with Multi-Resonant Forcing