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

TL;DR

This paper investigates how multi-resonant forcing influences pattern formation in systems near a Hopf bifurcation, revealing stability conditions for various patterns through analysis and simulations.

Contribution

It provides a weakly nonlinear analysis of pattern stability under multi-frequency forcing, highlighting limitations of third-order approximations.

Findings

Stripe and hexagon patterns are linearly stable at small amplitudes.

Hexagon patterns can be simultaneously stable in both up and down configurations.

Higher amplitude regimes show that some predicted patterns are unstable in simulations.

Abstract

We study the excitation of spatial patterns by resonant, multi-frequency forcing in systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. Using weakly nonlinear analysis we show that for small amplitudes only stripe or hexagon patterns are linearly stable, whereas square patterns and patterns involving more than three modes are unstable. In the case of hexagon patterns up- and down-hexagons can be simultaneously stable. The third-order, weakly nonlinear analysis predicts stable square patterns and super-hexagons for larger amplitudes. Direct simulations show, however, that in this regime the third-order weakly nonlinear analysis is insufficient, and these patterns are, in fact unstable.