Tristability between stripes, up-, and down-hexagons and snaking bifurcation branches of spatial connections between up- and down-hexagons

TL;DR

This paper extends amplitude equation analysis to fifth order to identify tristable and bistable regimes among stripes and hexagons in pattern-forming systems, revealing complex bifurcation structures and stationary front connections.

Contribution

It introduces fifth order amplitude equations for better prediction of multistability and bifurcations in hexagonal pattern systems, advancing beyond third order models.

Findings

Identification of tristable and bistable parameter ranges.

Discovery of stationary front connections between hexagon states.

Complex bifurcation structures in generalized Swift-Hohenberg systems.

Abstract

Third order amplitude equations on hexagonal lattices can be used for predicting the existence and stability of stripes, up- and down-hexagons in pattern forming systems. These amplitude equations predict the nonexistence of bistable ranges between up- and down-hexagons and tristable ranges between stripes, up- and down-hexagons. In the present work we use fifth order amplitude equations for finding such bistable and tristable ranges for a generalized Swift-Hohenberg equation and discuss stationary front connections between up- and down-hexagons.