Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg Reaction-Diffusion systems

TL;DR

This paper numerically investigates pattern-embedded-in-pattern structures in 2D reaction-diffusion systems, revealing snaking and non-snaking behaviors linked to bistability and energy differences, using bifurcation analysis and Ginzburg-Landau approximation.

Contribution

It provides the first detailed numerical analysis of embedded pattern branches and their snaking behavior in 2D reaction-diffusion systems, utilizing continuation methods and Ginzburg-Landau reduction.

Findings

Some branches exhibit snaking behavior in parameter space.

Non-snaking branches are associated with narrow bistable ranges.

Examples of fully localized patterns embedded in patterns are provided.

Abstract

For a Selkov--Schnakenberg model as a prototype reaction-diffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the Ginzburg-Landau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg-Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non-snaking behaviour appears to be related to too narrow bistable ranges with rather small Ginzburg-Landau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized atterns over patterns, for instance hexagon patches embedded in radial stripes, and fully localized hexagon patches over straight stripes.