Dihedral rings of patterns emerging from a Turing bifurcation

TL;DR

This paper investigates the formation of ring-like pattern configurations in reaction-diffusion systems near Turing bifurcations, revealing how strong interactions lead to dihedral pattern rings through analytical proofs and numerical methods.

Contribution

It provides the first analytical proof of strongly interacting ring-like patterns bifurcating from quiescence near Turing instabilities in two-component systems.

Findings

Existence of dihedral ring patterns near Turing bifurcation

Constructive methods for initial conditions in numerical continuation

Numerical validation of theoretical predictions

Abstract

Collective organisation of patterns into ring-like configurations has been well-studied when patterns are subject to either weak or semi-strong interactions. However, little is known numerically or analytically about their formation when the patterns are strongly interacting. We prove that approximate strongly interacting patterns can emerge in various ring-like dihedral configurations, bifurcating from quiescence near a Turing instability in generic two-component reaction-diffusion systems. The methods used are constructive and provide accurate initial conditions for numerical continuation methods to path-follow these ring-like patterns in parameter space. Our analysis is complemented by numerical investigations that illustrate our findings.