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Dihedral rings of patterns emerging from a Turing bifurcation

Dan J. Hill, Jason J. Bramburger, David J. B. Lloyd

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.

Dihedral rings of patterns emerging from a Turing bifurcation

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.
Paper Structure (12 sections, 10 theorems, 154 equations, 11 figures)

This paper contains 12 sections, 10 theorems, 154 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof of Theorem \ref{['thm:Ring']}
  4. The Core Manifold
  5. The Far-Field Manifold
  6. The Rescaling and Transition Charts
  7. Matching Core and Far Field
  8. Discussion
  9. Proof of Lemma \ref{['Lemma:Ntup;normal']}
  10. Properties of the Cubic Vector Function
  11. Properties of the Jacobian
  12. Symmetries of the Cubic Matching Equation

Key Result

Theorem 2.2

Assume Hypotheses R-D:hyp;1 and R-D:hyp;2. Fix $m,N \in \mathbb{N}$ and assume the constants $\{a_{n}\}_{n=0}^{N}$ are solutions of the nonlinear matching condition for each $n = 0,1,\dots, N$ and $-N \leq i,j,k \leq N$. Then, there exist constants $q_0,\mu_0,r_0,r_1 > 0$ such that the Galerkin system eqn:R-D;Galerk has a radially localised solution of the form for each $\mu\in(0,\mu_{0})$, $n\i

Figures (11)

  • Figure 1: Comparisons between axisymmetric (top) and dihedral (bottom) spots and rings in the Swift--Hohenberg equation \ref{['e:SH']} with $\gamma=1.6$. Spot solutions (left) have a global maximum at the origin, while ring solutions (right) have global extrema away from the origin, as can be observed in the radial profiles of each pattern provided in red.
  • Figure 2: A plethora of numerically identified dihedral ring solutions to the Swift--Hohenberg equation \ref{['e:SH']} with $\gamma = 1.6$. Dihedral group symmetries of each pattern are indicated by $\mathbb{D}_m$ as an inset on each panel. Here we adopt the notation $\mathbb{D}_\infty$ to denote an axisymmetric ring pattern since the radial symmetry endows it with invariance with respect to every dihedral symmetry.
  • Figure 3: Projected solutions $\langle\hat{U}_{0}^{*},\mathbf{u}\rangle_2$ when $N=1$, given by Theorem \ref{['thm:Ring']} for (R)hombic $(\mathbb{D}_{2})$ and (T)riangular $(\mathbb{D}_{3})$ rings are computed in a circular disc of radius 20 circumscribing the square region, with $\mu=0.08$.
  • Figure 4: Projected solutions $\langle\hat{U}_{0}^{*},\mathbf{u}\rangle_2$ when $N=2$, given by Theorem \ref{['thm:Ring']} for (R)hombic $(\mathbb{D}_{2})$ rings are computed in a circular disc of radius 25 circumscribing the square region, with $\mu=0.08$.
  • Figure 5: Projected solutions $\langle\hat{U}_{0}^{*},\mathbf{u}\rangle_2$ when $N=3$, given by Theorem \ref{['thm:Ring']} for (R)hombic $(\mathbb{D}_{2})$ and (T)riangular $(\mathbb{D}_{3})$ rings are computed in a circular disc of radius 30 circumscribing the square region, with $\mu=0.08$.
  • ...and 6 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1: hill2022approximate, Lemma 4.1
  • Remark 3.2
  • Lemma 3.3
  • Remark 3.4
  • Lemma 3.5: vandenberg2015Rigorous, Theorem 1.1
  • Lemma 3.6
  • proof
  • ...and 10 more