Approximate localised dihedral patterns near a Turing instability

Abstract

Fully localised patterns involving cellular hexagons or squares have been found experimentally and numerically in various continuum models. However, there is currently no mathematical theory for the emergence of these localised cellular patterns from a quiescent state. A key issue is that standard techniques for one-dimensional patterns have proven insufficient for understanding localisation in higher dimensions. In this work, we present a comprehensive approach to this problem by using techniques developed in the study of axisymmetric patterns. Our analysis covers localised patterns equipped with a wide range of dihedral symmetries, avoiding a restriction to solutions on a predetermined lattice. The context in this paper is a theory for the emergence of such patterns near a Turing instability for a general class of planar reaction-diffusion equations. Posing the reaction-diffusion system in polar coordinates, we carry out a finite-mode Fourier decomposition in the angular variable to yield a large system of coupled radial ordinary differential equations. We then utilise various radial spatial dynamics methods, such as invariant manifolds, rescaling charts, and normal form analysis, leading to an algebraic matching condition for localised patterns to exist in the finite-mode reduction. This algebraic matching condition is nontrivial, which we solve via a combination of by-hand calculations and Gröbner bases from polynomial algebra to reveal the existence of a plethora of localised dihedral patterns. These results capture the essence of the emergent localised hexagonal patterns witnessed in experiments. Moreover, we combine computer-assisted analysis and a Newton-Kantorovich procedure to prove the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions. This includes the localised hexagon patches that have been elusive to analytical treatment.

Figures (15)

• Figure 1: (a) Examples of localised planar patterns possessing different dihedral symmetries: including squares ($\mathbb{D}_{4}$, top-left), spots (radial, top-right), hexagons ($\mathbb{D}_{6}$, centre), and rhomboids ($\mathbb{D}_{2}$, bottom-right). Light yellow regions indicate peaks, while dark blue regions indicate depressions. (b) A localised $\mathbb{D}_{6}$ patch observed in the von Hardenberg model for dryland vegetation vonHardenberg2001, where dark regions indicate peaks in vegetation density and light regions indicate depressions. Here, solutions are found in the bistability region between uniform solutions and domain-covering hexagons, for parameters seen in Gowda2014. (c) An illustration of the geometry between the core and far-field manifolds $\mathcal{W}^{cu}_{-}(\mu)$ and $\mathcal{W}^{s}_{+}(\mu)$, respectively, for (left) $\mu=0$ and (right) $0<\mu\ll1$. (d) Localised $\mathbb{D}_6$ patches (blue) are found to bifurcate off the lower branch of a localised radial spot (black) for \ref{['e:SH']} when $\gamma=1$. Panels ($d_1$)-($d_3$) show surface plots of solutions at various points (red circles) on the curve. (e) For different choices of truncation order $N$, solutions bifurcate from the flat state along unique curves in parameter space which coincide as $\mu$ increases.
• Figure 2: Solutions when $N=1$, given by Theorem \ref{['thm:SmallPatch']}, for $(R)$hombic $\sim \mathbb{D}_{2}$, $(T)$riangular $\sim \mathbb{D}_{3}$, $(S)$quare $\sim \mathbb{D}_{4}$, and $(H)$exagonal $\sim \mathbb{D}_{6}$ patte rns are plotted in a circular disc of radius $20$ circumscribing the square region. Each panel presents a contour plot of the asymptotic solution $u(r,\theta)$ of \ref{['eqn:R-D;Galerk']} as $\mu\to0$. Light regions indicate peaks, while dark regions indicate depressions.
• Figure 3: Solutions when $N=2$, given by Theorem \ref{['thm:SmallPatch']}, are plotted in a circular disc of radius $30$ circumscribing the square region. Up to a half-period rotation, there are two solutions for $(R)$hombic $\sim \mathbb{D}_{2}$, $(T)$riangular $\sim \mathbb{D}_{3}$, $(S)$quare $\sim \mathbb{D}_{4}$, and $(H)$exagonal $\sim \mathbb{D}_{6}$ patterns.
• Figure 4: Solutions when $N=3$, given by Theorem \ref{['thm:SmallPatch']}, for $(R)$hombic $\sim \mathbb{D}_{2}$, $(T)$riangular $\sim \mathbb{D}_{3}$, $(S)$quare $\sim \mathbb{D}_{4}$, $(P)$entagonal $\sim \mathbb{D}_{5}$, and $(H)$exagonal $\sim \mathbb{D}_{6}$ patterns are plotted in a circular disc of radius $40$ circumscribing the square region.
• Figure 5: Solutions when $N=4$, given by Theorem \ref{['thm:SmallPatch']}, for $(H)$exagonal $\sim\mathbb{D}_{6}$ patterns are plotted in a circular disc of radius $50$ circumscribing the square region.

Theorems & Definitions (46)

• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Lemma 4.1
• proof
• Remark 4.2
• Remark 4.3
• Lemma 4.4
• proof