Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

Abstract

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc $\mathcal {D}$ whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in $\mathcal {D}$, which is invariant under the action of a lattice subgroup Γ of U(1,1), the group of isometries of ${\mathcal{D}}$. In our case Γ generates a tiling of $\mathcal {D}$ with regular octagons. This problem was introduced as an example of spontaneous pattern formation in a model of image feature detection by the visual cortex where the features are assumed to be represented in the space of structure tensors. Under ‘generic’ assumptions the bifurcation problem reduces to an ODE which is invariant by an irreducible representation of the group of automorphisms $\mathcal {G}$ of the compact Riemann surface $\mathcal {D}/\varGamma$. The irreducible representations of $\mathcal {G}$ have dimensions one, two, three and four. The bifurcation diagrams for the representations of dimensions less than four have already been described and correspond to well-known group actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, thus completing the classification. In one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.