Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

This paper completes the classification of bifurcation diagrams for H-planforms in the Poincaré disc D whose fundamental domain is a regular octagon. An H-planform is a steady solution of a PDE or integro-differential equation in D, which is invariant under the action of a lattice subgroup Γ of U(1, 1), the group of isometries of D. In our case Γ generates a tiling of D with regular octagons. This problem was introduced in [15] as an example of spontaneous pattern formation in a model of image features 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 G of the compact Riemann surface D/Γ. The irreducible representations of G have dimension one, two, three and four. The bifurcation diagrams for the representations of dimension less than four have been obtained in [15] and correspond to already well known goup actions. In the present work we compute the bifurcation diagrams for the remaining three irreducible representations of dimension four, henceforth completing this classification. An interesting finding is that in one of these cases, there is generic bifurcation of a heteroclinic network connecting equilibria with two different orbit types.