Bifurcation of Hyperbolic Planforms

Abstract

Motivated by a model for the perception of textures by the visual cortex in primates, we analyze the bifurcation of periodic patterns for nonlinear equations describing the state of a system defined on the space of structure tensors, when these equations are further invariant with respect to the isometries of this space. We show that the problem reduces to a bifurcation problem in the hyperbolic plane D (Poincaré disc). We make use of the concept of a periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/Γ , where Γ is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows us to use the machinery of equivariant bifurcation theory. Solutions which generically bifurcate are called "H-planforms", by analogy with the "planforms" introduced for pattern formation in Euclidean space. This concept is applied to the case of an octagonal periodic pattern, where we are able to classify all possible H-planforms satisfying the hypotheses of the Equivariant Branching Lemma. These patterns are, however, not straightforward to compute, even numerically, and in the last section we describe a method for computation illustrated with a selection of images of octagonal H-planforms.Communicated by M. Golubitsky.

Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

“…In this section we recall some basic facts about the Poincaré disc and its isometries and we summarize results of [14] which will be useful in subsequent analysis.…”

“…Remark that for all µ, the symmetric state V = 0 is a solution of equation (2) and its uniqueness has been discussed in [18]. As was shown in [14], we are able to neglect the dependence on δ ∈ R + * as it does not play a significant role in the analysis that follows. Therefore Equation (2) is posed on the 2D hyperbolic surface D from now on.…”

“…In the following we shall denote the irreducible representations by their character: χ j is the representation with this character. The following lemma is proved in [14].…”

“…In a recent paper [14], a similar problem arose, but instead of being posed in the Euclidean plane, it was posed on the hyperbolic plane or, more conveniently, on the Poincaré disc. The problem originates from modeling the cortical perception of visual features, which we now briefly describe.…”

“…This approach to the problem was presented in [14] and an example was studied, namely the case where the group Γ corresponds to a tiling of D with regular octagons. This particular choice was initially motivated and explained in [13] where families of subgroups of U (1, 1) were identified to naturally arise from the retinal input to the hypercolumns in the visual area V1 such as the group Γ.…”