Bifurcation of hyperbolic planforms

TL;DR

This paper investigates the bifurcation of periodic patterns called H-planforms on hyperbolic surfaces, motivated by visual cortex texture perception, and classifies solutions for octagonal patterns using equivariant bifurcation theory.

Contribution

It introduces the concept of H-planforms on hyperbolic spaces, reducing the bifurcation problem to a compact Riemann surface, and classifies solutions for octagonal patterns.

Findings

Classification of all possible H-planforms for octagonal patterns

Reduction of bifurcation problem to hyperbolic plane and Riemann surface

Development of a computational method for H-planforms

Abstract

Motivated by a model for the perception of textures by the visual cortex in primates, we analyse 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 (Poincar\'e disc). We make use of the concept of periodic lattice in D to further reduce the problem to one on a compact Riemann surface D/T, where T is a cocompact, torsion-free Fuchsian group. The knowledge of the symmetry group of this surface allows to carry out 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.