Journal of Nonlinear Science

, Volume 21, Issue 4, pp 465–498 | Cite as

Bifurcation of Hyperbolic Planforms

  • Pascal ChossatEmail author
  • Grégory Faye
  • Olivier Faugeras
Open Access


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 \(\mathcal {D}\) (Poincaré disc). We make use of the concept of a periodic lattice in \(\mathcal {D}\) to further reduce the problem to one on a compact Riemann surface \(\mathcal {D}/\varGamma\), 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.


Equivariant bifurcation analysis Neural fields Poincaré disc Periodic lattices Hyperbolic planforms Irreducible representations Laplace–Beltrami operator 

Mathematics Subject Classification (2000)

35B36 37G40 45G15 58C40 58J55 92B20 92C20 


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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Pascal Chossat
    • 1
    • 2
    • 3
    Email author
  • Grégory Faye
    • 1
    • 2
  • Olivier Faugeras
    • 1
    • 2
  1. 1.NeuroMathComp LaboratoryINRIASophia AntipolisFrance
  2. 2.CNRSENSParisFrance
  3. 3.UMR CNRS 6621University of Nice Sophia-AntipolisNice Cedex 02France

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