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Bifurcation of hyperbolic planforms

Pascal Chossat 1, 2, * Grégory Faye 2 Olivier Faugeras 2
* Corresponding author
2 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
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 prob- lem 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 bi- furcate 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 octag- onal periodic pattern, where we are able to classify all possible H-planforms satis- fying the hypotheses of the Equivariant Branching Lemma. These patterns are, how- ever, 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.
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Pascal Chossat, Grégory Faye, Olivier Faugeras. Bifurcation of hyperbolic planforms. Journal of Nonlinear Science, Springer Verlag, 2011, 21 (4), pp.465-498. ⟨10.1007/s00332-010-9089-3⟩. ⟨hal-00807355⟩

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