Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

Faye, Grégory; Chossat, Pascal

doi:10.1007/s00332-011-9118-x

Cited by 10 publications

(33 citation statements)

References 35 publications

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“…Recall, that (ρ 1 , θ 1 ) and (ρ 2 , θ 2 ) denote polar coordinates in H (out) 1 and H (in) 2 , respectively, such that v 1 = ρ 1 cos θ 1 , q 1 = ρ 1 sin θ 1 , w 2 = ρ 2 cos θ 2 and q 2 = ρ 2 sin θ 2 . The map φ 1 is given by (4). The map φ 2 is…”

Section: Proof Of Theoremmentioning

confidence: 99%

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$ R 4

Podvigina

Chossat

2016

J Nonlinear Sci

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Robust heteroclinic cycles in equivariant dynamical systems in R 4 have been a subject of intense scientific investigation because, unlike heteroclinic cycles in R 3 , they can have an intricate geometric structure and complex asymptotic stability properties that are not yet completely understood. In a recent work, we have compiled an exhaustive list of finite subgroups of O(4) admitting the so-called simple heteroclinic cycles, and have identified a new class which we have called pseudo-simple heteroclinic cycles. By contrast with simple heteroclinic cycles, a pseudo-simple one has at least one equilibrium with an unstable manifold which has dimension 2 due to a symmetry. Here, we analyse the dynamics of nearby trajectories and asymptotic stability of pseudo-simple heteroclinic cycles in R 4 .

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Section: Proof Of Theoremmentioning

confidence: 99%

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$ R 4

Podvigina

Chossat

2016

J Nonlinear Sci

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“…Hence one can write SU(1, 1) = KAN (Iwasawa decomposition, see [32]). The corresponding subgroups of SSPD(2) can be computed using (20).…”

Section: Isometries Of the Poincaré Discmentioning

confidence: 99%

“…As we shall see, this allows to further identify Ξ with R + * × D, D being the Poincaré disc, or pseudo-sphere, equipped with its hyperbolic structure. The analysis of the Wilson-Cowan equation and its bifurcations in this hyperbolic geometry setting has been undertaken in a series of publications [12,13,[20][21][22]. In this paper I give a synthetic presentation of these studies.…”

Section: Introductionmentioning

confidence: 99%

The hyperbolic model for edge and texture detection in the primary visual cortex

Chossat

2020

J. Math. Neurosc.

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The modeling of neural fields in the visual cortex involves geometrical structures which describe in mathematical formalism the functional architecture of this cortical area. The case of contour detection and orientation tuning has been extensively studied and has become a paradigm for the mathematical analysis of image processing by the brain. Ten years ago an attempt was made to extend these models by replacing orientation (an angle) with a second-order tensor built from the gradient of the image intensity, and it was named the structure tensor. This assumption does not follow from biological observations (experimental evidence is still lacking) but from the idea that the effectiveness of texture processing with the structure tensor in computer vision may well be exploited by the brain itself. The drawback is that in this case the geometry is not Euclidean but hyperbolic instead, which complicates the analysis substantially. The purpose of this review is to present the methodology that was developed in a series of papers to investigate this quite unusual problem, specifically from the point of view of tuning and pattern formation. These methods, which rely on bifurcation theory with symmetry in the hyperbolic context, might be of interest for the modeling of other features such as color vision or other brain functions.

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“…It turns out that in this case cubic order in X is enough. However here again the calculations are cumbersome and we refer the reader to [30] for details. We first need to choose suitable coordinates for X ∈ R 4 and we do so for the representation χ 12 (we know similar results hold for χ 13 .…”

Section: Classification Of H-planforms and Bifurcation Diagramsmentioning

confidence: 99%

“…The next order is 5 and one can show that there are 4 independant G-equivariant terms at this order. Details are provided in [30]. Among these terms, three are gradients while one is non gradient.…”

Section: Classification Of H-planforms and Bifurcation Diagramsmentioning

confidence: 99%

Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

Chossat¹,

Faye²

2013

J Dyn Diff Equat

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International audienceIn this paper we present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincaré disc D. Different types of patterns are considered: spatially periodicstationarysolutions,radialsolutionsandtraveling waves,howeverthereare significantdifferencesintheresultswiththeEuclideancase.Weapplyequivariantbifurcation theory to the study of spatially periodic solutions on a given lattice of D also called H- planforms in reference with the "planforms" introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of D and periodic in the "transverse" direction. We highlight our theoretical results with a selection of numerical simulations

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Bifurcation Diagrams and Heteroclinic Networks of Octagonal H-Planforms

Cited by 10 publications

References 35 publications

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$ R 4

Asymptotic Stability of Pseudo-simple Heteroclinic Cycles in $${\mathbb R}^4$$ R 4

The hyperbolic model for edge and texture detection in the primary visual cortex

Pattern Formation for the Swift-Hohenberg Equation on the Hyperbolic Plane

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