Pattern formation in three-dimensional reaction–diffusion systems

Callahan, T. K.; Knobloch, Edgar

doi:10.1016/s0167-2789(99)00041-x

Cited by 85 publications

(60 citation statements)

References 23 publications

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“…The point group of Γ y0 contains the 2π/3 rotations around the vertical axis, (α − ) 2 and (α − ) 4 , and the reflections on vertical planes (β − )(α − ), (β − )(α − ) 3 , (β − )(α − ) 5 . The point group of Σ y0 is α 2 , βα .…”

Section: Restrictionmentioning

confidence: 99%

On the projection of functions invariant under the action of a crystallographic group

Pinho

Labouriau

2014

Journal of Pure and Applied Algebra

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Abstract. We study functions defined in (n + 1)-dimensional domains that are invariant under the action of a crystallographic group. We give a complete description of the symmetries that remain after projection into an ndimensional subspace and compare it to similar results for the restriction to a subspace. We use the Fourier expansion of invariant functions and the action of the crystallographic group on the space of Fourier coefficients. Intermediate results relate symmetry groups to the dual of the lattice of periods.

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Section: Restrictionmentioning

confidence: 99%

On the projection of functions invariant under the action of a crystallographic group

Pinho

Labouriau

2014

Journal of Pure and Applied Algebra

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“…20 The Brusselator model was more frequently employed in investigations than the LE; the two have been compared in some cases and display common features. 21,22 Many pattern types are possible in two and three dimensions and multistability between patterns is exhibited. Prediction of the type of pattern selected is therefore a formidable task.…”

Section: Discussionmentioning

confidence: 99%

Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

Bánsági

Taylor

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The formation of Turing patterns was investigated in thin cylindrical layers using the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction. The influence of the width of the layer W and the diameter D of the inner cylinder on the pattern with intrinsic wavelength l were determined in simulations with initial random noise perturbations to the uniform state for W < l/2 and D $ l or lower. We show that the geometric constraints of the reaction domain may result in the formation of helical Turing patterns with parameters that give stripes (b ¼ 0.2) or spots (b ¼ 0.37) in two dimensions. For b ¼ 0.2, the helices were composed of lamellae and defects were likely as the diameter of the cylinder increased. With b ¼ 0.37, the helices consisted of semi-cylinders and the orientation of stripes on the outer surface (and hence winding number) increased with increasing diameter until a new stripe appeared.

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“…The system size dependence causes the existence of characteristic domain size of the Turing patterns. Callahan and Knobloch [9] revealed that patterns become less regular because many unstable modes exist and interact each other as the domain grows. It is not cleared yet how such dependences work on the spherical surfaces.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Turing Patterns on a Spherical Surface Using Polyhedron Approximation

Yoshino¹

2017

Forma

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We considered Turing patterns on a spherical surface from the viewpoint of polyhedron geometry. We restrict our consideration to a set of parameters that produces a pattern of spots. We obtained numerical solutions for the Turing system on a spherical surface and approximated the solutions to convex polyhedrons. The polyhedron structure was dependent on both the radius of the sphere R and the initial condition. The number n of faces of the polyhedron increased with an increase in R. For small values of R, highly ordered structures were observed. With an increase of the value of R, a variety of structures were observed for each n, and the symmetry property of the spots, which determined the regularity of the polyhedron structure, gradually disappeared. We classified the numerical results according to their symmetrical properties of the approximated polyhedrons. The results revealed that the obtained Turing patterns lost symmetrical properties and varied the structures within same number of spots.

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Pattern formation in three-dimensional reaction–diffusion systems

Cited by 85 publications

References 23 publications

On the projection of functions invariant under the action of a crystallographic group

On the projection of functions invariant under the action of a crystallographic group

Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

Analysis of Turing Patterns on a Spherical Surface Using Polyhedron Approximation

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