Symmetric attractors in three-dimensional space

Brisson, Gabriel F.; Gartz, Kaj M.; McCune, Benton; OʼBrien, Kevin P.; Reiter, Clifford A.

doi:10.1016/0960-0779(95)00094-1

Cited by 27 publications

(5 citation statements)

References 3 publications

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“…Consider next the IVP of FO (7) in the complex plane. With u = z, z ∈ C and z(0) = x(0) + iy(0), the IVP has the following form…”

Section: Logistic Map (2) Of Fomentioning

confidence: 99%

“…Other recent work on creating chaotic attractors with symmetry [6] has been based on demanding that the coefficients of general polynomial or trigonometric functions satisfy certain criterion in order to respect the symmetry group of interest. This has included chaotic attractors with cubic [7], tetrahedral [8], hypercube [9], frieze and wallpaper symmetries [10]. A comprehensive literature on chaotic attractors with symmetry is presented in [6] An example of a map on the complex plane, f : C → C with dihedral or cyclic symmetry considered in this paper is [4,11] (see also [12])…”

Section: Introductionmentioning

confidence: 99%

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Symmetry-breaking and bifurcation diagrams of fractional-order maps

Danca¹

2022

Preprint

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In this paper two important aspects related to Caputo's fractional-order discrete variant of a class of maps defined on the complex plane, are analytically and numerically revealed: attractors symmetry-broken induced by the fractional-order and the sensible problem of determining the right bifurcation diagram of discrete systems of fractional-order. It is proved that maps of integer order with dihedral symmetry or cycle symmetry loose their symmetry once they are transformed in fractional-order maps. Also, it is conjectured that, contrarily to integer-order maps, determining the bifurcation diagrams of fractional-order maps is far from being a clarified problem. Two examples are considered: dihedral logistic map and cyclic logistic map.

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“…Consider next the IVP of FO (7) in the complex plane. With u = z, z ∈ C and z(0) = x(0) + iy(0), the IVP has the following form…”

Section: Logistic Map (2) Of Fomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetry-breaking and bifurcation diagrams of fractional-order maps

Danca¹

2022

Preprint

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“…More recently, attractors arising from the iteration of functions in the plane have been studied and equivariant functions have been used to create attractors that appear chaotic while having rotational and/or reflectional symmetries [3][4][5] . Examples of attractors with the symmetry of the cube in 3space and n-cube in n-space have also been created [6] [7]; in addition, examples of attractors in 3-space with dueling planar symmetries have been studied [8f In this paper we create attractors in 3-space which empirically appear to be chaotic but which have the symmetry of the tetrahedron.…”

Section: Introductionmentioning

confidence: 99%

Chaotic attractors with the symmetry of a tetrahedron

Reiter

1997

Computers & Graphics

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International audienceFunctions that are equivariant with respect to the symmetries of a tetrahedron are determined. Linear combinations of these functions that give rise to chaotic attractors are used to create images in 3-space of attractors with the symmetry of the tetrahedron. These attractors are visually appealing because of the tension between the pattern forced by their symmetry and the randomness arising from their chaotic behavior

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“…Equation (19) has the following outstanding features. First, to create symmetrical patterns, one needs to construct mappings that meet certain requirements [12][13][14][15][17][18][19][20][21]. However, under certain circumstances, this kind of mapping is not easy to achieve.…”

Section: Colorful Spherical Patterns With [P Q] Symmetrymentioning

confidence: 99%

“…Field and Golubitsky first proposed equivariant mappings to yield aesthetic patterns with discrete planar symmetries [12]. This idea later inspired Reiter to create chaotic attractors with symmetries of the tetrahedron [13] and octahedron [14] in three-dimensional Euclidean space R 3 . Recently, Lu et al established several families of invariant mappings to generate similar images [15].…”

Section: Introductionmentioning

confidence: 99%