\(n\)-dimensional chaotic attractors with crystallographic symmetry
type: Item
instance of: scholarly article
title: \(n\)-dimensional chaotic attractors with crystallographic symmetry
zbMATH Open document ID: 1024.37018
DOI: 10.1016/S0960-0779(00)00040-0
author: Jeffrey P. Dumont
author: Flynn J. Heiss
author: Clifford A. Reiter
author: Lisa M. Vislocky
author: Kevin Charles Jones
published in: Chaos, Solitons and Fractals
publication date: 2003-03-19T00:00:00Z
review text: The title refers to discrete Euclidean dynamical systems showing some kind of chaotic behavior comprising at least sensitivity with respect to initial conditions, and possibly other properties not explicitly stated. This paper demonstrates the inadequacy of the misleading term ``chaos'' for what it is often used in mathematics in that it displays the high degree of order in form of symmetry ``chaotic'' system can have, a fact that can hardly surprise a mathematician. Starting with a fairly arbitrary function, a new function is constructed that is equivariant with respect to the translational symmetries of a given lattice and the symmetries of a position group of the lattice, which consists of affine transformations. This result is used to furnish a rich supply of candidates for the construction of chaotic systems each having an attractor invariant under a given symmetry group, e.g. a crystallographic group in dimension two or three. Numerical examples sustantiating this are provided, together with attractive pictures of the symmetric attractors.
reviewed by: Dieter Erle
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 1884583
zbMATH Keywords: discrete dynamical system
zbMATH Keywords: sensitivity with respect to initial conditions
zbMATH Keywords: crystallographic group
zbMATH Keywords: symmetric group
MaRDI profile type: Publication
cites work: Q5534201
cites work: Q3996318
cites work: Deterministic Nonperiodic Flow
cites work: Q4278709
cites work: Weak Chaos and Quasi-Regular Patterns
cites work: Symmetric attractors in three-dimensional space.
cites work: Chaotic attractors with discrete planar symmetries
cites work: Chaotic attractors near forbidden symmetry
cites work: Q4023703
cites work: Attractors with the Symmetry of the n-Cube
cites work: Chaotic attractors with the symmetry of the dodecahedron
cites work: Q5362971
cites work: Q3735714
cites work: Detecting the symmetry of attractors
cites work: Q4161848
cites work: Algebraic aspects of crystallography. II: Non-primitive translations in space groups
cites work: Q3232307
description: scientific article; zbMATH DE number 1884583