\(n\)-dimensional chaotic attractors with crystallographic symmetry
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.
sensitivity with respect to initial conditions
scientific article; zbMATH DE number 1884583
scientific article