Abstract Similarity, Fractals and Chaos
Dynamical Systems
2019-05-08 v1
Abstract
To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincare, Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered.
Cite
@article{arxiv.1905.02198,
title = {Abstract Similarity, Fractals and Chaos},
author = {Marat Akhmet and Ejaily Milad Alejaily},
journal= {arXiv preprint arXiv:1905.02198},
year = {2019}
}