ABSTRACT SIMILARITY, FRACTALS AND CHAOS

AKHMET M., Alejaily E. M.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B, vol.26, no.5, pp.2479-2497, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 26 Issue: 5
  • Publication Date: 2021
  • Doi Number: 10.3934/dcdsb.2020191
  • Journal Name: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH
  • Page Numbers: pp.2479-2497
  • Keywords: Abstract self-similarity, self-similar space, similarity map, fractals, chaos, multi-dimensional chaotic maps, SELF-SIMILAR SETS, HAUSDORFF DIMENSION, SYSTEMS
  • Middle East Technical University Affiliated: Yes

Abstract

A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincare, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.