Abstract similarity, fractals and chaos

Marat Akhmet; Ejaily Milad Alejaily

doi:10.3934/dcdsb.2020191

Article Contents

2021, Volume 26, Issue 5: 2479-2497. Doi: 10.3934/dcdsb.2020191

This issue Previous Article On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit Next Article A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale

Abstract similarity, fractals and chaos

Marat Akhmet^1, , and
Ejaily Milad Alejaily^2,

1.
Department of Mathematics, Middle East Technical University, Ankara, Turkey
2.
Department of Fundamental Sciences, College of Engineering Technology, Houn, Lybya

^* Corresponding author: Tel.: +90 312 210 5355, Fax: +90 312 210 2972
^* Corresponding author: Tel.: +90 312 210 5355, Fax: +90 312 210 2972

Received: July 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: May 2021

The first author has been supported by a grant (118F161) from TÜBİTAK, the Scientific and Technological Research Council of Turkey

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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 Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

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Mathematics Subject Classification: Primary: 28A80, 65P20; Secondary: 37B10.

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Figure 1. (a) The first step of abstract self-similar set construction. (b) The illustration of the boundary agreement

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Figure 2. Examples of the $ 2^{nd} $ and the $ 3^{rd} $ order subsets of the Sierpinski carpet

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Figure 3. A trajectory of the point under the similarity map

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Figure 4. The construction of abstract self-similar set corresponding to the Sierpinski gasket

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Figure 5. The construction of abstract self-similar set corresponding to the Koch curve

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Figure 6. The $ 1^{st} $ and the $ 2^{nd} $ order subsets for the Cantor set

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Figure 7. The construction of abstract self-similar set using the map (12)

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Figure 8. The first three iterations of DASS construction using the map (13)

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Figure 9. The two trajectories of the system (13)

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