Two New Types of Chaotic Maps and Minimal Systems

Mohammed Nokhas Murad Kaki; Sherko Hassan Abdurrahman

doi:doi:10.11648/j.pamj.s.2014030601.12

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Two New Types of Chaotic Maps and Minimal Systems

Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 6-1)

Received: 5 September 2014 Accepted: 16 September 2014 Published: 17 September 2014

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Abstract

In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 6-1) This article belongs to the Special Issue Mathematical Theory and Modeling
DOI	10.11648/j.pamj.s.2014030601.12
Page(s)	7-12
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Topologically θ - Transitive Map, α- Chaotic, Chaotic Amp, α- Transitive, θ- Dense

References

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Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. (2014). Two New Types of Chaotic Maps and Minimal Systems. Pure and Applied Mathematics Journal, 3(6-1), 7-12. https://doi.org/10.11648/j.pamj.s.2014030601.12

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ACS Style

Mohammed Nokhas Murad Kaki; Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl. Math. J. 2014, 3(6-1), 7-12. doi: 10.11648/j.pamj.s.2014030601.12

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AMA Style

Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl Math J. 2014;3(6-1):7-12. doi: 10.11648/j.pamj.s.2014030601.12

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@article{10.11648/j.pamj.s.2014030601.12,
  author = {Mohammed Nokhas Murad Kaki and Sherko Hassan Abdurrahman},
  title = {Two New Types of Chaotic Maps and Minimal Systems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {6-1},
  pages = {7-12},
  doi = {10.11648/j.pamj.s.2014030601.12},
  url = {https://doi.org/10.11648/j.pamj.s.2014030601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2014030601.12},
  abstract = {In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.},
 year = {2014}
}

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TY  - JOUR
T1  - Two New Types of Chaotic Maps and Minimal Systems
AU  - Mohammed Nokhas Murad Kaki
AU  - Sherko Hassan Abdurrahman
Y1  - 2014/09/17
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2014030601.12
DO  - 10.11648/j.pamj.s.2014030601.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 7
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2014030601.12
AB  - In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.
VL  - 3
IS  - 6-1
ER  -

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Author Information

Mohammed Nokhas Murad Kaki

University of Sulaimani, Faculty of Science and Science Education, School of Science, Math Department
Sherko Hassan Abdurrahman

University of Sulaimani, Faculty of physical and Basie Education, School of Basic Education, Department of Computer Science

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APA Style

Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. (2014). Two New Types of Chaotic Maps and Minimal Systems. Pure and Applied Mathematics Journal, 3(6-1), 7-12. https://doi.org/10.11648/j.pamj.s.2014030601.12

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ACS Style

Mohammed Nokhas Murad Kaki; Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl. Math. J. 2014, 3(6-1), 7-12. doi: 10.11648/j.pamj.s.2014030601.12

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AMA Style

Mohammed Nokhas Murad Kaki, Sherko Hassan Abdurrahman. Two New Types of Chaotic Maps and Minimal Systems. Pure Appl Math J. 2014;3(6-1):7-12. doi: 10.11648/j.pamj.s.2014030601.12

Copy | Download

@article{10.11648/j.pamj.s.2014030601.12,
  author = {Mohammed Nokhas Murad Kaki and Sherko Hassan Abdurrahman},
  title = {Two New Types of Chaotic Maps and Minimal Systems},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {6-1},
  pages = {7-12},
  doi = {10.11648/j.pamj.s.2014030601.12},
  url = {https://doi.org/10.11648/j.pamj.s.2014030601.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2014030601.12},
  abstract = {In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Two New Types of Chaotic Maps and Minimal Systems
AU  - Mohammed Nokhas Murad Kaki
AU  - Sherko Hassan Abdurrahman
Y1  - 2014/09/17
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2014030601.12
DO  - 10.11648/j.pamj.s.2014030601.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 7
EP  - 12
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.s.2014030601.12
AB  - In this paper, we introduce and study the relationship between two different notions of chaotic maps, namely topological α–chaotic maps, topological θ-chaotic maps and investigate some of their properties in two topological spaces (X, τα) and (X, τθ), τα denotes the α–topology(resp. τθ denotes the θ–topology) of a given topological space (X, τ). The two notions are defined by using the concepts of α-transitive map and θ-transitive map respectively Also, we define and study the relationship between two types of minimal mappings, namely, α - minimal mapping and θ-minimal mapping, The main results are the following propositions: 1). Every topologically α-chaotic map is a chaotic map which implies topologically θ- chaotic map, but the converse not necessarily true. 2). Every α-minimal map is a minimal map which implies θ- minimal map in topological spaces, but the converse not necessarily true.
VL  - 3
IS  - 6-1
ER  -

Copy | Download