Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems

This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</...

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Main Authors: Jie Zhou, Tianxiu Lu, Jiazheng Zhao
Format: Article
Language:English
Published: MDPI AG 2023-12-01
Series:Axioms
Subjects:
Devaney chaos
sensitivity
set-valued discrete dynamical systems
Online Access:https://www.mdpi.com/2075-1680/13/1/20
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author Jie Zhou
Tianxiu Lu
Jiazheng Zhao
author_facet Jie Zhou
Tianxiu Lu
Jiazheng Zhao
author_sort Jie Zhou
collection DOAJ
description This paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> and its induced set-valued discrete dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>,</mo><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula>. Specifically, it explores the chaotic properties of these systems. The main finding is that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>w</mi><mi>e</mi></msup></semantics></math></inline-formula>-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.
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spelling doaj.art-f56f94516a1a49ec9efd0483ba24fba82024-01-26T15:03:00ZengMDPI AGAxioms2075-16802023-12-011312010.3390/axioms13010020Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical SystemsJie Zhou0Tianxiu Lu1Jiazheng Zhao2College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, ChinaCollege of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, ChinaCollege of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, ChinaThis paper focuses on the relationship between a non-autonomous discrete dynamical system (NDDS) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>H</mi><mo>,</mo><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> and its induced set-valued discrete dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>,</mo><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula>. Specifically, it explores the chaotic properties of these systems. The main finding is that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic if and only if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mrow><mn>1</mn><mo>,</mo><mo>∞</mo></mrow></msub></semantics></math></inline-formula> is Devaney chaotic in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>w</mi><mi>e</mi></msup></semantics></math></inline-formula>-topology. The paper also provides similar conclusions for weak mixing, mixing, mild mixing, chain-transitivity, and chain-mixing in non-autonomous set-valued discrete dynamical systems (NSDDSs). Additionally, the paper proves that weak mixing implies sensitivity in NSDDSs.https://www.mdpi.com/2075-1680/13/1/20Devaney chaossensitivityset-valued discrete dynamical systems
spellingShingle Jie Zhou
Tianxiu Lu
Jiazheng Zhao
Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
Axioms
Devaney chaos
sensitivity
set-valued discrete dynamical systems
title Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
title_full Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
title_fullStr Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
title_full_unstemmed Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
title_short Chaotic Characteristics in Devaney’s Framework for Set-Valued Discrete Dynamical Systems
title_sort chaotic characteristics in devaney s framework for set valued discrete dynamical systems
topic Devaney chaos
sensitivity
set-valued discrete dynamical systems
url https://www.mdpi.com/2075-1680/13/1/20
work_keys_str_mv AT jiezhou chaoticcharacteristicsindevaneysframeworkforsetvalueddiscretedynamicalsystems
AT tianxiulu chaoticcharacteristicsindevaneysframeworkforsetvalueddiscretedynamicalsystems
AT jiazhengzhao chaoticcharacteristicsindevaneysframeworkforsetvalueddiscretedynamicalsystems

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