Chaos on Fuzzy Dynamical Systems
Given a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semant...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/20/2629 |
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| author | Félix Martínez-Giménez Alfred Peris Francisco Rodenas |
| author_facet | Félix Martínez-Giménez Alfred Peris Francisco Rodenas |
| author_sort | Félix Martínez-Giménez |
| collection | DOAJ |
| description | Given a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> on a metric space, it induces the maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>f</mi><mo>¯</mo></mover><mo>:</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the hyperspace of nonempty compact subspaces of <i>X</i>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>:</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the space of normal fuzzy sets, consisting of the upper semicontinuous functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>:</mo><mi>X</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <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>X</mi><mo>)</mo></mrow><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>,</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, we considered several dynamical properties related to chaos: Devaney chaos, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics). |
| first_indexed | 2024-03-10T06:25:32Z |
| format | Article |
| id | doaj.art-fe80a921cc4046acbd4d1fbdcd469f9b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T06:25:32Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-fe80a921cc4046acbd4d1fbdcd469f9b2023-11-22T19:02:54ZengMDPI AGMathematics2227-73902021-10-01920262910.3390/math9202629Chaos on Fuzzy Dynamical SystemsFélix Martínez-Giménez0Alfred Peris1Francisco Rodenas2Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, SpainInstitut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, SpainInstitut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, SpainGiven a continuous map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></semantics></math></inline-formula> on a metric space, it induces the maps <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>f</mi><mo>¯</mo></mover><mo>:</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">K</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the hyperspace of nonempty compact subspaces of <i>X</i>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>:</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, on the space of normal fuzzy sets, consisting of the upper semicontinuous functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>:</mo><mi>X</mi><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <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>X</mi><mo>)</mo></mrow><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">F</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>,</mo><mover accent="true"><mi>f</mi><mo>^</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula>. In particular, we considered several dynamical properties related to chaos: Devaney chaos, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).https://www.mdpi.com/2227-7390/9/20/2629chaotic operatorshypercyclic operatorshyperspaces of compact setsspaces of fuzzy sets<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm446" display="block"> <mml:semantics> <mml:mi mathvariant="script">A</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-transitivity |
| spellingShingle | Félix Martínez-Giménez Alfred Peris Francisco Rodenas Chaos on Fuzzy Dynamical Systems Mathematics chaotic operators hypercyclic operators hyperspaces of compact sets spaces of fuzzy sets <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm446" display="block"> <mml:semantics> <mml:mi mathvariant="script">A</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-transitivity |
| title | Chaos on Fuzzy Dynamical Systems |
| title_full | Chaos on Fuzzy Dynamical Systems |
| title_fullStr | Chaos on Fuzzy Dynamical Systems |
| title_full_unstemmed | Chaos on Fuzzy Dynamical Systems |
| title_short | Chaos on Fuzzy Dynamical Systems |
| title_sort | chaos on fuzzy dynamical systems |
| topic | chaotic operators hypercyclic operators hyperspaces of compact sets spaces of fuzzy sets <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm446" display="block"> <mml:semantics> <mml:mi mathvariant="script">A</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-transitivity |
| url | https://www.mdpi.com/2227-7390/9/20/2629 |
| work_keys_str_mv | AT felixmartinezgimenez chaosonfuzzydynamicalsystems AT alfredperis chaosonfuzzydynamicalsystems AT franciscorodenas chaosonfuzzydynamicalsystems |