A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme
In this research, we look at the Julia set patterns that are linked to the entire transcendental function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/7/397 |
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| author | Darshana J. Prajapati Shivam Rawat Anita Tomar Mohammad Sajid R. C. Dimri |
| author_facet | Darshana J. Prajapati Shivam Rawat Anita Tomar Mohammad Sajid R. C. Dimri |
| author_sort | Darshana J. Prajapati |
| collection | DOAJ |
| description | In this research, we look at the Julia set patterns that are linked to the entire transcendental function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>e</mi><msup><mi>z</mi><mi>n</mi></msup></msup><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function. |
| first_indexed | 2024-03-09T03:23:19Z |
| format | Article |
| id | doaj.art-2f06445fe4b940da9cea0e4f6f6569a5 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T03:23:19Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-2f06445fe4b940da9cea0e4f6f6569a52023-12-03T15:04:58ZengMDPI AGFractal and Fractional2504-31102022-07-016739710.3390/fractalfract6070397A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative SchemeDarshana J. Prajapati0Shivam Rawat1Anita Tomar2Mohammad Sajid3R. C. Dimri4M.B. Patel Institute of Technology, New Vallabh Vidyanagar 388120, IndiaDepartment of Mathematics, H.N.B. Garhwal University, Srinagar 246174, IndiaPt.L.M.S. Campus, Sri Dev Suman Uttarakhand University, Rishikesh 249201, IndiaDepartment of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, H.N.B. Garhwal University, Srinagar 246174, IndiaIn this research, we look at the Julia set patterns that are linked to the entire transcendental function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>e</mi><msup><mi>z</mi><mi>n</mi></msup></msup><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, using the Mann iterative scheme, and discuss their dynamical behavior. The sophisticated orbit structure of this function, whose Julia set encompasses the entire complex plane, is described using symbolic dynamics. We also present bifurcation diagrams of Julia sets generated using the proposed iteration and function, which altogether contain four parameters, and discuss the graphical analysis of bifurcation occurring in the family of this function.https://www.mdpi.com/2504-3110/6/7/397fixed pointMann orbitJulia setbifurcation |
| spellingShingle | Darshana J. Prajapati Shivam Rawat Anita Tomar Mohammad Sajid R. C. Dimri A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme Fractal and Fractional fixed point Mann orbit Julia set bifurcation |
| title | A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme |
| title_full | A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme |
| title_fullStr | A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme |
| title_full_unstemmed | A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme |
| title_short | A Brief Study on Julia Sets in the Dynamics of Entire Transcendental Function Using Mann Iterative Scheme |
| title_sort | brief study on julia sets in the dynamics of entire transcendental function using mann iterative scheme |
| topic | fixed point Mann orbit Julia set bifurcation |
| url | https://www.mdpi.com/2504-3110/6/7/397 |
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