An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets
This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreove...
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MDPI AG
2025-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/40 |
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| author | Khairul Habib Alam Yumnam Rohen Anita Tomar Naeem Saleem Maggie Aphane Asima Razzaque |
| author_facet | Khairul Habib Alam Yumnam Rohen Anita Tomar Naeem Saleem Maggie Aphane Asima Razzaque |
| author_sort | Khairul Habib Alam |
| collection | DOAJ |
| description | This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply <i>s</i>-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">sin</mo><mo>(</mo><msup><mi>u</mi><mi>m</mi></msup><mo>)</mo><mo>−</mo><mi>α</mi><mi>u</mi><mo>+</mo><mi>β</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>,</mo><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</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>m</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. |
| format | Article |
| id | doaj-art-0396417c98bd4a3185e38428c584da31 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
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| series | Fractal and Fractional |
| spelling | doaj-art-0396417c98bd4a3185e38428c584da312025-01-24T13:33:28ZengMDPI AGFractal and Fractional2504-31102025-01-01914010.3390/fractalfract9010040An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot SetsKhairul Habib Alam0Yumnam Rohen1Anita Tomar2Naeem Saleem3Maggie Aphane4Asima Razzaque5Department of Mathematics, National Institute of Technology Manipur, Imphal 795004, Manipur, IndiaDepartment of Mathematics, Manipur University, Imphal 795003, Manipur, IndiaPt. L. M. S. Campus, Sridev Suman Uttarakhand University, Rishikesh 249201, Uttarakhand, IndiaDepartment of Mathematics, University of Management and Technology, Lahore 54770, PakistanDepartment of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South AfricaDepartment of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi ArabiaThis study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply <i>s</i>-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">sin</mo><mo>(</mo><msup><mi>u</mi><mi>m</mi></msup><mo>)</mo><mo>−</mo><mi>α</mi><mi>u</mi><mo>+</mo><mi>β</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>,</mo><mi>α</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</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>m</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure.https://www.mdpi.com/2504-3110/9/1/40efficiencystabilityescape criterionfractalsJulia setMandelbrot set |
| spellingShingle | Khairul Habib Alam Yumnam Rohen Anita Tomar Naeem Saleem Maggie Aphane Asima Razzaque An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets Fractal and Fractional efficiency stability escape criterion fractals Julia set Mandelbrot set |
| title | An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets |
| title_full | An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets |
| title_fullStr | An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets |
| title_full_unstemmed | An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets |
| title_short | An Effective Iterative Process Utilizing Transcendental Sine Functions for the Generation of Julia and Mandelbrot Sets |
| title_sort | effective iterative process utilizing transcendental sine functions for the generation of julia and mandelbrot sets |
| topic | efficiency stability escape criterion fractals Julia set Mandelbrot set |
| url | https://www.mdpi.com/2504-3110/9/1/40 |
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