Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme
Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as w...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/1/76 |
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| author | Asifa Tassaddiq Muhammad Tanveer Muhammad Azhar Muhammad Arshad Farha Lakhani |
| author_facet | Asifa Tassaddiq Muhammad Tanveer Muhammad Azhar Muhammad Arshad Farha Lakhani |
| author_sort | Asifa Tassaddiq |
| collection | DOAJ |
| description | Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as well as complex sine function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mo form="prefix">sin</mo><mrow><mo>(</mo><msup><mi>z</mi><mi>m</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> and complex exponential function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><msup><mi>e</mi><msup><mi>z</mi><mi>m</mi></msup></msup><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We use this to analyze the dynamical behavior of specific fractals namely Julia set and Mandelbrot set. This is achieved by generalizing the existing algorithms, which led to the visualization of beautiful fractals for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> and 4. Moreover, the image generation time in seconds using different values of input parameters is also computed. |
| first_indexed | 2024-03-09T12:40:19Z |
| format | Article |
| id | doaj.art-ebda0b544cc8461f99bae179811ae800 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T12:40:19Z |
| publishDate | 2023-01-01 |
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| series | Fractal and Fractional |
| spelling | doaj.art-ebda0b544cc8461f99bae179811ae8002023-11-30T22:19:59ZengMDPI AGFractal and Fractional2504-31102023-01-01717610.3390/fractalfract7010076Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative SchemeAsifa Tassaddiq0Muhammad Tanveer1Muhammad Azhar2Muhammad Arshad3Farha Lakhani4Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al-Majmaah 11952, Saudi ArabiaDepartment of Mathemtics and Statistics, Sub-Campus Depalpur, University of Agriculture, Faisalabad 38040, PakistanDepartment of Mathemtics, Government College University Lahore, Lahore 54000, PakistanDepartment of Mathemtics and Statistics, Sub-Campus Depalpur, University of Agriculture, Faisalabad 38040, PakistanSchool of Computing, University of Leeds, Leeds LS2 9JT, UKFractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we establish an escape criteria using DK-iteration as well as complex sine function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mo form="prefix">sin</mo><mrow><mo>(</mo><msup><mi>z</mi><mi>m</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> and complex exponential function <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><msup><mi>e</mi><msup><mi>z</mi><mi>m</mi></msup></msup><mo>+</mo><mi>c</mi><mo>;</mo><mspace width="3.33333pt"></mspace><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We use this to analyze the dynamical behavior of specific fractals namely Julia set and Mandelbrot set. This is achieved by generalizing the existing algorithms, which led to the visualization of beautiful fractals for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> and 4. Moreover, the image generation time in seconds using different values of input parameters is also computed.https://www.mdpi.com/2504-3110/7/1/76fractalssine functionexponential functionDK-iterationJulia set (J-set)Mandelbrot set (M-set) |
| spellingShingle | Asifa Tassaddiq Muhammad Tanveer Muhammad Azhar Muhammad Arshad Farha Lakhani Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme Fractal and Fractional fractals sine function exponential function DK-iteration Julia set (J-set) Mandelbrot set (M-set) |
| title | Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme |
| title_full | Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme |
| title_fullStr | Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme |
| title_full_unstemmed | Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme |
| title_short | Escape Criteria for Generating Fractals of Complex Functions Using DK-Iterative Scheme |
| title_sort | escape criteria for generating fractals of complex functions using dk iterative scheme |
| topic | fractals sine function exponential function DK-iteration Julia set (J-set) Mandelbrot set (M-set) |
| url | https://www.mdpi.com/2504-3110/7/1/76 |
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