Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration
The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or...
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MDPI AG
2023-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/10/768 |
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| author | Ashish Bhoria Anju Panwar Mohammad Sajid |
| author_facet | Ashish Bhoria Anju Panwar Mohammad Sajid |
| author_sort | Ashish Bhoria |
| collection | DOAJ |
| description | The majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mi>s</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><msup><mi>z</mi><mi>r</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula> and complex exponential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>e</mi><msup><mi>z</mi><mi>r</mi></msup></msup><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula> functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘<i>a</i>’ and ‘<i>b</i>’, and the parameters involved in the series expansion of the sine and exponential functions. |
| first_indexed | 2024-03-10T21:13:21Z |
| format | Article |
| id | doaj.art-0de28e3577de4aee8e5b28200d77248d |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T21:13:21Z |
| publishDate | 2023-10-01 |
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| series | Fractal and Fractional |
| spelling | doaj.art-0de28e3577de4aee8e5b28200d77248d2023-11-19T16:35:01ZengMDPI AGFractal and Fractional2504-31102023-10-0171076810.3390/fractalfract7100768Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur IterationAshish Bhoria0Anju Panwar1Mohammad Sajid2Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, IndiaDepartment of Mathematics, Maharshi Dayanand University, Rohtak 124001, IndiaDepartment of Mechanical Engineering, College of Engineering, Qassim University, Buraydah 51452, Saudi ArabiaThe majority of fractals’ dynamical behavior is determined by escape criteria, which utilize various iterative procedures. In the context of the Julia and Mandelbrot sets, the concept of “escape” is a fundamental principle used to determine whether a point in the complex plane belongs to the set or not. In this article, the fractals of higher importance, i.e., Julia sets and Mandelbrot sets, are visualized using the Picard–Thakur iterative procedure (as one of iterative methods) for the complex sine <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mi>s</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><msup><mi>z</mi><mi>r</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula> and complex exponential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>c</mi></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>e</mi><msup><mi>z</mi><mi>r</mi></msup></msup><mo>+</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>c</mi></mrow></semantics></math></inline-formula> functions. In order to obtain the fixed point of a complex-valued sine and exponential function, our concern is to use the fewest number of iterations possible. Using MATHEMATICA 13.0, some enticing and intriguing fractals are generated, and their behavior is then illustrated using graphical examples; this is achieved depending on the iteration parameters, the parameters ‘<i>a</i>’ and ‘<i>b</i>’, and the parameters involved in the series expansion of the sine and exponential functions.https://www.mdpi.com/2504-3110/7/10/768algorithmsfractalsJulia setMandelbrot setPicard–Thakur iterationescape criterion |
| spellingShingle | Ashish Bhoria Anju Panwar Mohammad Sajid Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration Fractal and Fractional algorithms fractals Julia set Mandelbrot set Picard–Thakur iteration escape criterion |
| title | Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration |
| title_full | Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration |
| title_fullStr | Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration |
| title_full_unstemmed | Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration |
| title_short | Mandelbrot and Julia Sets of Transcendental Functions Using Picard–Thakur Iteration |
| title_sort | mandelbrot and julia sets of transcendental functions using picard thakur iteration |
| topic | algorithms fractals Julia set Mandelbrot set Picard–Thakur iteration escape criterion |
| url | https://www.mdpi.com/2504-3110/7/10/768 |
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