Fractal generation and analysis using modified fixed-point iteration

Asifa Tassaddiq; Muhammad Tanveer; Muhammad Arshad; Rabab Alharbi; Ruhaila Md Kasmani; Asifa Tassaddiq; Muhammad Tanveer; Muhammad Arshad; Rabab Alharbi; Ruhaila Md Kasmani

doi:10.3934/math.2025437

AIMS Mathematics

2025, Volume 10, Issue 4: 9462-9492. doi: 10.3934/math.2025437

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Research article Topical Sections

Fractal generation and analysis using modified fixed-point iteration

1.
Department of Computer Science, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia
2.
Department of Mathematics and Statistics, University of Agriculture Faisalabad, Faisalabad 38000, Pakistan
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, Malaysia

Received: 16 January 2025 Revised: 15 March 2025 Accepted: 26 March 2025 Published: 24 April 2025
MSC : 37Fxx, 30D05, 37C25, 28A80

Fractals exhibit self-similarity across scales and have significant mathematical and artistic appeal. This study proposed a modified fixed-point iteration (MFPI) to generate fractals for the complex polynomial $ y^k + c $. The escape criterion for MFPI was established, enabling the construction of Mandelbrot and Julia sets. Comparative image analysis with M, Picard-Mann, and Mann iterations highlighted visual differences. Numerical metrics, including non-escape area index (NAI), average escape time (AET), fractal dimension (FD), and execution time, assessed the efficiency and performance of MFPI against existing methods.
- fixed-points,
- iterative methods,
- Mandelbrot and Julia sets,
- non-escape area index,
- average escape time,
- fractal dimension
Citation: Asifa Tassaddiq, Muhammad Tanveer, Muhammad Arshad, Rabab Alharbi, Ruhaila Md Kasmani. Fractal generation and analysis using modified fixed-point iteration[J]. AIMS Mathematics, 2025, 10(4): 9462-9492. doi: 10.3934/math.2025437

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Abstract

Fractals exhibit self-similarity across scales and have significant mathematical and artistic appeal. This study proposed a modified fixed-point iteration (MFPI) to generate fractals for the complex polynomial $ y^k + c $. The escape criterion for MFPI was established, enabling the construction of Mandelbrot and Julia sets. Comparative image analysis with M, Picard-Mann, and Mann iterations highlighted visual differences. Numerical metrics, including non-escape area index (NAI), average escape time (AET), fractal dimension (FD), and execution time, assessed the efficiency and performance of MFPI against existing methods.

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