Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration
Abstract
:1. Introduction
2. Basic Definitions and Preliminaries
3. Escape Criterion
4. Rich and Exquisite Patterns of the Fractal Mandelbrot Sets
Algorithm 1: Creation of the Mandelbrot sets using the hybrid Picard S-orbit |
Algorithm 2: Creation of the Mandelbrot sets using the S-orbit |
4.1. Rich and Exquisite Patterns of the Mandelbrot Sets Using Hybrid Picard S-Iteration vs. S-Iteration and Quadratic Functions
4.2. Rich and Exquisite Patterns of the Mandelbrot Sets Using Hybrid Picard S-Iteration vs. S-Iteration and Cubic Functions
5. Conclusions and Discussion
- Similarly in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23, we fixed parameter and varied parameter to obtain quadratic Mandelbrot sets. Mandelbrot sets created using the hybrid Picard S-iteration procedure are quite spontaneous from those created via S-iteration.
- The cubic Mandelbrot sets created in the hybrid Picard S-orbit and S-orbit with fixed parameter also have an obvious variation in figures for varying parameter .
- Quadratic Mandelbrot sets are symmetrical along the x-axis whereas cubic Mandelbrot sets are symmetrical along both the x-axis and y-axis.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MS | Mandelbrot set |
MSs | Mandelbrot sets |
SO | S Orbit |
PSO | Picard S-Orbit |
PSIP | Picard S-Iteration Process |
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Time for the Hybrid Picard S-Iteration | Time for S-Iteration | |
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Time for the Hybrid Picard S-Iteration | Time for S-Iteration | |
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Time for Picard S-Iteration | Time for S-Iteration | |
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Srivastava, R.; Tassaddiq, A.; Kasmani, R.M. Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration. Fractal Fract. 2024, 8, 116. https://doi.org/10.3390/fractalfract8020116
Srivastava R, Tassaddiq A, Kasmani RM. Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration. Fractal and Fractional. 2024; 8(2):116. https://doi.org/10.3390/fractalfract8020116
Chicago/Turabian StyleSrivastava, Rekha, Asifa Tassaddiq, and Ruhaila Md Kasmani. 2024. "Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration" Fractal and Fractional 8, no. 2: 116. https://doi.org/10.3390/fractalfract8020116
APA StyleSrivastava, R., Tassaddiq, A., & Kasmani, R. M. (2024). Escape Criteria Using Hybrid Picard S-Iteration Leading to a Comparative Analysis of Fractal Mandelbrot Sets Generated with S-Iteration. Fractal and Fractional, 8(2), 116. https://doi.org/10.3390/fractalfract8020116