In this manuscript, we investigate the dynamics of higher-order polynomials with complex coefficients by employing the Jungck–Noor iteration scheme (one of the iterative methods) in conjunction with $ s $-convexity, which controls the weighting of previous iterates versus current polynomial evaluations, tuning convergence speed, escape dynamics, and fractal density from compact, high-brightness patterns to intricate, detailed structures. This framework enables us to establish new escape criteria and to visualize nonclassical deviations of the celebrated Mandelbrot and Julia sets. The resulting fractal structures display intricate geometries that not only enrich the theoretical study of complex dynamics but also resemble patterns observed in natural systems. To highlight the novelty of our work, we provide both graphical and numerical illustrations that demonstrate how variations in polynomial parameters and iteration settings influence shape transformations, symmetry, color distributions, and computational complexity. A key observation is that each point in the Mandelbrot set encodes detailed information about the corresponding Julia fractal, reinforcing the deep interplay between the two families. Moreover, when real-valued parameters are considered in the polynomial map and iteration process, some fractals exhibit striking motifs suggestive of potential applications, for example, in pattern design within the textile industry. Our study also opens pathways for extending this framework to noise-perturbed systems and physical models, which will be explored in future work.
Citation: Anita Tomar, Swati Antal, Mohammad Sajid, Darshana J. Prajapati. Role of $ s $-convexity in the generation of fractals as Julia and Mandelbrot sets via three-step fixed point iteration[J]. AIMS Mathematics, 2025, 10(11): 26077-26105. doi: 10.3934/math.20251148
In this manuscript, we investigate the dynamics of higher-order polynomials with complex coefficients by employing the Jungck–Noor iteration scheme (one of the iterative methods) in conjunction with $ s $-convexity, which controls the weighting of previous iterates versus current polynomial evaluations, tuning convergence speed, escape dynamics, and fractal density from compact, high-brightness patterns to intricate, detailed structures. This framework enables us to establish new escape criteria and to visualize nonclassical deviations of the celebrated Mandelbrot and Julia sets. The resulting fractal structures display intricate geometries that not only enrich the theoretical study of complex dynamics but also resemble patterns observed in natural systems. To highlight the novelty of our work, we provide both graphical and numerical illustrations that demonstrate how variations in polynomial parameters and iteration settings influence shape transformations, symmetry, color distributions, and computational complexity. A key observation is that each point in the Mandelbrot set encodes detailed information about the corresponding Julia fractal, reinforcing the deep interplay between the two families. Moreover, when real-valued parameters are considered in the polynomial map and iteration process, some fractals exhibit striking motifs suggestive of potential applications, for example, in pattern design within the textile industry. Our study also opens pathways for extending this framework to noise-perturbed systems and physical models, which will be explored in future work.
| [1] | M. F. Barnsley, Fractals everywhere, 2 Eds., San Diego: Academic Press, 1993. https://doi.org/10.1016/c2013-0-10335-2 |
| [2] | R. L. Devaney, A first course in chaotic dynamical systems: Theory and experiment, 2 Eds., New York: Chapman and Hall/CRC, 2020. https://doi.org/10.1201/9780429280665 |
| [3] |
J. Hu, H. Shen, X. Liu, J. Wang, RDMA transports in datacenter networks: Survey, IEEE Netw., 38 (2024), 380–387. https://doi.org/10.1109/mnet.2024.3397781 doi: 10.1109/mnet.2024.3397781
|
| [4] | B. B. Mandelbrot, The fractal geometry of nature, New York: W. H. Freeman and Company, 1982. |
| [5] | G. M. Julia, Mémoire sur l'itération des fonctions rationnelles, J. Math. Pures Appl., 8 (1918), 47–245. |
| [6] |
S. Antal, A. Tomar, D. J. Prajapati, M. Sajid, Fractals as Julia sets of complex sine function via fixed point iterations, Fractal Fract., 5 (2021), 272. https://doi.org/10.3390/fractalfract5040272 doi: 10.3390/fractalfract5040272
|
| [7] |
D. J. Prajapati, S. Rawat, A. Tomar, M. Sajid, R. C. Dimri, A brief study of dynamics of Julia sets for entire transcendental function using Mann iterative scheme, Fractal Fract., 6 (2022), 397. https://doi.org/10.3390/fractalfract6070397 doi: 10.3390/fractalfract6070397
|
| [8] |
A. Tomar, V. Kumar, U. S. Rana, M. Sajid, Fractals as Julia and Mandelbrot sets of complex cosine functions via fixed point iterations, Symmetry, 15 (2023), 478. https://doi.org/10.3390/sym15020478 doi: 10.3390/sym15020478
|
| [9] | E. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, J. Math. Pures Appl., 6 (1890), 145–210. |
| [10] |
H. Afshari, H. Aydi, Some results about Krosnoselski–Mann iteration process, J. Nonlinear Sci. Appl., 9 (2016), 4852–4859. https://doi.org/10.22436/jnsa.009.06.120 doi: 10.22436/jnsa.009.06.120
|
| [11] | M. O. Olatinwo, Some stability and strong convergence results for the Jungck–Ishikawa iteration process, Creat. Math. Inform., 17 (2008), 33–42. |
| [12] |
S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. https://doi.org/10.2307/2039245 doi: 10.2307/2039245
|
| [13] |
Y. C. Kwun, A. A. Shahid, W. Nazeer, M. Abbas, S. M. Kang, Fractal generation via CR-iteration scheme with $s$-convexity, IEEE Access, 7 (2019), 69986–69997. https://doi.org/10.1109/access.2019.2919520 doi: 10.1109/access.2019.2919520
|
| [14] |
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.2307/2032162 doi: 10.2307/2032162
|
| [15] |
M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042 doi: 10.1006/jmaa.2000.7042
|
| [16] |
S. Y. Cho, A. A. Shahid, W. Nazeer, S. M. Kang, Fixed point results for fractal generation in Noor orbit and $s$-convexity, SpringerPlus, 5 (2016), 1843. https://doi.org/10.1186/s40064-016-3530-5 doi: 10.1186/s40064-016-3530-5
|
| [17] |
A. A. Shahid, W. Nazeer, K. Gdawiec, The Picard–Mann iteration with s-convexity in the generation of Mandelbrot and Julia sets, Monatsh. Math., 195 (2021), 565–584. https://doi.org/10.1007/s00605-021-01591-z doi: 10.1007/s00605-021-01591-z
|
| [18] |
M. Tanveer, S. Kang, W. Nazeer, Y. Kwun, New tricorns and multicorns antifractals in Jungck–Mann orbit, Int. J. Pure Appl. Math., 111 (2016), 287–302. https://doi.org/10.12732/ijpam.v111i2.13 doi: 10.12732/ijpam.v111i2.13
|
| [19] |
N. Adhikari, W. Sintunavarat, Exploring the Julia and Mandelbrot sets of $z^p + \log c^t$ using a four-step iteration scheme extended with $s$-convexity, Math. Comput. Simul., 220 (2024), 357–381. https://doi.org/10.1016/j.matcom.2024.01.010 doi: 10.1016/j.matcom.2024.01.010
|
| [20] |
S. Rawat, D. Prajapati, A. Tomar, K. Gdawiec, Generation of Mandelbrot and Julia sets as fractals for generalized rational maps using SP-iteration process equipped with $s$-convexity, Math. Comput. Simul., 220 (2024), 148–169. https://doi.org/10.1016/j.matcom.2023.12.040 doi: 10.1016/j.matcom.2023.12.040
|
| [21] |
S. Kumari, M. Kumari, R. Chugh, Generation of new fractals via SP-orbit with $s$-convexity, Int. J. Eng. Technol., 9 (2017), 2491–2504. https://doi.org/10.21817/ijet/2017/v9i3/1709030282 doi: 10.21817/ijet/2017/v9i3/1709030282
|
| [22] |
N. Özgür, S. Antal, A. Tomar, Julia and Mandelbrot sets of transcendental functions via Fibonacci–Mann iteration, J. Funct. Spaces, 2022 (2022), 2592573. https://doi.org/10.1155/2022/2592573 doi: 10.1155/2022/2592573
|
| [23] | M. Ashish, M. Rani, R. Chugh, Study of cubic Julia sets in NO, J. Comput. Sci. Technol., 1 (2013), 13–17. |
| [24] |
K. Gdawiec, W. Kotarski, A. Lisowska, Biomorphs via modified iterations, J. Nonlinear Sci. Appl., 9 (2016), 2305–2315. https://doi.org/10.22436/jnsa.009.05.33 doi: 10.22436/jnsa.009.05.33
|
| [25] | M. Rani, Cubic superior Julia sets, In: Proceedings of the 5th European conference on European computing conference, 2011, 80–84. |
| [26] | V. V. Strotov, S. A. Smirnov, S. E. Korepanov, A. V. Cherpalkin, Object distance estimation algorithm for real-time FPGA-based stereoscopic vision system, In: Proceedings Volume 10792, High-performance computing in Geoscience and remote sensing VIII, 2018. https://doi.org/10.1117/12.2324851 |
| [27] |
J. S. Pan, S. Q. Zhang, S. C. Chu, C. C. Hu, J. Wu, Efficient FPGA implementation of sine cosine algorithm using high-level synthesis, J. Internet Technol., 25 (2024), 865–876. https://doi.org/10.70003/160792642024112506007 doi: 10.70003/160792642024112506007
|
| [28] |
Y. Wang, S. Liu, H. Li, D. Wang, On the spatial Julia set generated by fractional Lotka–Volterra system with noise, Chaos Soliton Fract., 128 (2019), 129–138. https://doi.org/10.1016/j.chaos.2019.07.044 doi: 10.1016/j.chaos.2019.07.044
|
| [29] |
L. O. Jolaoso, S. H. Khan, Some escape time results for general complex polynomials and biomorph generation by a new iteration process, Mathematics, 8 (2020), 2172. https://doi.org/10.3390/math8122172 doi: 10.3390/math8122172
|
| [30] |
W. Nazeer, S. M. Kang, M. Tanveer, A. A. Shahid, Fixed point results in the generation of Julia and Mandelbrot sets, J. Inequal. Appl., 2015 (2015), 298. https://doi.org/10.1186/s13660-015-0820-3 doi: 10.1186/s13660-015-0820-3
|
| [31] |
I. Andreadis, T. E. Karakasidis, On a topological closeness of perturbed Julia sets, Appl. Math. Comput., 217 (2010), 2883–2890. https://doi.org/10.1016/j.amc.2010.08.024 doi: 10.1016/j.amc.2010.08.024
|
| [32] |
Y. Sun, P. Li, Z. Lu, Generalized quaternion M sets and Julia sets perturbed by dynamical noises, Nonlinear Dynam., 82 (2015), 143–156. https://doi.org/10.1007/s11071-015-2145-7 doi: 10.1007/s11071-015-2145-7
|
| [33] |
D. Li, M. Tanveer, W. Nazeer, X. Guo, Boundaries of filled Julia sets in generalized Jungck–Mann orbit, IEEE Access, 7 (2019), 76859–76867. https://doi.org/10.1109/access.2019.2920026 doi: 10.1109/access.2019.2920026
|
| [34] | A. Tomar, D. J. Prajapati, S. Antal, S. Rawat, Variants of Mandelbrot and Julia fractals for higher-order complex polynomials, Math. Methods Appl. Sci., 2022, 1–13. https://doi.org/10.1002/mma.8262 |
| [35] |
S. M. Kang, W. Nazeer, M. Tanveer, A. A. Shahid, New fixed point results for fractal generation in Jungck–Noor orbit with $s$-convexity, J. Funct. Spaces, 2015 (2015), 963016. https://doi.org/10.1155/2015/963016 doi: 10.1155/2015/963016
|
| [36] |
S. Antal, A. Tomar, D. J. Prajapati, M. Sajid, Variants of Julia and Mandelbrot sets as fractals via Jungck–Ishikawa fixed point iteration system with $s$-convexity, AIMS Mathematics, 7 (2022), 10939–10957. https://doi.org/10.3934/math.2022611 doi: 10.3934/math.2022611
|
| [37] |
K. Gdawiec, A. A. Shahid, Fixed point results for the complex fractal generation in the S-iteration orbit with $s$-convexity, Open J. Math. Sci., 2 (2018), 56–72. http://dx.doi.org/10.30538/oms2018.0017 doi: 10.30538/oms2018.0017
|
| [38] |
H. Zhang, M. Tanveer, Y. X. Li, Q. Peng, N. A. Shah, Fixed point results of an implicit iterative scheme for fractal generations, AIMS Mathematics, 6 (2021), 13170–13186. https://doi.org/10.3934/math.2021761 doi: 10.3934/math.2021761
|
| [39] |
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
|
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Anita Tomar, Swati Antal, Mohammad Sajid, Darshana J. Prajapati. Role of $ s $-convexity in the generation of fractals as Julia and Mandelbrot sets via three-step fixed point iteration[J]. AIMS Mathematics, 2025, 10(11): 26077-26105. doi: 10.3934/math.20251148
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