1. Introduction
Fractals are frequently found in nature because they are good descriptions of things like rivers, clouds, crystals, lightning, electricity, tree branches, and leaf patterns. The study of many natural or living frameworks, such as the fractality in water distribution networks, benefits greatly from the use of fractals [
1]. To understand and predict violent streams, fractals are also used in liquid mechanics. Fractal geometries play a crucial role in determining where water quality sensors should be placed in the water delivery network [
2] and fractal river network [
3]. Additionally, fractal theory is frequently applied in several fields, including engineering models, video compression [
4], and computational architectural design [
5]. Ring-shaped quantum dots play an important role in confining the electrons along a circular orbit. Remarkably, the effect of Mandelbrot fractality is explored by considering different types of Mandelbrot rings [
6]. According to research by cognitive neuroscientists, computer-generated fractals can reduce stress in viewers in precisely the same way as fractals seen in nature [
7]. These facts serve as the authors’ inspiration to use the concepts from fixed-point theory to build a new escape criterion. Then, using the escape radius, we present a comparative analysis of the fractals generated by two different iterations. It turned out that the recently suggested hybrid iteration method creates new, more interesting, and spontaneous visuals than the S-iteration does. Therefore, we first examine and comprehend several related concepts in the following paragraphs before presenting the main results.
In 1980, Mandelbrot introduced the Mandelbrot set when he was studying complex quadratic function
He utilizes
c as a complex parameter and invented the word “Fractal” for such best-known illustrations generated by a very basic computational method [
8]. The Mandelbrot set can be identified using a fractal artistic design [
9]. Fractal art is commonly settled with the help of fractal-creating software. In certain examples, graphics tools are utilized to additionally adjust the patterns produced. It is well known that fractal images cannot be created without computer fractal art because a computer has the extraordinary ability to calculate [
10]. Almost certainly, fractal art is recognized from other digital activities [
11,
12]. Fractals are generated using iterative techniques to solve polynomial equations or non-linear equations. Generating fractals can be an artistic endeavour, a mathematical model, or just a soothing diversion.
Various properties and extensions of the Mandelbrot set have been studied extensively in the literature. The initial and most evident generalization included the application of the function
rather than the second-degree polynomial [
13,
14]. Other categories of functions were also examined in the literature such as anti-polynomials [
15], transcendental [
16], rational [
17], elliptic [
18], etc. Another extension in the study of Mandelbrot sets is from complex number system [
19] to bicomplex numbers [
20], quaternions [
21], octonions [
22], etc. Some fixed-point techniques were also used for the extension and generalization of Mandelbrot sets. For the creation of fractals using fixed-point theory, different cyclical methods for locating fixed points in an identified map are used; for example, inversion fractals [
23], v-variable fractals [
24], iterated function systems [
25], and biomorphs [
26]. A new and novel iteration scheme is applied to solve image deblurring and signal recovery problems [
27] using polynomiography. The Mann iteration was used by Rani and Kumar [
28,
29] to visualize superior Julia and Mandelbrot sets. Afterwards, the Ishikawa iteration was used in [
30,
31] for the visualization of relative superior Julia and Mandelbrot sets. The authors in [
32] proved that the convergence rate of S-iteration is higher than that of the Ishikawa iteration and presented relatively superior Mandelbrot sets through the S-iteration scheme. In [
33], the Noor orbit was used by Rani et al. to visualize Mandelbrot sets. Li et al. utilize the Jungck–Mann orbit creation of Mandelbrot sets in [
34]. Different iterative schemes were used by different researchers, such as in [
35,
36] Jungck–CR iterative formulas having a specific convexity. Similarly, in [
37], S-iteration orbit with s-convexity was used. Then in [
38], Jungck–Mann and Jungck–Ishikawa iterations with s-convexity were used. Noor orbit and s-convexity were also used in [
39]. Recently, Zou et al. [
40] introduced the Mandelbrot and Julia sets via hybrid Picard–Mann iteration. For more recent and updated studies on related iterations and techniques, the interested reader is referred to [
41,
42,
43]
From this review of the literature, one can observe that most fractals have escape requirements that dictate their dynamical behaviour, and they do so through a variety of iterative strategies. One key idea is escape when figuring out if a point in the complex plane is contained in a Mandelbrot set. Thus, it is believed that accurately creating a high-level “Mandelbrot” is challenging and intricate [
6]. One especially unique feature of fractals that is lacking in other systems is scaling invariance. They are very suitable for real-world situations where researchers may conduct studies on multiple dimensions because of this feature. Since creating fractals is crucial from several perspectives indicated above, a lot of focus has been placed on doing it in recent years, utilizing a variety of methods. Multiple iterative procedures discussed above have also been used to create some fractals with generic properties [
44,
45,
46,
47]. In these situations, a decision must be made between a few different iteration techniques while considering crucial factors. For instance, an iteration method is more effective than the others based on two primary criteria: simplicity and convergence speed. Under such circumstances, the following issues inevitably surface: Which of these iteration techniques is accelerating convergence? Hence, it was demonstrated that the Picard S-iteration method converges more quickly than the CR iteration method and, consequently, more quickly than any other known iteration method, as well as all of the Picard, Mann, Ishikawa, Noor, SP, and S methods [
48]. To the best of our knowledge, this recently developed three-step iteration in the literature has not been applied to fractals. Hence this research article fills this gap, which is significant because many iteration systems [
49,
50,
51] provide variants for the same function in terms of shape, size, colour, and additional attributes. Furthermore, the results of research conducted at various scales might paint somewhat distinct images even though the geometrical features of the structure itself are the same. This is true for systems with diverse structures. Conversely, a mathematical fractal is a kind of feedback that depends on recursion; it is constructed around an iterated equation. Iteration is essential to fully appreciating the aesthetic value and beauty of fractals. It is simpler to visualize self-similar behaviour as consistently carrying out a task. Each time a step is finished, we replace each initiator copy with a smaller generator copy, rotating as necessary. Taking motivation from these facts, we use this newly developed iteration scheme to prove an escape criterion. It proved useful for generating fractals by analyzing their dynamic patterns in the presence of complex polynomials. The fact that a new iteration strategy was successfully implemented emphasizes how significant this research is compared to the previous ones. The dependence of time on variations in the involved parameters of the iteration scheme is analyzed numerically and graphically. The three-stage, fast-convergence Picard S-iterative technique is therefore more notable and unique than the corpus of existing work.
The plan of this article is as follows: We discuss some necessary definitions and preliminaries related to different iterations in
Section 2. Then, we prove a general escape criterion to generate the fractals in
Section 3. This criterion is proved using a new hybrid Picard S-iteration method for general complex polynomials. Further refinement of these criteria is also presented in the form of corollaries. The visualization of Mandelbrot sets is provided in
Section 4. The pseudocode to create the Mandelbrot sets in Picard S-orbit is also exhibited in this section. Based upon that
Section 4.1 and
Section 4.2 contain a variety of images for Mandelbrot sets by considering different parameter values in the proposed iteration technique. A comparison of fractal images produced by using the new proposed iteration and the classical S-iteration is also provided in this section. Time analysis is performed for seconds using tabular and graphical illustrations. This comparison proved that the Mandelbrot set images produced by the hybrid Picard S-iteration are far better than those produced by S-iteration. This comparison is discussed and concluded in
Section 5, leading to some future directions of this work.