Comparative Study of Some Fixed-Point Methods in the Generation of Julia and Mandelbrot Sets

Fractal is a geometrical shape with property that each point of the shape represents the whole. Having this property, fractals procured the attention in computer graphics, engineering, biology, mathematics, physics, art, and design. ,e fractals generated on highest priorities are the Julia andMandelbrot sets. So, in this paper, we develop some necessary conditions for the convergence of sequences established for the orbits of M, M∗, and K-iterative methods to generate these fractals. We adjust algorithms according to the develop conditions and draw some attractive Julia and Mandelbrot sets with sequences of iterates from proposed fixed-point iterative methods. Moreover, we discuss the self-similarities with input parameters in each graph and present the comparison of images with proposed methods.


Introduction
e Latin word fractal (means fractured, divided, or broken) is commonly used for an image having the property of selfsimilarity in complex graphics [1]. Fractals have many applications in social sciences and engineering. In computer engineering, fractals are used to establish the security system, computer networking, image encryption, image compression, and cryptography [2]. In biology, fractals are used to study the culture of microorgans, nerve system, etc. [3]. In physics, fractals are used in fluid mechanics to understand the nature of fluids and their properties. Fractals are used in electrical and electronics engineering (i.e., in the fabricating of antennae, radar system, capacitors, security control system, radio, and antennae for wireless system) [4,5]. Moreover, architectural patterns and designs are also fractals [6]. Fractals have application in many other emerging fields [7][8][9].
Before the invention of computer, the researchers sketched aesthetic patterns, images, graphs, and geometries manually. e graph of cantor set, Koch snowflake, and Sierpinski's triangles are the patterns that can be generated manually. In 1918, Gaston Julia and Pierre Fatou defined two complementary sets (i.e., Julia set and Fatou set). But they could not sketch the graphs of Julia set and Fatou set. After the invention of computers, Mandelbrot made it possible to draw the graphs of Julia set with help of computers in 1970. He studied the Julia set for a polynomial Q a 0 (z m ) � z 2 m + a 0 , where z is a complex variable and a 0 is a complex parameter. Mandelbrot presented the characteristics of Julia set in [10] and explained that Julia set had great diversity of aesthetic designs [11]. e Mandelbrot set for Q a 0 (z m ) � z p m + a 0 , where z is a complex variable and a 0 is a complex parameter, was discussed in [12]. e images resembled with Julia and Mandelbrot sets for rational and transcendental complex functions were visualized in [13]. Some 4D and 3D fractals for quaternions and bicomplex and tricomplex functions were studied in [14,15] and [16]. To generalize Julia and Mandelbrot sets, initially Rani et al. used fixed-point theory in the generation of fractals (refer in [17,18]). Some generalized fractals via explicit fixed-point iterative methods were studied in [19][20][21][22][23][24]. e implicit iterative methods were used to develop convergence criterion for fractals in [25][26][27][28][29][30].
ere are some well-known criterions to generate the fractals such as distance estimator [37], potential function algorithms [38], and escape criteria [39]. In this paper, we use escape criterion conditions to sketch some bewitching Julia and Mandelbrot sets. In this paper, we develop some necessary conditions for the convergence of |Q m a 0 | to generate fractals (i.e., especially for Julia and Mandelbrot sets) via some fixed-point iterative methods. We use proposed conditions in algorithms to sketch Julia and Mandelbrot sets. Furthermore, we present some graphs to compare the images. e influence of input parameters on images is also discussed.
is paper is composed of five sections: Section 2 deals with some basic concepts about fractals and fixed-point iterative methods; in Section 3, we develop some convergence conditions to generate fractals; we establish comparison among Julia sets and Mandelbrot sets via proposed methods in Section 4, and at the end, we conclude this paper in Section 5.

Some Basic Concepts
In this section, we discuss some basic concepts.
Definition 1 (Julia set [40]). Let Q a 0 (z m ) � z p m + a 0 be a complex polynomial with p ≥ 2. en, the set of points J Q a 0 in C is named as the filled Julia set, when the orbits of the points in J Q a 0 does not move to ∞ as m ⟶ ∞, i.e., where Q m a 0 is the m − th iterate of z. e set of boundary points of J Q a 0 is called the simple Julia set.
Definition 2 (Mandelbrot set [41]). e collection of all connected Julia sets is defined as the Mandelbrot set M, i.e., M � a 0 ∈ C: J Q a 0 is connected .
Equivalently, the Mandelbrot set is defined as [42] M � a 0 ∈ C: Q m Since the critical point of Q a 0 is 0, so the authors set z 0 � 0 as an initial guess. ere are many fixed-point iterative methods in literature that can be used to generate fractals. For each method, the authors prove escape criterion to generate fractals. In this paper, we use M, M * , and Kiterative methods to visualize Julia and Mandelbrot sets. e proposed fixed-point iterative methods are defined as follows.

Convergence Analysis
Here, we prove some convergence conditions (i.e., escape criterion) for complex polynomial Q a 0 (z) � z p + a 0 , where p ≥ 2 and a 0 ∈ C via M, M * , and K-iterative methods, respectively. Without necessary conditions, we cannot generate fractal because the convergence condition is the basic key to run the algorithm. roughout this section, we use Q(z) as Q a 0 (z) and z 0 � z, u 0 � u, v 0 � v, and w 0 � w in the following way.

Applications of Fractals
To visualize the fractals, some convergence conditions are required, and actually, these are the main tools to execute the algorithm properly and sketch the desired type of fractals. In literature, the authors fixed maximum number of iterations up to hundred. To check self-similarity and get better results, we fixed the maximum number of iterations at 1000. In this section, we adjust two algorithms: one for the Julia set and other for the Mandelbrot set to generate fractals via proposed methods. We visualize some Julia and Mandelbrot sets for different involve parameters.

Julia Sets.
Julia is known as the pioneer of complex fractals. In this subsection, we sketch some graphs of Julia set at different input parameters. We generate Julia sets for M, M * , and K-iterative methods by using Algorithm 1 and compare the images of Julia set for proposed methods. Example 1. In this example, we present the Julia sets for a polynomial Q(z) � z 2 + a 0 , where a 0 ∈ C in the orbits of M, M * , and K-iterative methods, respectively. e graphs in Figures 1-3 for a 0 � − 0.05 − 0.63i, a � 0.01, b � 0.9, and A � [− 1.5, 1.5] 2 are quadratic Julia sets in the orbits of M, M * , and K-iterative methods, respectively. e images in Figures 1 and 3 are Julia sets resembling Chinese dragon having two repelling fixed points: one is at the right end spiral, and other is in spiral on the left side. e image in Figure 2 is a filled connected quadratic Julia set. e graphs in Figures 4-6  Example 2. In second example, we visualize some cubic Julia sets for a polynomial Q(z) � z 3 + a 0 , where a 0 ∈ C in the orbits of M, M * , and K-iterative methods, respectively. e images in Figures 10-12 are like the cubic Douady rabbits.
We observe the image in Figure 10 is a smart Douady rabbit, in Figure 11 is a fat Douady rabbit, and in Figure 12 is a relatively weak but more attractive Douady rabbit for cubic complex polynomial. e main body of graphs in Figures 13-15 is like a circular saw having three teeth. e Input: Q a 0 � z p + a 0 -a complex polynomial, A-covered area, M � 1000, a, b ∈ (0, 1]), a 0 ∈ C-involved parameters, coloursmap [0..h − 1] with h colours. Output: sketched Julia set.

Mandelbrot Sets.
Mandelbrot examined the graph of complex polynomial Q(z) � z 2 + a 0 and observed that the main body of image is a cardioid having a large bulb symmetry along x-axis and two small bulbs symmetry along y-axis. e image of Q(z) is usually called the classical Mandelbrot set, and it is also called God's thumb. In this subsection, we sketch some graphs of Mandelbrot set at different input parameters for M, M * , and K-iterative methods by using Algorithm 2 and compare the images of Mandelbrot set for proposed methods.
Example 4. In this example, we visualize some graphs of Mandelbrot sets for a polynomial Q(z) � z 2 + a 0 , where a 0 ∈ C in the orbits of M, M * , and K-iterative methods,        (1) for a 0 ∈ A do (2) R-convergence condition for proposed method (3) m � 0 (4) z 0 -initial guess for Q a 0 (5) while m ≤ K do (6) Proposed iterative method (7) if |z m+1 | > R then (8) break (9) m � m + 1     number of bulbs in different sizes, but if we magnify any bulb of image, it reflects the shape of whole image. In the generation of graphs in Figures 22-24, we change input             Example 6. e last example demonstrates the ochto Mandelbrot sets for a polynomial Q(z) � z 8 + a 0 , where a 0 ∈ C in the orbits of M, M * , and K-iterative methods, respectively. All images for the graphs in Figures 36-38 have the same inputs as a � 0.01, b � 0.9, and A � [− 1.5, 1.5] 2 . We notice that 7 large bulbs appear on the main body of each ochto Mandelbrot set. e shape of bulbs for each method is also different in images.

Conclusions
We analyzed M, M * , and K-iterative methods in the generation of Julia and Mandelbrot sets. We established some convergence conditions for the orbits of M, M * , and Kiterative methods, respectively. We used the established convergence conditions in algorithms to sketch some Julia and Mandelbrot sets. Fascinating Julia and Mandelbrot sets were generated for different input parameters and compared the images. We observed that, for each proposed method, image is slightly different in shape from other two methods. Furthermore, we noticed that, for a very small change in any input parameter, the images drastically changed. Moreover, we concluded that the complex graphs of Julia and Mandelbrot sets generated in this research were the application of fractal geometry.