Fixed Point Results for Fractal Generation of Complex Polynomials Involving Sine Function via Non-Standard Iterations

Due to the uniqueness and self-similarity, fractals became most attractive and charming research field. Nowadays researchers use different techniques to generate beautiful fractals for a complex polynomial <inline-formula> <tex-math notation="LaTeX">$z^{n}+c$ </tex-math></inline-formula>. This article demonstrates some fixed point results for a sine function (i.e. <inline-formula> <tex-math notation="LaTeX">$\sin (z^{n}) +c$ </tex-math></inline-formula>) via non-standard iterations (i.e. Mann, Ishikawa and Noor iterations etc.). Since each two steps iteration (i.e. Ishikawa and S iterations) or each three steps iteration (i.e. Noor, CR and SP iterations) have same escape radii for any complex polynomial, so we use these results for S, CR and SP iterations also to apply for the generation of Julia and Mandelbrot sets with <inline-formula> <tex-math notation="LaTeX">$\sin (z^{n}) +c$ </tex-math></inline-formula>. At some fixed input parameters, we observe the engrossing behavior of Julia and Mandelbrot sets for different <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>.


I. INTRODUCTION
To draw the graphs via escape time algorithms in the form of unique and self-similar images by using some electronic tools became an attractive field named as fractals. The work on fractals was started in early twentieth century when Pierre Fatou and Gaston Julia tried to find the successive approximations of f : z −→ z 2 + c where z, c ∈ C. In 1919 Julia [1] was succeeded to iterate this function but failed to sketch it. Mandelbrot [2] is known as the father of fractals because he used the word fractal for the complex graphs of f : z −→ z 2 + c. In 1985, he sketched the Julia set and studied its features. He observed that for different values of c the Julia sets have diversity in their nature. Moreover, he changed the roles of z and c in Julia set and defined a new set called Mandelbrot set. In Julia set we study the behaviour of the iterates for each z, and in the Mandelbrot set we study connectedness of Julia set for each c defining those sets. His work was then extended by Lakhtakia et al. [3] in 1987. They generalized the Mandelbrot set for The associate editor coordinating the review of this manuscript and approving it for publication was Yeliz Karaca .
f : z −→ z p + c where p ≥ 2. Crowe et al. [4] defined the anti Julia and anti Mandelbrot sets in 1989 and discussed their connected locus. They generated complex graphs for z 2 + c and later on these graphs called ''tricorn'' [5]. The property to have uniqueness and self-similarity caused that fractals found applications in image encryption [6] or compression [7], cryptography [8], art and design [9]. The applications of fractal theory in the fields of electrical and electronics engineering revolutionized the industry of security control system, radar system, capacitors, radio and antennae for wireless system [10], [11]. Moreover, architects and engineers sketched and designed the maps of different projects on the basis of fractal theory [12]. Many generalizations have been made in fractals via various fixed point iterations. Fractals for rational and transcendental complex functions were elaborated in [13]. Higher dimensional fractals were discussed in [14], [15] and [16]. Interesting generalized Julia and Mandelbrot sets visualized with various iterations can be found in the literature, e.g, Mann iteration [17], Ishikawa-iteration [18], S-iteration [19], Noor-iteration [20], CR-iteration [21] and SP-iteration in [22]. Moreover, the Jungck-type iterations were used in [23]- [28] and [29]. Various iterations VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ were also used to generate biomorphs [30], [31] and multi-corns [19], [32]. Throughout the history of fractals, researchers proved escape criteria for complex polynomials z n +c and z n +az+c. During this research, we found some images on Internet and papers in which authors just generated Julia and Mandelbrot sets with sin(z n ) + c, but they could not prove the escape criteria for this complex function. In this article, we prove the escape criteria for Mann, Ishikawa and Noor iterations for the complex function sin(z n ) + c. We use the proved criteria in algorithms to generate Julia and Mandelbrot sets. At some fixed input parameters we present the image comparisons of Julia and Mandelbrot sets in orbits of some nonstandard iterations (i.e.In Mann-orbit, Ishikawa-orbit, S-orbit, Noororbit, CR-orbit and in SP-orbit).
The rest of paper is organized as follows. Some important definitions of Mandelbrot and Julia sets and various iterations are presented in Sec. II. Section III presents some fixed point results in the generation of fractals for a complex function sin(z n ) + c. We discuss the behavior of Julia and Mandelbrot sets for different n via proposed algorithms in Sec. IV. We conclude this article in Sec. V.

II. PRELIMINARIES
In this section we present definitions of sets and iterations used in this article.
Definition 1 (Julia set [1]): Let T c : C → C be a complex mapping, where c ∈ C is a parameter. Then the set of points where T k c (z) is the k-th iterate of z is called the filled Julia set. The set of boundary points of J T c is called simple Julia set.
Definition 2 (Mandelbrot set [33]): Let T c : C → C be a complex mapping, where c ∈ C is a parameter. Then the collection of constants c for which the corresponding Julia set J T c is connected is called as Mandelbrot set M , i.e., equivalently Mandelbrot set can be defined as [34]: where θ is any critical point of T c , this is true for T (z) = z n +c, because its critical point is z = 0 and T (0) = c, so after the first iteration we get z 1 = c, so we can omit the first iteration and take c as the z 0 . There are many fixed point iterations in literature, but the basic one was introduced by Charles Emile Picard.
To generate fractals, it is necessary to define the orbit of the iteration. The orbits of our proposed iterations are defined and generalized as follows: Definition 10 (Orbit for any proposed iteration): Let T (z k ) = sin(z n k ) + c be a complex sine function with n ≥ 2. Then the sequence of iterates {z k } k∈N from any proposed iterations (i.e. from iterations (4)- (10)) is called orbit of that iteration.

III. FIXED POINT RESULTS
In this section we prove some fixed point results (i.e. escape criterion or limitations) for complex sine function T (z) = sin(z n ) + c where n ≥ 2 and c ∈ C via proposed-iterations. Algorithms are important to generate fractals and escape limitations are the basic key to run the algorithms. Since where x, y, z ∈ C. Throughout this section we use T (z) as T c (z), z 0 = z, y 0 = y and x 0 = x. Furthermore, to generate Julia and Mandelbrot sets, we assume that the sums of series Theorem 1: Assume that T c (z) = sin(z n ) + c where n ≥ 2 and c ∈ C be a complex sine function with |z| ≥ |c| > Picard-iteration is defined as follows: Proof: Since T (z) = sin(z n ) + c and z 0 = z, then Picard-iteration is: For k = 0, we have For k = 1, we have Theorem 2: Assume that T c (z) = sin(z n ) + c where n ≥ 2 and c ∈ C be a complex sine function with |z| ≥ |c| > 2 α 1 |a 1 | Mann-iteration is defined as follows: where α 1 ∈ (0, 1] and k = 0, 1, 2, . . ., then |z k | −→ ∞ when k −→ ∞. VOLUME 8, 2020 Proof: Since T (z) = sin(z n )+c, then the Mann-iteration is: For k = 0, we have and z ∈ C except those values of z wherefore |a 1 | = 0.

IV. APPLICATIONS IN FRACTALS
There are many types of complex fractals (i.e. Julia sets, Mandelbrot sets, Multibrot sets, Multicorns sets, Biomorphs and root finding fractals etc.). To generate such fractals some methods needed to execute the algorithms. In literature the most popular methods to generate the fractals are as follows: • Distance Estimator [45], • Potential Function Algorithms [46] and • Escape Criterion [47]. We use escape criteria in Algorithms 1 and 2 to fascinate the Julia and Mandelbrot sets in graphs. The specification of the computer used to generate the examples was the following:                                 (i.e. α 1 , α 2 , α 3 , |a 1 |, |a 2 |, |a 3 |). The image generation time is also calculated for each iteration.
Example 2: In this example we generate the Julia sets n = 3. In Figs.36-42 the parameters are α 1 = α 2 = α 3 = 0.5,                          have same geometry but slightly differ in Julia points. Each iterations have also miner difference in shape with others.

B. MANDELBROT SETS
Mandelbrot coined the word fractal for a self-similar complex graph of z 2 + c. He determined the properties of quadratic complex fractal and he observed that a classical quadratic Mandelbrot set has two main parts, one is the primary part (i.e. Belly of the Mandelbrot set or Cardiod) and other is the secondary part that contains one large and two small bulbs. Later on Mandelbrot set was generalized in such a way that each Mandelbrot set has n small bulbs, n − 1 cardioids and small bulbs for n ≥ 2. In this subsection we sketch some VOLUME 8, 2020                   Example 6: The last example presents the higher degree Mandelbrot sets of complex sine function T (z) = sin(z 10 ) + c. In this example we also fixed the parameters as α 1 = 0.9, α 2 = 0.5, α 3 = 0.9 (i.e. in Figs.71-77)

V. CONCLUSION
In this research we proved fixed point results for complex sine function T (z) = sin(z n ) + c where n ≥ 2 and c ∈ C via Picard, Mann, Ishikawa and Noor iterations. We applied these results in algorithms and generalized the Julia and Mandelbrot sets. We demonstrated some examples for Julia and Mandelbrot sets of a complex sine function by using the proved results. We also compared the images at some fixed parameters and calculated the image generation time in second for each image. Moreover, we analyzed that images of Julia and Mandelbrot sets slightly different for each iteration.