Abstract
In this paper, utilizing the Fibonacci-Mann iteration process, we explore Julia and Mandelbrot sets by establishing the escape criteria of a transcendental function, , ; here, is a complex variable, and and are complex numbers. Also, we explore the effect of involved parameters on the deviance of color, appearance, and dynamics of generated fractals. It is well known that fractal geometry portrays the complexity of numerous complicated shapes in our surroundings. In fact, fractals can illustrate shapes and surfaces which cannot be described by the traditional Euclidean geometry.
1. Introduction and Preliminaries
Let us consider the well-known Fibonacci sequence defined recursively by with the initial conditions . Recently, a novel iteration process, Fibonacci-Mann iteration, is introduced as where and (see [1] for more details). It is worth mentioning here that a fixed point iteration performs a significant role in the generation of geometrical pictures of classical Julia and Mandelbrot sets (for instance, see [2–4], and the references therein). In [2], by establishing the escape criteria for a complex function where is a complex variable and and are complex numbers; new Julia sets were studied by providing new algorithms for exploring Julia sets utilizing four distinct iterations (the Picard iteration [5], the Mann iteration [6], the Ishikawa iteration [7], and the Noor-iteration [8]). Also, the effects of change in values of parameters on the deviance of color appearance and dynamics of fractals were investigated in the sequel.
Motivated by these recent studies, our aim in this paper is to develop escape criteria for a function of the form (3) using a new algorithm via the Fibonacci-Mann iteration process (2) for visualizing the stunning fractals. It is well known that the escape criterion [9] is indispensable for exploring the Mandelbrot and Julia sets. We furnish some graphical illustrations of the generated complex fractals using the MATLAB software, algorithm, and colormap to demonstrate the variation in images and explore the effect of the involved parameters on the deviance of color, appearance, and dynamics of generated fractals. Also, we observe that as we zoom in at the edges of the petals of the Mandelbrot set, we come across the Julia set meaning thereby each point of the Mandelbrot set includes massive image data of a Julia set.
A filled Julia set is the set of complex numbers so that the orbits do not converge to a point at infinity ([10, 11]). For the polynomial of degree , we denote it by , that is,
The boundary of is the Julia set; that is,
The set of parameters so that the filled Julia set of the polynomial is connected is known as the Mandelbrot set ([12, 13]), that is, or
2. An Escape Criteria via Fibonacci-Mann Iteration Process
In this section, we establish an escape criterion for the complex transcendental function (3). We take , , , and as . Suppose that where , except for the values of , , and so that Then, we have and so
except for the value of so that ,
Theorem 1. Let , , and the sequence of iterates be the Fibonacci-Mann iteration. Suppose and where . Then, we have as .
Proof. Let , . Now
For , since we have and , considering inequality (9), we get
Hence, we obtain
Let . Since , following similar steps and using the inequality (13), we obtain
and so
Because, by inequality (13) and the fact that , it is easy to see that , and this implies
Again, using the inequality and (14), we find .
Let and set . By inequality (10), it is easy to see that
Using this last inequality and inequality (9), we get
and this implies
Since , we have
and hence,
Similarly, by inequalities (15), (19), and (21), we get
Repeating this process till th term, we find
Then, because of inequality (10), we have
where . This implies that the orbit of tends to infinity; that is, we find as .
Corollary 2. If we consider , then the Fibonacci-Mann orbit escapes to infinity.
Remark 3. The motivation for choosing the Fibonacci-Mann iteration method in the generation of Julia and Mandelbrot fractal sets is the fact that for , both Mann iteration, as well as Fibonacci-Mann iteration, converge to a fixed point. However, the Fibonacci-Mann iteration converges faster than the Mann iteration. But for , Mann iteration needs not converge to a fixed point; however, the Fibonacci-Mann iteration converges for all the initial values. By taking in inequality (2), we get the Mann iteration [6]. Also, for and , we get the Picard iteration [5]. It neither reduces to Ishikawa-iteration [7], nor to Noor-iteration [8] since Ishikawa-iteration is a two-step process and Noor-iteration is a three-step process. On the other hand, Antal et al. [2] used the Picard iteration, the Mann iteration, the Ishikawa iteration, and the Noor-iteration to explore and compare the fractals as Julia sets. It is well known that Banach [14] utilized Picard iteration [5] to approximate a fixed point for underlying contraction mapping. But when we use slightly weaker mapping, then Picard iteration needs not converge. Consequently, Mann iteration [6], Ishikawa iteration [7], Krasnosel’ski iteration [15], modified Mann iteration [16], and so on have been introduced by distinct researchers to solve this issue for different contractions.
Remark 4. In Theorem 1, we proved the conclusion by symmetry by starting with taking , then and repeating the process till the term. The parameters selected have not been studied in this point of view till now and are new. We refer the interested reader to [17, 18] for a detailed information about the Fibonacci sequence. It is well-known that the golden ratio and the Fibonacci sequence have numerous applications which range from the description of plant growth, the crystallographic structure of certain solids to music, and the development of computer algorithms for searching data bases. This fascinating sequence of numbers is named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci. It is interesting to recall that the Fibonacci sequence is initially explored by an ancient Indian mathematician and poet Acharya Pingala (450 BC-200 BC), the author of the Chandaśāstra (the earliest known treatise on Sanskrit prosody).
3. Generation of Julia and Mandelbrot Sets
We use MATLAB 8.5.0 (R2015a) for developing fractals for transcendental complex sine function (3) via the Fibonacci-Mann iteration (2) process. We develop Algorithms 1 and 2 to explore the geometry of Julia and Mandelbrot sets, respectively. It is interesting to notice that the structure of the fractals is very much dependent on the selection of iterative processes. During the simulation process, we have obtained and analyzed many fractals but included a limited number of fractals to discuss the behavior for the different parameter values associated with it. The parameters , and perform a very significant role in giving vibrant colors and exploring the characteristics of the associated Julia sets and Mandelbrot sets. Throughout the paper, we use the standard “jet” colormap (as shown in Figure 1).
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3.1. Julia Set
As we change the value of (see Table 1), keeping other parameters fixed, we get amazing fractals, which are visible in Figures 2(a)–2(f). As the value of increases, the fractal takes a circular shape. For , we obtain a Julia set that is similar to a circular saw or colorful teething ring (Figure 2(f)).
(a) Quadratic
(b) Quartic
(c) Quintic
(d) Sextic
(e) Octic
(f) Decic
The parameter gives rotational symmetry when it is purely real (imaginary) and changes the sign. For the same set of parameters and only changing the sign of real and complex parameter as in Table 2, the resultant fractals can be seen in Figures 3(a)–3(d).
(a)
(b)
(c)
(d)
The parameter also adds beauty to the fractals. As the absolute value of increases keeping other parameters the same (as in Table 3), the more aesthetic fractals can be seen (Figures 4(a)–4(d)).
(a)
(b)
(c)
(d)
The impact of change in the values of parameters and simultaneously (see Table 4) on the cubic Julia set can be seen in Figures 5(a)–5(c). Noticeably, cubic Julia set in Figure 5(a) is symmetrical about both the axes; however, in Figures 5(b) and 5(c), it is symmetrical only about -axis. Changes in the values of and from complex to real as well as a decrease in absolute value add beauty to resulting fractals.
(a)
(b)
(c)
The parameter is responsible for the volume of the fractal (see Table 5). Even a slight decrease in from 0.35 to 0.20 expands the quintic Julia set which are symmetrical about -axis as shown in Figures 6(a)–6(c).
(a)
(b)
(c)
3.2. Mandelbrot Set
Like Julia set, Mandelbrot also becomes rounded (see Figures 7 and 8) as increases (Tables 6 and 7). Noticeably, the number of branches in Figures 7(a)–7(e) is while the number of branches in Figures 8(a)–8(f) is () (unlike Figure 7).
(a) Quadratic
(b) Cubic
(c) Quartic
(d) Quintic
(e) Septic
(a) Quartic
(b) Quintic
(c) Sextic
(d) Septic
(e) Octic
(f) Nonic
Figure 9 demonstrates the effect of change in sign as well change in real to the complex value of parameter on quadratic Mandelbrot set (see Table 8).
(a)
(b)
(c)
Lower values (Table 9) of give more beautiful, artistic, and larger fractals which are symmetrical about -axis (Figures 10(a)–10(c)).
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(b)
(c)
Figure 11 demonstrates the effect of change in parameters and simultaneously on the quintic Mandelbrot set (see Table 10).
(a)
(b)
(c)
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Figure 12 demonstrates the effect of change in parameters and simultaneously on the cubic Mandelbrot set (see Table 11). Figures 12(a) and 12(b) appear like a pair of duck which are mirror images of each other.
(a)
(b)
Remark 5. (i)During the generation of fractals, it is surprising to see that, for the same parameter set values, the effect of even minor changes in one parameter causes a major impact on the appearance of the resultant fractal. Consequently, it is significant to select appropriate parameters to obtain the desired fractal pattern.(ii)The majority of Julia and Mandelbrot sets generated by the sine function are symmetrical about the -axis except Figures 2(a)–2(c) and Figures 3(a) and 3(b).(iii)The change in the sign of the value of parameters leads to reflexive and rotational symmetry.(iv)The Julia and Mandelbrot fractals explored in this work are aesthetic, novel, and pleasing because the complex sine function contains a lot of attributes in it. The motivation behind this is the fact that on altering the iteration process, the dynamics and behavior of the fractals are also altered, which are significant from the graphical as well as applications viewpoint.(v)We have displayed just the zoomed kind of fractals since the transcendental function is unbounded so that the fractals which occupy the infinite area may lie in. But due to the unboundedness of only on a real and imaginary axis, it can be observable.(vi)Almost all the fractals occupy the area from to .
4. Conclusion
We have generated Mandelbrot and Julia sets of various transcendental complex sine functions to demonstrate the significance of the newly developed Fibonacci-Mann iteration process. We have analyzed the behavior of variants of the Julia and Mandelbrot sets for different parameter values after obtaining fascinating nonclassical variants of classical Mandelbrot and Julia fractals using the MATLAB software. We have noticed that the role of each parameter is distinct. Therefore, we have restricted our discussion to a limited type of combination of parameters. However, we have tried to cover the maximum possible combination of parameters involved in developing the algorithm (escape criterion) in the Corollary 2. Also, we have observed that as we zoom in on the edges of the petals of the Mandelbrot set, we come across the Julia set meaning thereby each point of the Mandelbrot set includes massive image data of a Julia set. Also, the size of fractals relies on the value of parameter . As the value of parameter increases, the area captured by the fractals decreases, and its shape becomes circular. On the other hand, the shape as well as the symmetry of each fractal relies on the values of parameters and . We have explored a new technique via Fibonacci-Mann iteration for visualizing the filled-in Julia and Mandelbrot sets.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.