# Julia and Mandelbrot Sets of Transcendental Function via Fibonacci-Mann Iteration

Authors: Nihal Özgür; Swati Antal; Anita Tomar
Journal: Journal of Function Spaces
 (2022)

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## 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,sinzn+az+c, n≥2; here, z is a complex variable, and a and c 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.


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## Body

## 1. Introduction and Preliminaries

Let us consider the well-known Fibonacci sequencefn defined recursively by
(1)fn+1=fn+fn−1,n≥1,with the initial conditions f0=f1=1. Recently, a novel iteration process, Fibonacci-Mann iteration, is introduced as
(2)zn+1=tnTfnzn+1−tnzn,where tn∈0,1 and n∈ℕ (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
(3)Tz=sinzn+az+c,n≥2,where z is a complex variable and a and c 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 T:ℂ⟶ℂ of degree ≥2, we denote it by FT, that is,
(4)FT=z∈ℂ:Tzkk=0∞isbounded.The boundary ofJT is the Julia set; that is, JT=∂FT.The set of parametersc∈ℂ so that the filled Julia set JTc of the polynomial Tcz=z2+c is connected is known as the Mandelbrot set ([12, 13]), that is,
(5)M=c∈ℂ:JTcisconnected,or
(6)M=c∈ℂ:Tczk↛∞ask⟶∞.


## 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 x0=x, y0=y, z0=z, and Tz as Ta,cz. Suppose that
(7)1−z2n3!+z4n5!−⋯≥u1,1−y2n3!+y4n5!−⋯≥u2,1−x2n3!+x4n5!−⋯≥u3,where ui∈0,1, 1≤i≤3 except for the values of x, y, and z so that u1=u2=u3=0. Then, we have
(8)sinzn=zn−z3n3!+z5n5!−⋯=zn1−z2n3!+z4n5!−⋯,and so
(9)sinzn≥znu1,z∈ℂ except for the value of z so that u1=0, u1∈0,1.Theorem 1.
LetTa,cz=sinzn+az+c,n≥2, a,c∈ℂ, and the sequence of iterates zkk∈ℕ be the Fibonacci-Mann iteration. Suppose t=inftn>0 and
(10)z≥c>2Ta+1tu11/n−1,where T=suptn. Then, we have zk⟶∞ as k⟶∞.Proof.
Letz0=z, Ta,cz=sinzn+az+c. Now
(11)zk+1=tkTa,cfkzk+1−tkzk.
Fork=0, since we have z≥c and f0=1, considering inequality (9), we get
(12)z1=t0Ta,cf0z+1−t0z=t0sinzn+az+c+1−t0z≥t0sinzn−t0az−t0c−1−t0z=t0sinzn−t0az−t0c−1−t0z≥t0u1zn−t0az−t0z−1−t0z=t0u1zn−zt0a+t0+1−t0=t0u1zn−zt0a+1≥tu1zn−zTa+1≥zTa+1tu1zn−1Ta+1−1.
Hence, we obtain(13)z1≥z1Ta+1≥ztu1zn−1Ta+1−1.
Letk=1. Since f1=1, following similar steps and using the inequality (13), we obtain
(14)z2≥z1tu1z1n−1Ta+1−1≥ztu1zn−1Ta+1−1tu1z1n−1Ta+1−1≥ztu1zn−1Ta+1−12,and so
(15)z2≥ztu1zn−1Ta+1−12.
Because, by inequality (13) and the fact that z≥c>2Ta+1/tu11/n−1, it is easy to see that z1≥z, and this implies
(16)z1tu1z1n−1Ta+1−1≥ztu1zn−1Ta+1−1.
Again, using the inequalityz1≥z≥c>2Ta+1/tu11/n−1 and (14), we find z2≥z1.
Letk=2 and set ω1=Ta,cz2. By inequality (10), it is easy to see that
(17)z2n−1u1≥a+2.
Using this last inequality and inequality (9), we get
(18)ω1z2=sinz2n+az2+cz2≥sinz2n−az2−cz2≥z2nu1−az2−z2z2≥z2n−1u1−a−1≥1,
and this implies(19)ω1≥z2.
Sincef2=2, we have
(20)z3=t2Ta,cf2z2+1−t2z2=t2sinω1n+aω1+c+1−t2z2≥t2sinω1n−t2aω1−t2c−1−t2z2=t2sinω1n−t2aω1−t2c−1−t2z2≥t2u1ω1n−t2aω1−t2ω1−1−t2ω1≥t2u1ω1n−ω1t2a+1≥tu1ω1n−ω1Ta+1≥ω1Ta+1tu1ω1n−1Ta+1−1,and hence,
(21)z3≥z3Ta+1≥ω1tu1ω1n−1Ta+1−1.
Similarly, by inequalities (15), (19), and (21), we get
(22)z3≥ztu1zn−1Ta+1−13.
Repeating this process tillkth term, we find
(23)zk≥ztu1zn−1Ta+1−1k.
Then, because of inequality (10), we have
(24)tu1zn−1Ta+1−1>1,where u1∈0,1. This implies that the orbit of z tends to infinity; that is, we find zk⟶∞ as k⟶∞.Corollary 2.
If we considerc>2Ta+1/tu11/n−1, 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 fortn∈0,.5, 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 tn∈0.5,1, Mann iteration needs not converge to a fixed point; however, the Fibonacci-Mann iteration converges for all the initial values. By taking fn=1 in inequality (2), we get the Mann iteration [6]. Also, for fn=1 and tn=1, 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 Theorem1, we proved the conclusion by symmetry by starting with taking k=0, then k=1,k=2, and repeating the process till the kth 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 a,c,n,u1,t,t, and T 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).Algorithm 1:Geometry of Julia set.
Input:Tz=sinzn+az+c, where a,c∈ℂ and n=2,3,⋯; A⊂ℂ− area; K−maximum number of iterations; tn,u1∈0,1−Parameters of the generalized Fibonacci-Mann iteration; colormap0..C−1 -color map with C colors.Output: Julia set for area A.1:forz∈Ado2:R1=2Ta+1/tu11/n−13:R=maxc,R14:n≥15:z=06:whilen≤Kdo7:f0=18:f1=19:fn+1=fn+fn−110:zn+1=tnTfnzn+1−tnzn11:ifzn+1>Rthen12:       break13:     end if14:n=n+115:end while16:i=C−1n/K17:   colorz with colormapi18:end forAlgorithm 2: Geometry of Mandelbrot set.
Input:Tz=sinzn+az+c, where a,c∈ℂ and n=2,3,⋯; A⊂ℂ− area; K−maximum number of iterations; tn,u1∈0,1−Parameters of the generalized Fibonacci-Mann iteration; colormap0..C−1 -color map with C colors.Output: Mandelbrot set for area A.1:forc∈Ado2:R1=2Ta+1/tu11/n−13:R=maxc,R14:n≥15:whilen≤Kdo6:f0=17:f1=18:fn+1=fn+fn−19:zn+1=tnTfnzn+1−tnzn10:ifzn+1>Rthen11:        break12:end if13:n=n+114:end while15:i=C−1n/K16:   colorz with colormapi17:end forFigure 1
Colormap used in the graphical examples.


### 3.1. Julia Set

As we change the value ofn (see Table 1), keeping other parameters fixed, we get amazing fractals, which are visible in Figures 2(a)–2(f). As the value of n increases, the fractal takes a circular shape. For n=10, we obtain a Julia set that is similar to a circular saw or colorful teething ring (Figure 2(f)).Table 1
Parameters for generation of Julia set for different values ofn.
actTtu1n(i)19i−0.835−0.2321i0.00090.10.10.22(ii)19i−0.835−0.2321i0.00090.10.10.24(iii)19i−0.835−0.2321i0.00090.10.10.25(iv)19i−0.835−0.2321i0.00090.10.10.26(v)19i−0.835−0.2321i0.00090.10.10.28(vi)19i−0.835−0.2321i0.00090.10.10.210Figure 2
Effect ofn on Julia set.
(a)
Quadratic(b)
Quartic(c)
Quintic(d)
Sextic(e)
Octic(f)
DecicThe parametera 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 a as in Table 2, the resultant fractals can be seen in Figures 3(a)–3(d).Table 2
Parameters for generation of quartic Julia set for different values ofa.
actTtu1n(i)19i−0.835−0.2321i0.00090.10.10.24(ii)-19i−0.835−0.2321i0.00090.10.10.24(iii)-19−0.835−0.2321i0.00090.10.10.24(iv)19−0.835−0.2321i0.00090.10.10.24Figure 3
Effect of change in sign in the real and complex parametera of quartic Julia set.
(a)(b)(c)(d)The parametera also adds beauty to the fractals. As the absolute value of a increases keeping other parameters the same (as in Table 3), the more aesthetic fractals can be seen (Figures 4(a)–4(d)).Table 3
Parameters for generation of quadratic Julia set for different values ofa.
actTtu1n(i)103.140.000299010.01050.01050.92(ii)203.140.000299010.01050.01050.92(iii)−10+50i3.140.000299010.01050.01050.92(iv)50−50i3.140.000299010.01050.01050.92Figure 4
Effect of increase in the absolute value ofa on quadratic Julia set.
(a)(b)(c)(d)The impact of change in the values of parametersa and c 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 x-axis. Changes in the values of a and c from complex to real as well as a decrease in absolute value add beauty to resulting fractals.Table 4
Parameters for generation of cubic Julia set for different values ofa and c.
actTtu1n(i)40−40i−3.25+3.50i0.00199909140.01910.01910.0123(ii)5.77.50.00199909140.01910.01910.0123(iii)1.82.7180.00199909140.01910.01910.0123Figure 5
Effect of decrease in the absolute value of parametersa and c simultaneously on cubic Julia set.
(a)(b)(c)The parametert is responsible for the volume of the fractal (see Table 5). Even a slight decrease in t from 0.35 to 0.20 expands the quintic Julia set which are symmetrical about x-axis as shown in Figures 6(a)–6(c).Table 5
Parameters for generation of quintic Julia set for different values oft.
actTtu1n(i)2.20.00350.350.1150250.1150250.925(ii)2.20.00350.250.1150250.1150250.925(iii)2.20.00350.200.1150250.1150250.925Figure 6
Effect of decrease in parametert on quintic Julia set.
(a)(b)(c)


### 3.2. Mandelbrot Set

Like Julia set, Mandelbrot also becomes rounded (see Figures7 and 8) as n increases (Tables 6 and 7). Noticeably, the number of branches in Figures 7(a)–7(e) is 2n while the number of branches in Figures 8(a)–8(f) is (n−1) (unlike Figure 7).Figure 7
Effect ofn on Mandelbrot set.
(a)
Quadratic(b)
Cubic(c)
Quartic(d)
Quintic(e)
SepticFigure 8
Effect of change inn on Mandelbrot set.
(a)
Quartic(b)
Quintic(c)
Sextic(d)
Septic(e)
Octic(f)
NonicTable 6
Parameters for generation of Mandelbrot set for different values ofn.
atTtu1n(i)-1.878970.0000260.21050.21050.09322(ii)-1.878970.0000260.21050.21050.09323(iii)-1.878970.0000260.21050.21050.09324(iv)-1.878970.0000260.21050.21050.09325(v)-1.878970.0000260.21050.21050.09327Table 7
Parameters for generation of Mandelbrot set for different values ofn.
atTtu1n(i)-2.20.15939110.1150250.1150250.924(ii)-2.20.15939110.1150250.1150250.925(iii)-2.20.15939110.1150250.1150250.926(iv)-2.20.15939110.1150250.1150250.927(v)-2.20.15939110.1150250.1150250.928(vi)-2.20.15939110.1150250.1150250.929Figure9 demonstrates the effect of change in sign as well change in real to the complex value of parameter a on quadratic Mandelbrot set (see Table 8).Figure 9
Effect of change in sign as well change in real to complex parametera on quadratic Mandelbrot set.
(a)(b)(c)Table 8
Parameters for generation of quadratic Mandelbrot set for different values ofa.
atTtu1n(i)1.878970.0000260.21050.21050.09322(ii)-1.878970.0000260.21050.21050.09322(iii)1.87897i0.0000260.21050.21050.09322Lower values (Table9) of t give more beautiful, artistic, and larger fractals which are symmetrical about x-axis (Figures 10(a)–10(c)).Table 9
Parameters for generation of cubic Mandelbrot set for different values oft.
atTtu1n(i)-2.20.0590.1150250.1150250.923(ii)-2.20.15939110.1150250.1150250.923(iii)-2.20.910.1150250.1150250.923Figure 10
Effect of change in parametert on cubic Mandelbrot set.
(a)(b)(c)Figure11 demonstrates the effect of change in parameters a and t simultaneously on the quintic Mandelbrot set (see Table 10).Figure 11
Effect of change in parametersa and t simultaneously on quintic Mandelbrot set.
(a)(b)(c)(d)Table 10
Parameters for generation of quintic Mandelbrot set for different values ofa and t.
atTtu1n(i)-2i0.130.90250.90250.925(ii)-0.50.15939110.90250.90250.925(iii)00.910.90250.90250.925(iv)-2.2i0.0310.90250.90250.925Figure12 demonstrates the effect of change in parameters T,t, and u1 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.Figure 12
Effect of change in parametersT,t, and u1 simultaneously on cubic Mandelbrot set.
(a)(b)Table 11
Parameters for generation of cubic Mandelbrot set for different values ofT,t, and u1.
atTtu1n(i)-50.05250.20.010.053(ii)-50.05250.30.050.0053Remark 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 thex-axis except Figures 2(a)–2(c) and Figures 3(a) and 3(b).(iii)
The change in the sign of the value of parametersa 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 functionTz=sinzn+az+c 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 functionsinz is unbounded so that the fractals which occupy the infinite area may lie in. But due to the unboundedness of sinz only on a real and imaginary axis, it can be observable.(vi)
Almost all the fractals occupy the area from−0.1,0.1×−0.1,0.1 to −10,10×−10,10.


## 3.1. Julia Set

As we change the value ofn (see Table 1), keeping other parameters fixed, we get amazing fractals, which are visible in Figures 2(a)–2(f). As the value of n increases, the fractal takes a circular shape. For n=10, we obtain a Julia set that is similar to a circular saw or colorful teething ring (Figure 2(f)).Table 1
Parameters for generation of Julia set for different values ofn.
actTtu1n(i)19i−0.835−0.2321i0.00090.10.10.22(ii)19i−0.835−0.2321i0.00090.10.10.24(iii)19i−0.835−0.2321i0.00090.10.10.25(iv)19i−0.835−0.2321i0.00090.10.10.26(v)19i−0.835−0.2321i0.00090.10.10.28(vi)19i−0.835−0.2321i0.00090.10.10.210Figure 2
Effect ofn on Julia set.
(a)
Quadratic(b)
Quartic(c)
Quintic(d)
Sextic(e)
Octic(f)
DecicThe parametera 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 a as in Table 2, the resultant fractals can be seen in Figures 3(a)–3(d).Table 2
Parameters for generation of quartic Julia set for different values ofa.
actTtu1n(i)19i−0.835−0.2321i0.00090.10.10.24(ii)-19i−0.835−0.2321i0.00090.10.10.24(iii)-19−0.835−0.2321i0.00090.10.10.24(iv)19−0.835−0.2321i0.00090.10.10.24Figure 3
Effect of change in sign in the real and complex parametera of quartic Julia set.
(a)(b)(c)(d)The parametera also adds beauty to the fractals. As the absolute value of a increases keeping other parameters the same (as in Table 3), the more aesthetic fractals can be seen (Figures 4(a)–4(d)).Table 3
Parameters for generation of quadratic Julia set for different values ofa.
actTtu1n(i)103.140.000299010.01050.01050.92(ii)203.140.000299010.01050.01050.92(iii)−10+50i3.140.000299010.01050.01050.92(iv)50−50i3.140.000299010.01050.01050.92Figure 4
Effect of increase in the absolute value ofa on quadratic Julia set.
(a)(b)(c)(d)The impact of change in the values of parametersa and c 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 x-axis. Changes in the values of a and c from complex to real as well as a decrease in absolute value add beauty to resulting fractals.Table 4
Parameters for generation of cubic Julia set for different values ofa and c.
actTtu1n(i)40−40i−3.25+3.50i0.00199909140.01910.01910.0123(ii)5.77.50.00199909140.01910.01910.0123(iii)1.82.7180.00199909140.01910.01910.0123Figure 5
Effect of decrease in the absolute value of parametersa and c simultaneously on cubic Julia set.
(a)(b)(c)The parametert is responsible for the volume of the fractal (see Table 5). Even a slight decrease in t from 0.35 to 0.20 expands the quintic Julia set which are symmetrical about x-axis as shown in Figures 6(a)–6(c).Table 5
Parameters for generation of quintic Julia set for different values oft.
actTtu1n(i)2.20.00350.350.1150250.1150250.925(ii)2.20.00350.250.1150250.1150250.925(iii)2.20.00350.200.1150250.1150250.925Figure 6
Effect of decrease in parametert on quintic Julia set.
(a)(b)(c)


## 3.2. Mandelbrot Set

Like Julia set, Mandelbrot also becomes rounded (see Figures7 and 8) as n increases (Tables 6 and 7). Noticeably, the number of branches in Figures 7(a)–7(e) is 2n while the number of branches in Figures 8(a)–8(f) is (n−1) (unlike Figure 7).Figure 7
Effect ofn on Mandelbrot set.
(a)
Quadratic(b)
Cubic(c)
Quartic(d)
Quintic(e)
SepticFigure 8
Effect of change inn on Mandelbrot set.
(a)
Quartic(b)
Quintic(c)
Sextic(d)
Septic(e)
Octic(f)
NonicTable 6
Parameters for generation of Mandelbrot set for different values ofn.
atTtu1n(i)-1.878970.0000260.21050.21050.09322(ii)-1.878970.0000260.21050.21050.09323(iii)-1.878970.0000260.21050.21050.09324(iv)-1.878970.0000260.21050.21050.09325(v)-1.878970.0000260.21050.21050.09327Table 7
Parameters for generation of Mandelbrot set for different values ofn.
atTtu1n(i)-2.20.15939110.1150250.1150250.924(ii)-2.20.15939110.1150250.1150250.925(iii)-2.20.15939110.1150250.1150250.926(iv)-2.20.15939110.1150250.1150250.927(v)-2.20.15939110.1150250.1150250.928(vi)-2.20.15939110.1150250.1150250.929Figure9 demonstrates the effect of change in sign as well change in real to the complex value of parameter a on quadratic Mandelbrot set (see Table 8).Figure 9
Effect of change in sign as well change in real to complex parametera on quadratic Mandelbrot set.
(a)(b)(c)Table 8
Parameters for generation of quadratic Mandelbrot set for different values ofa.
atTtu1n(i)1.878970.0000260.21050.21050.09322(ii)-1.878970.0000260.21050.21050.09322(iii)1.87897i0.0000260.21050.21050.09322Lower values (Table9) of t give more beautiful, artistic, and larger fractals which are symmetrical about x-axis (Figures 10(a)–10(c)).Table 9
Parameters for generation of cubic Mandelbrot set for different values oft.
atTtu1n(i)-2.20.0590.1150250.1150250.923(ii)-2.20.15939110.1150250.1150250.923(iii)-2.20.910.1150250.1150250.923Figure 10
Effect of change in parametert on cubic Mandelbrot set.
(a)(b)(c)Figure11 demonstrates the effect of change in parameters a and t simultaneously on the quintic Mandelbrot set (see Table 10).Figure 11
Effect of change in parametersa and t simultaneously on quintic Mandelbrot set.
(a)(b)(c)(d)Table 10
Parameters for generation of quintic Mandelbrot set for different values ofa and t.
atTtu1n(i)-2i0.130.90250.90250.925(ii)-0.50.15939110.90250.90250.925(iii)00.910.90250.90250.925(iv)-2.2i0.0310.90250.90250.925Figure12 demonstrates the effect of change in parameters T,t, and u1 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.Figure 12
Effect of change in parametersT,t, and u1 simultaneously on cubic Mandelbrot set.
(a)(b)Table 11
Parameters for generation of cubic Mandelbrot set for different values ofT,t, and u1.
atTtu1n(i)-50.05250.20.010.053(ii)-50.05250.30.050.0053Remark 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 thex-axis except Figures 2(a)–2(c) and Figures 3(a) and 3(b).(iii)
The change in the sign of the value of parametersa 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 functionTz=sinzn+az+c 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 functionsinz is unbounded so that the fractals which occupy the infinite area may lie in. But due to the unboundedness of sinz only on a real and imaginary axis, it can be observable.(vi)
Almost all the fractals occupy the area from−0.1,0.1×−0.1,0.1 to −10,10×−10,10.


## 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 Corollary2. 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 n. As the value of n 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 a and c. We have explored a new technique via Fibonacci-Mann iteration for visualizing the filled-in Julia and Mandelbrot sets.



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