Fractals as Julia Sets for a New Complex Function via a Viscosity Approximation Type Iterative Methods

Abstract

In this article, we examine and investigate various variants of Julia set patterns for complex exponential functions $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ and $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ (which are analytic except at $z=0$) where $n\ge 2,$ $m,n\in \mathbb{N},$ $\alpha ,\beta ,\gamma \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ by employing a viscosity approximation-type iterative method. We employ the proposed iterative method to establish an escape criterion for visualizing Julia sets. We provide graphical illustrations of Julia sets that emphasize their sensitivity to different iteration parameters. We present graphical illustrations of Julia sets; the color, size, and shape of the images change with variations in the iteration parameters. With fixed input parameters, we observe the intriguing behavior of Julia sets for different values of n and m. We hope that the conclusions of this study will inspire researchers with an interest in fractal geometry.

1. Introduction

The fascinating world of fractal mathematics has captivated mathematicians, scientists, and artists, providing profound insights into the delicate interplay between order and chaos in the natural world. Julia sets are a prominent subject of study among various mathematical shapes and patterns, renowned for their captivating visual representations and fascinating mathematical properties. Named after the French mathematician Gaston Julia, these sets are a key concept in complex dynamics, resulting from the iterations of complex functions that investigate specific values within the complex plane. Building on this work, P. Fatou [1] introduced the Fatou set, which serves as the complement of Julia sets within the domain, as discussed further in subsequent studies such as [2,3,4,5,6,7].

The word “fractal” is derived from the Latin term meaning “to divide” or “to break”, which reflects the concept of self-similar patterns in complex structures. Fractals are infinitely intricate, repeating patterns with numerous real-world applications, frequently observed in nature. They effectively describe phenomena such as tree branches, leaf arrangements, clouds, lightning, electricity, crystals, and rivers. Fractals also play a vital role in analyzing various natural and biological structures, including microbial cultures (see, e.g., [8,9,10,11]).

Julia sets are intriguing mathematical constructs that can be created through a process known as iteration. By repeatedly applying a simple function to a complex number, we generate captivating fractal patterns. To create a Julia set, we first select a complex number, typically denoted as c, which plays a crucial role in defining the unique characteristics of each set. The iterations begin with an initial value, often represented as z, and involve repeatedly applying the function $f\left(z\right)=z^2+c,$ for a complex constant c and investigating the chaotic behavior by employing iteration schemes from fixed point theory; see [12]. The resulting Julia set consists of complex numbers that do not diverge to infinity after a specified number of iterations. The vibrant colors and intricate shapes within the Julia set reflect the varying convergence properties of these complex numbers. An escape criterion helps determine when to stop iterating and decide if a point belongs to the Julia set. A common approach is to set a maximum number of iterations: if a point reaches this limit without diverging to infinity, it is included in the set.

Subsequently, many mathematicians employed various iterative processes, including Mann, Picard, Ishikawa, Noor, Junkck-Ishikawa, Junkck-CR iterations, and Junkck-SP iterations with c-convexity. These approaches have been used to explore the behavior and patterns of different polynomials, as well as complex sine and cosine functions, and transcendental functions. It is recognized that the color, shape, and other properties can change based on the iterative methods applied to the same functions (see [13,14,15,16,17]). Iterative schemes are not limited to the generation of Julia sets; they also have applications in generating other types of fractals, including biomorphs, iterated function system fractals, inverse fractals, and root-finding fractals (see [18] and its references).

This study, inspired by the work of Kumari et al. [19], explores the application of viscosity-type approximation methods in fractal generation. Further, Tanveer et al. [20] investigated fractals for complex polynomial functions with the incorporation of a logarithmic function for the constant term c. Similarly, our paper extends this approach by replacing the constant c with $log\left(c^t\right),$ where $t\in \mathbb{R}$ and $t\ge 1.$ Later on, Rawat et al. [21] investigated a generalized viscosity approximation methodology, emphasizing its potential in generating Mandelbrot and Julia sets. In their subsequent study [22,23], the researchers utilized the viscosity approximation iteration process to analyze and visualize the fractals as Julia sets, further enriching the understanding of these fascinating fractal structures. We first adapt the existing viscosity approximation-type iterative method to create escape criteria for the new complex functions $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ and $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ (which are analytic except at $z=0$) where $n\ge 2,$ $m,n\in \mathbb{N},$ $\alpha ,\beta ,\gamma \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1.$ We then provide graphical examples based on these escape criteria, develop an escape time algorithm, and generate fractals as Julia sets.

The plan of this article is structured as follows: Section 2 presents the basic definitions and results necessary to achieve the goal of this article. In Section 3, we establish the main theorems to derive a general escape criterion, which is crucial for constructing Julia sets using a viscosity approximation-type iterative method for the considered complex functions. Section 4 details the algorithms used and provides graphical examples of Julia sets generated with MATLAB software, illustrating the sets for various parameters. This section also offers a comprehensive discussion of the results. Finally, the last section concludes our work.

2. Preliminaries

This section contains some basic definitions and results necessary to achieve the goal of this article.

Definition 1

(Julia set [1]). Let $T:\mathbb{C}\to \mathbb{C}$ be a complex function. Then, the set of points

$$\begin{array}{c}\hfill f_T=\{z\in \mathbb{C}:\{|T^k{\left(z\right)\left|\right\}}_{k=0}^{\infty }\mathit{is}\mathit{bounded}\}.\end{array}$$

where $T^k$ is the $k^{th}$ iteration of T known as the filled Julia set. The boundary of the filled Julia set $f_T$ is known as Julia sets.

In 2000, Moudafi [24] introduced the viscosity approximation method. In the context of the complex plane, this method can be defined as follows:

Definition 2

([24]). Let $p,W:\mathbb{C}\to \mathbb{C}$ be complex functions, where p is a contraction mapping. The sequence $\left\{z_k\right\}$ of iterates is known as the viscosity approximation method for an initial point $z_0\in \mathbb{C}$ and ${\mu }_k\in (0,1)$ if it can be expressed as

$$\begin{array}{c}\hfill z_{k+1}={\mu }_{k}p\left(z_k\right)+(1-{\mu }_k)W\left(z_k\right),k\ge 0.\end{array}$$

(1)

Consider the following complex functions $W,T:\mathbb{C}\to \mathbb{C}$ defined as

$$\left\{\begin{array}{c}W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,\hfill \\ T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma ,\hfill \end{array}\right.$$

where $n\ge 2, \alpha ,\beta ,\gamma \in \mathbb{C}, c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1.$ These mappings are analytic except at $z=0.$ Moreover, let $p\left(z\right)=az+b$ be a complex contraction with $a,b\in \mathbb{C}$ and $\left|a\right|<1.$

Let us assume that ${\mu }_k=\mu ,$ where $\mu \in (0,1).$ For any $z_0\in \mathbb{C},$ the sequence $\left\{z_k\right\}$ is generated by

$$z_{k+1}=\mu p\left(z_k\right)+(1-\mu )W\left(z_k\right).$$

(2)

and

$$z_{k+1}=\mu p\left(z_k\right)+(1-\mu )T\left(z_k\right).$$

(3)

The iterative method given in (2) and (3) is called the viscosity approximation type method.

Remark 1.

The viscosity approximation-type iterative method reduces to the Halpern iteration when $p\left(z\right)=b$, and the Mann iteration when $p\left(z\right)=z.$

• To generate fractals and escape, limitations are the basic key to running the algorithms. For $n\ge 2$ and $z\in \mathbb{C},$ the series expansion for the exponential functions is as follows:

$$\begin{array}{c}\hfill |e^{z^n}|=|\sum _{k=0}^{\infty }{\displaystyle \frac{z^{nk}}{k!}}|>|\sum _{k=1}^{\infty }{\displaystyle \frac{z^{nk}}{k!}}|=|z^n||\sum _{k=1}^{\infty }{\displaystyle \frac{z^{n(k-1)}}{k!}}|>\left|\omega \right||z^n|\end{array}$$

(4)

where $\left|\omega \right|\in (0,1)$ and $z\in \mathbb{C}$ except those values z wherefore $\left|\omega \right|=0$ (see the details in [17]).

3. Main Results

In this section, we establish the escape time algorithm via the viscosity approximation-type iterative method for the considered new exponential functions. Throughout the manuscript, let $\eta ={\displaystyle \frac{log{c}^t}{c}}.$ Thus, we have $log{c}^t=\eta c.$ As a result, we establish an escape radius and utilize it to visualize non-classical variants of fractals, as presented in the following results.

3.1. Escape Criterion for $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t$

In this subsection, we prove the escape criteria for a new exponential function $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ where $n\ge 2,$$\alpha ,\beta \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ and $m,n\in \mathbb{N}.$ via the viscosity approximation-type iterative method given in (2).

Theorem 1.

Assume that $\left|z_0\right|{\ge max\left\{\right|c|,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}}\}>{\left({\displaystyle \frac{(1+|a|+(1-\mu \left)\right|\eta \left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$, where $n\ge 2,$$\alpha ,\beta \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ and $m,n\in \mathbb{N}$. If the sequence $\left\{z_k\right\}$ is defined by the viscosity approximation-type iterative method defined in (2), and $p\left(z\right)=az+b$ is a contraction mapping with $\left|a\right|<1$, then $|z_k|\to \infty ,$ as $k\to \infty .$

Proof of Theorem 1.

For $k=0,$ consider from (2) that

$$\begin{array}{ccc}\hfill |z_1|& =& |\mu p\left(z_0\right)+(1-\mu )W\left(z_0\right)|\hfill \\ & =& |\mu (a{z}_0+b)+(1-\mu )\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t\right)|\hfill \\ & \ge & (1-\mu )\left|\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\eta c\right)\right|-\mu \left|(a{z}_0+b)\right|\hfill \\ & \ge & (1-\mu )|\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}\right)|-(1-\mu )|\eta c|-\mu |a{z}_0|+\mu |b|\hfill \\ & \ge & (1-\mu )\left|\alpha \right||e^{z^n}|-(1-\mu ){\displaystyle \frac{\left|\beta \right|}{|z^m|}}-(1-\mu )\left|\eta \right|\left|c\right|-\mu \left|a\right||z_0|+\mu |b|\hfill \end{array}$$

Using (4), and our assumption $\left|z_0\right|\ge max\{\left|c\right|,\left|b\right|,{\left|\beta \right|}^{{\displaystyle \frac{1}{n+m}}}\}$ yields that $-\left|b\right|\ge -|z_0|,$ and $-\left|c\right|\ge -|z_0|,$ and $-\left|\beta \right|\ge -|z_0{|}^{n+m},$ and we have

$$\begin{array}{ccc}\hfill |z_1|& \ge & (1-\mu )\left|\alpha \right|\left|\omega \right||z^n|-(1-\mu ){\displaystyle \frac{|z_0{|}^{n+m}}{|z^m|}}-(1-\mu )\left|\eta \right||z_0|-\mu |a\left|\right|z_0|+\mu \left|z_0\right|\hfill \\ & \ge & (1-\mu )\left|\alpha \right|\left|\omega \right||z^n|-(1-\mu )|z_0^n|-(1-\mu )|\eta \left|\right|z_0|-|a\left|\right|z_0|,\mu <1\hfill \\ & \ge & |z_0|\left((1-\mu )\left(\right|\alpha \left|\right|\omega |-1\left)\right|z_0^{n-1}|-(1-\mu )|\eta |-|a|\right).\hfill \end{array}$$

$\left|z_0\right|{\ge max\left\{\right|c|,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}}\}>{\left({\displaystyle \frac{(1+|a|+(1-\mu \left)\right|\eta \left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$, which implies $(1-\mu )\left(\right|\alpha \left|\right|\omega |-1\left)\right|z_0^{n-1}|-(1-\mu )|\eta |-|a|>1.$ Hence, $|z_1|>|z_0|,$ which implies that there exists a real number $\mathrm{\Omega }>0$ so that $|z_1|>(1+\mathrm{\Omega })|z_0|$. On continuing the above procedure, we obtain $|z_k{|>(1+\mathrm{\Omega })}^k\left|z_0\right|.$ Hence, $|z_k|\to \infty ,$ as $n\to \infty .$ □

In the proof of Theorem 1, we have used only the fact that $\left|z_1\right|\ge {\left({\displaystyle \frac{(1+|a|+(1-\mu \left)\right|\eta \left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$ and $\left|z_0\right|{\ge max\left\{\right|c|,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}}\}.$ So, we can refine it and obtain the following corollary.

Corollary 1.

Let $\left|z_0\right|>max\left\{|c|,|b|,{\left|\beta \right|}^{{\displaystyle \frac{1}{m+n}}},{\left({\displaystyle \frac{(1+|a|+(1-\mu \left)\right|\eta \left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}\right\}$, where $n\ge 2,$ $\alpha ,\beta ,a,b\in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ and $m,n\in \mathbb{N},$ and $\mu \in (0,1)$ with $\left|a\right|<1.$ Then, $|z_k|\to \infty ,$ as $n\to \infty .$

3.2. Escape Criterion for $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$

We establish the escape criteria for a new exponential function $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma ,$ where $n\ge 2,$ $\alpha ,\beta ,\gamma \in \mathbb{C}$, and $m,n\in \mathbb{N}$ via the viscosity approximation-type iterative method given in (3).

Theorem 2.

Assume that $\left|z_0\right|{\ge max\left\{\right|\gamma |,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}}\}>{\left({\displaystyle \frac{(2+\mu |a\left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$, where $n\ge 2,$$\alpha ,\beta ,\gamma \in \mathbb{C},$ and $m,n\in \mathbb{N}$. If the sequence $\left\{z_k\right\}$ is defined by the viscosity approximation-type iterative method defined in (3), and $p\left(z\right)=az+b$ is contraction mapping with $\left|a\right|<1$, then $\left|z_k\right|\to \infty ,$ as $k\to \infty .$

Proof of Theorem 2.

For $k=0,$ consider from (3) that

$$\begin{array}{ccc}\hfill \left|z_1\right|& =& |\mu p\left(z_0\right)+(1-\mu )T\left(z_0\right)|\hfill \\ & =& |\mu (a{z}_0+b)+(1-\mu )\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma \right)|\hfill \\ & \ge & (1-\mu )\left|\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma \right)\right|-\mu \left|(a{z}_0+b)\right|\hfill \\ & \ge & (1-\mu )|\left(\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}\right)|-(1-\mu )|\gamma |-\mu |a{z}_0|+\mu |b|\hfill \\ & \ge & (1-\mu )\left|\alpha \right||e^{z^n}|-(1-\mu ){\displaystyle \frac{\left|\beta \right|}{|z^m|}}-(1-\mu )\left|\gamma \right|-\mu \left|a\right||z_0|+\mu |b|\hfill \end{array}$$

Using (4), and our assumption $\left|z_0\right|{\ge max\left\{\right|\gamma |,|b|,|\beta |}^{{\displaystyle \frac{1}{n+m}}}\}$ yields that $-\left|b\right|\ge -|z_0|,$ and $-\left|\gamma \right|\ge -|z_0|,$ and $-\left|\beta \right|\ge -|z_0{|}^{n+m},$ and we have

$$\begin{array}{ccc}\hfill |z_1|& \ge & (1-\mu )\left|\alpha \right|\left|\omega \right||z^n|-(1-\mu ){\displaystyle \frac{|z_0{|}^{n+m}}{|z^m|}}-(1-\mu )|z_0|-\mu |a\left|\right|z_0|+\mu \left|z_0\right|\hfill \\ & \ge & (1-\mu )\left|\alpha \right|\left|\omega \right||z^n|-(1-\mu )|z_0^n|-\mu |a\left|\right|z_0|+2\mu |z_0|-|z_0|\hfill \\ & \ge & (1-\mu )\left|\alpha \right|\left|\omega \right||z^n|-(1-\mu )|z_0^n|-\mu |a\left|\right|z_0|-|z_0|\hfill \\ & \ge & |z_0|\left((1-\mu )\left(\right|\alpha \left|\right|\omega |-1\left)\right|z_0^{n-1}|-\mu |a|-1\right).\hfill \end{array}$$

$|z_0{|\ge max\{\left|\gamma \right|,\left|b\right|,\left|\beta \right|}^{{\displaystyle \frac{1}{m+n}}}\}>{\left({\displaystyle \frac{(2+\mu |a\left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$, which implies $(1-\mu )\left(\right|\alpha \left|\right|\omega |-1\left)\right|z_0^{n-1}|-\mu |a|-1>1.$ Hence, $|z_1|>|z_0|,$ which implies that there exists a real number $\mathrm{\Omega }>0$ so that $|z_1|>(1+\mathrm{\Omega })|z_0|$. On continuing the above procedure, we obtain $|z_k{|>(1+\mathrm{\Omega })}^k\left|z_0\right|.$ Hence, $|z_k|\to \infty ,$ as $n\to \infty .$ □

In the proof of Theorem 2, we have used only the fact that $\left|z_1\right|\ge {\left({\displaystyle \frac{(2+\mu |a\left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}$ and $\left|z_0\right|{\ge max\left\{\right|\gamma |,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}}\}.$ So, we can refine it and obtain the following corollary.

Corollary 2.

Let $|z_0|>max\{{\left|\gamma \right|,\left|b\right|,\left|\beta \right|}^{{\displaystyle \frac{1}{m+n}}},{\left({\displaystyle \frac{(2+\mu |a\left|\right))}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}\}$, where $n\ge 2,$ $\alpha ,\beta ,\gamma ,a,b\in \mathbb{C},$ and $m,n\in \mathbb{N},$ and $\mu \in (0,1)$ with $\left|a\right|<1.$ Then, $|z_k|\to \infty ,$ as $n\to \infty .$

4. Graphical Examples

In this section, we refine the viscosity approximation-type iterative method. To visualize the fractals, certain convergence conditions are required, which serve as essential tools for effectively executing the algorithm and accurately rendering the desired fractal patterns. We generate Julia sets using the viscosity approximation-type iterative method, setting a maximum number of iterations 70 throughout the paper. Using $\mathrm{MATLAB} \mathrm{R}2019\mathrm{a}$ $(9.6.0.1072779,$ 64-bit), we aim to obtain non-classical Julia sets for distinct parameters. For fractal visualization, we design two algorithms: one for the Julia set of $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t$ and another for $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$, employing the viscosity approximation-type iterative method. By sketching Julia sets across varying input parameters and values of n and m, we demonstrate the results generated through this method using Algorithms 1 and 2.

Algorithm 1. Julia set generation for $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t$.

Input: $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ where $n\ge 2,$ $\alpha ,\beta \in \mathbb{C},c\in \mathbb{C}\setminus \left\{0\right\}$ and $t\in \mathbb{R},t\ge 1,$ and $m,n\in \mathbb{N}$—parameters for $W;$ K—maximal number of iterations; A—area; $\mu \in (0,1)$ parameter in (2); $p\left(z\right)=az+b$, where $a,b\in \mathbb{C}$ and $\left|a\right|<1;$ color map $[0..C$-$1]$—color with C colors.

Output: Julia set for area A.

for $z_0\in A$ do

$\eta ={\displaystyle \frac{log{c}^t}{c}}$

$R=max\{{\left|c\right|,\left|b\right|,\left|\beta \right|}^{{\displaystyle \frac{1}{m+n}}},{\left({\displaystyle \frac{(1+|a|+(1-\mu \left)\right|\eta \left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}\}$

k = 0

while $|z_{k+1}| >R$ and $k\le K$ do

$z_{k+1}=\mu p\left(z_k\right)+(1-\mu )W\left(z_k\right),$

$k=k+1$

$i=\left[(C-1){\displaystyle \frac{k}{K}}\right]$

color $z_0$ with color map $\left[i\right]$

Algorithm 2. Julia set generation for $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$.

Input: $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma ,$ where $n\ge 2,$ $\alpha ,\beta ,\gamma \in \mathbb{C},$ and $m,n\in \mathbb{N}$—parameters for $T;$ K—maximal number of iterations; A—area; $\mu \in (0,1)$—parameter in (3); $p\left(z\right)=az+b$, where $a,b\in \mathbb{C}$ and $\left|a\right|<1;$ color map $[0..C$-$1]$—color with C colors.

Output: Julia set for area A.

for $z_0\in A$ do

$R={max\left\{\right|c|,|b|,|\beta |}^{{\displaystyle \frac{1}{m+n}}},{\left({\displaystyle \frac{(1+\mu |a\left|\right)}{(1-\mu )\left(\right|\alpha \left|\right|\omega |-1)}}\right)}^{{\displaystyle \frac{1}{n-1}}}\}$

k = 0

while $|z_{k+1}| >R$ and $k\le K$ do

$z_{k+1}=\mu p\left(z_k\right)+(1-\mu )W\left(z_k\right),$

$k=k+1$

$i=\left[(C-1){\displaystyle \frac{k}{K}}\right]$

color $z_0$ with color map $\left[i\right]$

4.1. Julia Sets for $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t$

The Julia sets corresponding to the complex exponential function associated with the logarithmic function for different input values are illustrated in this subsection. Note that the image execution times (IETs) are recorded in seconds. For consistency, we utilized the same color map presented in Figure 1 for both examples of Julia sets associated with $W\left(z\right)$ and $T\left(z\right).$

Example 1.

In this example, we consider two cases. In the first case, we use integer values of $t,$ specifically $t=2,7,13,$ while in the second case, we use non-integer values, specifically $t=0.5,4.5,9.5.$ For $n=2,$ and $m=1,$ Julia sets for $W\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+log{c}^t,$ are generated using the following input parameters (

Table 1. The parameters $\alpha ,\beta$, and c are fixed as purely real and t varies.

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	1	2	0.2	0.65	9	0.8	0.7	0.35	0.18	3.61 s
$\left(ii\right)$	2	1	7	0.2	0.65	9	0.8	0.7	0.35	0.18	3.81 s
$\left(iii\right)$	2	1	13	0.2	0.65	9	0.8	0.7	0.35	0.18	4.09 s
$\left(iv\right)$	2	1	0.5	0.2	0.65	9	0.8	0.7	0.35	0.18	4.76 s
$\left(v\right)$	2	1	4.5	0.2	0.65	9	0.8	0.7	0.35	0.18	4.96 s
$\left(vi\right)$	2	1	9.5	0.2	0.65	9	0.8	0.7	0.35	0.18	5.16 s

):

In Figure 2, the values of $\alpha ,\beta$ and c are fixing as purely real, while the value of t is vary. In the first case, integer values of t are used, specifically $t=2,7,13$ (Figure 2i–iii). In the second case, non-integer values are considered, specifically $t=0.5,4.5,9.5$ (Figure 2iv–vi). The images clearly demonstrate that the value of t significantly affects the shape, size, and color of the fractals, with the sets exhibiting symmetry about the real axis. The resulting Julia sets resemble intricate patterns, such as Rangoli designs or stained-glass artwork.

In Figure 2, Figure 3 and Figure 4i–vi, for $n=2$ and $m=1$, we observe Julia sets generated by fixing two of the parameters $\alpha$, $\beta$, or c while varying the third. In Figure 2, $\alpha$, $\beta$, and c are fixed as purely real values; in Figure 3, they are fixed as purely imaginary values; and in Figure 4, they are fixed as complex values. These variations illustrate that the parameters $\alpha$, $\beta$, and c significantly influence the shape, size, and color of each set, particularly along the edges of the leaf-like structures and the patterns of the arms, which exhibit subtle differences as the value of t increases, whether for integer or non-integer values. Notably, the floral size increases outward from the center as the value of t increases. The obtained images display intricate and aesthetically appealing fractal shapes, with the Julia sets resembling traditional Rangoli patterns, floral designs, or ornate glass art. The image generation time is also recorded, showing an increase with each iteration in the last column of the tables (

Table 2. The parameters $\alpha ,\beta$, and c are fixed as purely imaginary and t varies.

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	1	2	1.2i	65i	7i	0.8	0.7	0.35	0.75	4.81 s
$\left(ii\right)$	2	1	7	1.2i	65i	7i	0.8	0.7	0.35	0.75	4.98 s
$\left(iii\right)$	2	1	13	1.2i	65i	7i	0.8	0.7	0.35	0.75	5.21 s
$\left(iv\right)$	2	1	0.5	1.2i	65i	7i	0.8	0.7	0.35	0.75	5.61 s
$\left(v\right)$	2	1	4.5	1.2i	65i	7i	0.8	0.7	0.35	0.75	5.91 s
$\left(vi\right)$	2	1	9.5	1.2i	65i	7i	0.8	0.7	0.35	0.75	6.11 s

and

Table 3. The parameters $\alpha ,\beta$ and c are fixed as complex and t varies.

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	1	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	4.67 s
$\left(ii\right)$	2	1	9	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	4.76 s
$\left(iii\right)$	2	1	17	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	4.81 s
$\left(iv\right)$	2	1	0.5	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	4.89 s
$\left(v\right)$	2	1	2.5	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	4.97 s
$\left(vi\right)$	2	1	6.5	$-2-1.2i$	$-21-65i$	$6-7i$	0.8	0.7	0.35	0.75	5.08 s

).

In Figure 5, we fix the values of all parameters and vary the values of n and m. The images show Julia sets with $m=n=$ even/odd and the other input parameters fixed as given above in $\left(i\right)$–$\left(ix\right)$. From these images, we observe that as n and m increase, the sets take on circular shapes, with petal counts corresponding to the values of n and m, and the number of petals increases with increased values of n and m. The obtained images show very beautiful and complex fractal shapes, and the resulting Julia sets resemble floral shapes, or intricate glass art. The image generation time is also recorded and provided in the last column of the

Table 4. Parameters for different values of n and $m.$

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	3	2	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	5.17 s
$\left(ii\right)$	3	3	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	5.76 s
$\left(iii\right)$	5	3	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	6.01 s
$\left(iv\right)$	5	5	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	6.29 s
$\left(v\right)$	8	7	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	6.97 s
$\left(vi\right)$	2	7	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	7.18 s
$\left(vii\right)$	5	9	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	7.89 s
$\left(viii\right)$	3	8	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	8.97 s
$\left(ix\right)$	71	104	2	$-2-1.2i$	$-21-65i$	$6-7i$	0.4	$0.7i$	0.35	0.85	9.08 s

.

In Figure 6, for $n=3$ and $m=2,$ we vary the values of the parameter $\mu$. It becomes evident that the parameter $\mu$ significantly influences the set’s shape, size, and color. As shown in Figure 6, as either parameter $\mu$ increases, the set expands and undergoes alterations in its form. Additionally, axial symmetry is a notable feature in the resulting sets. In each Julia set image, specific lashes show symmetry along the x-axis and extend symmetrically along the positive x-axis from each edge of a leaf-like shape. Additionally, in Figure 6i–iv, we can observe some spiral structures in the generated Julia sets, but the spiral pattern arms and the size of the lashes or bunches slightly decrease as the value of $\mu$ increases. The generated Julia sets look like traditional Rangoli patterns or could be likened to intricate glass paintings. The image generation time is also calculated for each iteration (

Table 5. Effect of parameter $\mu .$

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	3	2	2	−2	−21	6	41	−0.15	0.15	0.05	5.17 s
$\left(ii\right)$	3	2	2	−2	−21	6	41	−0.1	0.15	0.15	5.87 s
$\left(iii\right)$	3	2	2	−2	−21	6	41	−0.1	0.15	0.35	6.47 s
$\left(iv\right)$	3	2	2	−2	−21	6	41	−0.1	0.15	0.65	7.07 s
$\left(v\right)$	3	2	2	−2	−21	6	41	−0.1	0.15	0.85	7.67 s
$\left(vi\right)$	3	2	2	−2	−21	6	41	−0.1	0.15	0.97	8.11 s

).

As shown in Figure 7, the values of a and b have a significant impact on the shape, size, and color of the fractals as Julia sets. The generated Julia sets resemble Rangoli designs or may be viewed as glass paintings, with pattern arms differing slightly for different values of a and $b.$ Additionally, axial symmetry is a notable feature in the resulting sets. In each Julia set image except $\left(iv\right)$, specific lashes show symmetry along the x-axis and extend symmetrically along the positive x-axis from each edge of a leaf-like shape, with some of these lashes spiraling outward. The image generation time is also recorded and provided in the last column of the

Table 6. Effect of parameters a and $b.$

	n	m	t	$\mathit{\alpha }$	$\mathit{\beta }$	c	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	3	2	2	−2	−21	6	0.05	0.15	3.47 s
$\left(ii\right)$	3	2	2	−2	−21	6	0.15	0.15	4.03 s
$\left(iii\right)$	3	2	2	−2	−21	6	0.35	0.15	4.57 s
$\left(iv\right)$	3	2	2	−2	−21	6	0.65	0.15	5.13 s
$\left(v\right)$	3	2	2	−2	−21	6	0.85	0.15	5.71 s
$\left(vi\right)$	3	2	2	−2	−21	6	0.97	0.15	6.27 s

.

4.2. Julia Sets for $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$

In this subsection, the Julia sets corresponding to the new complex exponential function for different input values are illustrated.

Example 2.

In this example, we note that the last column in all the tables displays the image execution time (IET) in seconds. For $n=2,$ and $m=1,$ Julia sets for $T\left(z\right)=\alpha e^{z^n}+{\displaystyle \frac{\beta }{z^m}}+\gamma$ are generated here with the following inputs (

Table 7. Effect of parameters $\alpha ,\beta$, and $\gamma$.

	n	m	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	1	0.2	0.65	9	0.8	0.7	0.35	0.17	3.61 s
$\left(ii\right)$	2	1	0.2i	0.65	9	0.8	0.7	0.35	0.17	3.81 s
$\left(iii\right)$	2	1	$-1+0.2i$	0.65	9	0.8	0.7	0.35	0.17	4.09 s
$\left(iv\right)$	2	1	0.1	3	6	0.8	−0.1	0.2	0.17	4.76 s
$\left(v\right)$	2	1	0.1	3i	6	0.8	−0.1	0.2	0.17	4.76 s
$\left(vi\right)$	2	1	0.1	$0.3-0.3i$	6	0.8	−0.1	0.2	0.17	4.76 s
$\left(vii\right)$	2	1	0.1	0.3	3	0.8	−0.1	0.2	0.17	4.76 s
$\left(viii\right)$	2	1	0.1	0.3	0.3i	0.8	−0.1	0.2	0.17	4.76 s
$\left(ix\right)$	2	1	0.1	0.3	$2+0.3i$	0.8	−0.1	0.2	0.17	4.76 s

):

In Figure 8, we varied the values of the parameters $\alpha ,\beta$ and $\gamma$, and fixed the values of the remaining parameters. From the images, it is evident that the values of $\alpha ,\beta$ and $\gamma$ significantly impact the shape, size, and color of the fractals as Julia sets. The resulting Julia sets resemble flowers, Rangoli designs, or intricate glass paintings, with the pattern arms showing slight variations for different values of $\alpha ,\beta$ and $\gamma .$ Additionally, specific lashes exhibit symmetry along the x-axis and extend symmetrically along the positive x-axis from each edge of a leaf-like shape, with some of these lashes spiraling outward. The image execution time strictly increases, as shown in the last column of the above table.

In Figure 9, it is evident that the values of a and b significantly impact the shape, size, and color of the fractals as Julia sets. The generated Julia sets resemble flowers, Rangoli designs, or intricate glass paintings. Additionally, the pattern arms change subtly with different values of a and b (

Table 8. Effect of parameters a and $b.$

	n	m	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	1	0.2	5	9	−0.1	0.7	0.35	0.17	3.61 s
$\left(ii\right)$	2	1	0.2	5	9	−0.1i	0.7	0.35	0.17	3.81 s
$\left(iii\right)$	2	1	0.2	5	9	$0.5-0.2i$	0.7	0.35	0.17	4.09 s
$\left(iv\right)$	2	1	0.1	5	6	0.3	−0.1	0.35	0.17	4.76 s
$\left(v\right)$	2	1	0.1	5	6	0.3	7i	0.35	0.17	5.26 s
$\left(vi\right)$	2	1	0.1	5	6	0.3	11−7i	0.35	0.17	5.93 s

).

As shown in Figure 10, we fixed the values of all parameters and varied the values of n and m. From these images, we observe that as n and m increase, the sets take on circular shapes, with petal counts corresponding to the values of n and m, and the number of petals increases with increased values of n and m. The obtained images display beautiful and complex fractal shapes, and the resulting Julia sets resemble floral shapes or intricate glass art (

Table 9. Effect of parameters n and $m.$

	n	m	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	2	1.2	4	2.8	0.2	1.1	0.9	0.17	4.31 s
$\left(ii\right)$	2	5	1.2	4	2.8	0.2	1.1	0.9	0.17	4.83 s
$\left(iii\right)$	5	9	1.2	4	2.8	0.2	1.1	0.9	0.17	4.33 s
$\left(iv\right)$	5	1	1.2	4	2.8	0.2	1.1	0.9	0.17	4.61 s
$\left(v\right)$	7	2	1.2	4	2.8	0.2	1.1	0.9	0.17	5.11 s
$\left(vi\right)$	13	7	1.2	4	2.8	0.2	1.1	0.9	0.17	5.76 s

).

It can be seen that as the value of $\mu$ increases, the resulting Julia sets evolve into more complex and detailed fractal patterns (Figure 11,

Table 10. Effect of parameter $\mu .$

	n	m	$\mathit{\alpha }$	$\mathit{\beta }$	c	a	b	$\mathit{\mu }$	$\mathit{\omega }$	IET (in s)
$\left(i\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.1	0.35	4.63 s
$\left(ii\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.25	0.35	5.43 s
$\left(iii\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.45	0.35	6.21 s
$\left(iv\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.65	0.35	6.96 s
$\left(v\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.85	0.35	7.61 s
$\left(vi\right)$	2	4	1.1i	4i	2.8i	0.02	1.1	0.95	0.35	8.78 s

). The growth in $\mu$ directly affects the overall shape, size, and structure, with the patterns becoming more intricate and dynamic. The transformations are reflected in the figures, which range from floral-like structures to swirling, symmetrical arrangements resembling Rangoli designs, colorful teething rings, or even circular saw blades. These diverse shapes also bear resemblance to stained-glass artworks due to their complexity and vivid color contrasts.

Finally, as the parameters vary, the Julia points representing the set’s boundaries shift, adding further uniqueness to each fractal. The interplay of these parameters highlights the sensitivity of Julia sets to even slight changes in input values, illustrating the delicate balance between mathematical properties and visual aesthetics in fractal geometry. Each variation not only increases the fractal’s complexity but also amplifies its aesthetic beauty, making each generated Julia set a visually captivating and mathematically rich creation. The following have been observed:

• The parameters $\alpha ,\beta ,\gamma ,c,a,b,\mu$ and t are crucial in determining the shape, size, and color of the fractals.
• The convergence criteria established for the fractals play a crucial role in determining the resolution and pixel richness of the fractal images.
• All the fractals presented in this paper are highly innovative, aesthetically appealing, and visually captivating, resulting from the complex functions W(z) and T(z).

5. Conclusions

We have established an escape criterion using the viscosity approximation-type iterative method for newly proposed complex functions. Leveraging these results, we generated intricate fractal structures as Julia sets. These findings were implemented in Algorithms 1 and 2 to visualize the Julia sets. Utilizing MATLAB software, we analyzed and discussed the behavior of various Julia set variants under different parameter values, uncovering fascinating non-classical variations of Julia fractals. Our study reveals that the size and complexity of these fractals are significantly influenced by parameters $\alpha ,\beta ,\gamma ,c,a,b,\mu$ and $t,$ as well as the exponents n and $m.$ An arbitrarily small change in these parameters can lead to significant alterations in the shape, color, and size of the fractals. In future work, we aim to extend this exploration to generate fractals as Mandelbrot sets by adapting the complex exponential functions into complex sine or cosine functions. We also plan to introduce metrics such as generation time and ANI in these analyses. Additionally, the results of this paper hold practical potential for applications in the textile industry, particularly for designing and printing purposes.