A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods

Abstract

This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.

1. Introduction

The research community has shown a significant amount of interest in chaotic circuits over the past few decades because they have been applied in a variety of fields, including emulating economic models, random bit generators, image processing, robotics, neural networks, secure communications, and the design of electronic circuits. As a result, a significant number of chaotic systems, each implemented using circuitry, have been documented in published works. However, one of the most important fields of research is building robust chaotic oscillators with structures that are as simple as feasible, either from the perspective of circuitry or mathematics. This is one of the most essential research directions.

Fractional differential equations have gained significant recognition in modeling and chaos [1,2], leading to the development of various alternative methods for solving such equations [3,4]. A wide range of disciplines, spanning from electrical engineering to biology and physics, have harnessed the utility of modeling chaotic and hyper-chaotic systems [5,6,7,8,9,10,11]. Numerous research studies explore the practical applications of chaos in areas such as the modeling of electrical circuits. The use of symmetrical chaotic models is acceptable because it is difficult to forecast a wide range of real-world occurrences. Numerous novel methods for assessing chaotic systems have appeared in recent years [12,13,14,15]. Two of these methods, asymptotic stability and Lyapunov exponents, shed light on how the parameters of the model affect the dynamics of the chaotic model. Numerous mathematical and scientific fields make use of fractional calculus. For specific cutting-edge research and applications, scientists, mathematicians, biologists, and those from other fields [16,17,18,19,20,21,22,23,24] are increasingly turning to fractional calculus.

We have seen several examples of nonlinear chaotic circuits, both autonomous and nonautonomous. A varactor, inductor, and resistor diode form the oscillator [25], whereas a capacitor and a linear resistor make up Dean’s circuit [26] and others [27,28,29,30,31,32,33,34].

The simple chaotic jerk system is discussed in [35] and shown in Figure 1. It can be modeled mathematically as shown below:

$$\left\{\begin{array}{l}{C}_1\frac{\mathrm{d}{v}_{C_1}}{\mathrm{d}t}=-\frac{1}{R}{v}_{C_2}\\ C_2\frac{\mathrm{d}{v}_{C_2}}{\mathrm{d}t}=-\frac{1}{R}{v}_{C_3}\\ C_3\frac{\mathrm{d}{v}_{C_3}}{\mathrm{d}t}=-\frac{1}{R}{v}_{C_1}-\frac{1}{R_b}{v}_{C_3}+\frac{R_a}{R}2I_{\mathrm{S}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{v_{C_2}}{n{V}_{\mathrm{T}}}\right)\end{array}\right.$$

(1)

In system (1), $v_{C_1},v_{C_2}$ and $v_{C_3}$ are the voltages across the three capacitors $C_1,C_2$ and $C_3$, respectively, and $v_{C_2}=v_{AB}$. It is noted that these capacitors have the same values $\left(C_1=C_2=C_3=C\right)$ in this work.

Figure 1 presents a graphical depiction of the electronic circuit encompassing the chaotic jerk system, which is essential in our research. Incorporating this circuit diagram plays a crucial role in facilitating the shift from abstract mathematical models to tangible applications within chaotic dynamics. Figure 1 and Equation (1) are inherently interconnected in our research. Equation (1) serves as the fundamental mathematical framework upon which our research is constructed, while Figure 1 represents the concrete manifestation of this equation as an electronic circuit. The correlation between theory and practice facilitates observing and manipulating chaotic phenomena inside the circuit system. System (1) is rescaled by using dimensionless variables and parameters given by $x=\frac{v_{C_1}}{n{V}_{\mathrm{T}}},y=\frac{v_{C_2}}{n{V}_{\mathrm{T}}},z=\frac{v_{C_3}}{n{V}_{\mathrm{T}}}$, $\tau =\frac{t}{RC},a=\frac{2R_a{I}_{\mathrm{S}}}{n{V}_{\mathrm{T}}}$ and $b=\frac{R}{R_b}$. As a result, system (1) is rewritten as:

$$\left\{\begin{array}{l}\frac{\mathrm{d}x}{\mathrm{d}t}=-y\\ \frac{\mathrm{d}y}{\mathrm{d}t}=-z\\ \frac{\mathrm{d}z}{\mathrm{d}t}=-x-bz+a\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(y\right)\end{array}\right.$$

(2)

Despite the fact that the value of the parameter is determined by the properties of the diode, we are still able to easily obtain the value we desire by adjusting the value of the resistor $R_a$. By choosing $R_a=10 \mathrm{k}\mathsf{\Omega },{ I}_S=1 \mathrm{n}\mathrm{A},V_{\mathrm{T}}=26 \mathrm{m}\mathrm{V}$ and $n=2$, the value of $a$ is fixed as $3.846\times {10}^{-4}$, and we choose b = 0.7 (see [35]) with initial conditions $\left(x_0,y_0,z_0\right)=\left(\mathrm{0.1,0.1,0.2}\right)$. One of the new jerk circuit’s most important advantages is its anti-monotonicity, coexisting attractors, and successful application of the simple chaotic jerk system in the suggested sound encryption approach.

Here, we consider the symmetry chaotic jerk system in which we replace the integer-order derivatives with a fractional order. We introduce the fractional symmetry chaotic jerk system with $0<q<1$ as follows.

$$\begin{array}{l}{}_0{}^{ABC}{D}_t^{\alpha }x=-y\\ {}_0{}^{ABC}{D}_t^{\alpha }y=-z\\ {}_0{}^{ABC}{D}_t^{\alpha }z=-x-bz+a\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(y\right)\end{array}$$

(3)

where ${}_0{}^{ABC}{D}_t^{\alpha }(.)$ is the Atangana–Baleanu Caputo fractional derivative.

The new iterative Laplace method (NILM), introduced in [36], is an improved version of the ADM method, which is used to solve nonlinear equations [37,38,39]. This does not need any calculation of Adomian polynomials in all its iterations. The new iterative Laplace approach has the following advantages: improvements in accuracy, convergence for complex cases, flexibility in problem types, convenience for numerical implementation, reduction in analytical complexity, automation potential, insights into solution behavior, convergence monitoring, adaptability to boundary conditions, and potential for further development.

This novel scheme (ABC-FD) was presented in [40] for the first time. Introducing a novel scheme for this derivative can offer several advantages and improve its application in various fields. Here are some potential advantages of such a novel scheme: improved accuracy and flexibility, better handling of complex systems, and applicability to real-world phenomena.

This study examines numerical solutions of the simple chaotic jerk system with fractional order by using two different techniques. The fractional model is an extension of the classical model, and one of the advantages of using it is the flexibility to choose fractional order values as required. The novelty of performing a comparative study using two different numerical methods enhances the novelty. The comparison itself can reveal the strengths and limitations of each method when applied to the modified system. This helps researchers understand how different numerical techniques capture and represent chaotic behavior. The study’s results may provide unique control strategies or methods for utilizing chaotic behavior for signal processing, secure communications, and cryptography applications. The work closes the gap between theoretical notions of chaotic systems and their practical implications by investigating the behavior of the changed system through numerical simulations.

The research reported in this work is motivated by the urgent need to tackle the intricate dynamics inherent in symmetric chaotic jerk systems. We aim to offer a more profound comprehension of the underlying drive behind our study and its wider ramifications. Symmetry chaotic systems present a significant obstacle in diverse scientific and technical domains. The comprehension and proficient administration of such systems are essential across several disciplines, including physics, engineering, biology, and environmental science. The existence of chaos can pose challenges to predictive modeling and decision-making procedures, underscoring the need to devise novel approaches to address these complex dynamics. The primary aim of this study is to develop practical solutions for chaotic jerk systems exhibiting symmetry. This objective is accomplished by employing the Atangana–Baleanu Caputo fractional derivative (ABC-FD) scheme in conjunction with the novel iterative Laplace technique (NILM). The purpose of these strategies is to provide novel perspectives and efficient resources for understanding and controlling the intricacies of chaotic dynamics. The results of this work have implications that go beyond the scope of symmetry chaotic jerk systems. They hold significance for addressing the prevalence of chaos in diverse scientific and engineering contexts. By providing novel approaches to handling chaotic behavior, this research advances the field of nonlinear dynamics and fractional calculus, enabling the development of more accurate models in various domains.

The motivation behind this study stems from the urgent requirement to address symmetrical chaotic systems. It has the potential to make substantial contributions to resolving these systems and enhancing the comprehension and control of chaos within a wider scientific and engineering framework. The new ABC-FD scheme, due to its capacity to detect chaotic behavior, highlights the distinctiveness of our approach. The methods described in this study lay the foundation for exploring and evaluating potential fractional models, thereby contributing to the advancement of science and engineering.

An outline of this paper is as follows. In Section 2, we begin by providing some preliminaries and basic definitions that are used in our study. A numerical scheme for the ABC fractional derivative is presented in Section 3. Applications of the ABC-FD scheme are illustrated in Section 4. The new iterative Laplace method (NILM) and its application are provided in Section 5 and Section 6, respectively. Section 7 covers the numerical results and discussions. Finally, Section 8 includes the conclusions of our study.

2. Preliminaries and Basic Definitions

The fractional operators that are needed in this study are presented in a concise summary.

Definition 1. 

Let $q\in [1,\infty )$ and  $\omega$  be an open subset of  $R$;  then, the Sobolev space  $H^q\left(\omega \right)$ is defined by [41]:

$$H^q\left(\omega \right)=\{f\in L^2\left(\omega \right):D^{\beta }f\in L^2\left(\omega \right), \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{l}\mathrm{l} \left|\beta \right|\le q\}$$

(4)

Definition 2. 

The Atangana–Baleanu Caputo (ABC) fractional derivative of a function  $y\left(\tau \right)\in H^1(0,c),c>0$ with  $\alpha \in \left(\mathrm{0,1}\right]$ is defined as follows [42]:

$${}_0{}^{ABC}{D}_t^{\alpha }y\left(t\right)=\frac{B\left(\alpha \right)}{1-\alpha }\frac{d}{dt}{\int }_0^{t}y\left(\tau \right)E_{\alpha }\left(\frac{-\alpha }{1-\alpha }{\left(t-\tau \right)}^{\alpha }\right)d\tau . 0<\alpha <1,$$

(5)

Definition 3. 

The Mittag–Leffler function can be expressed as follows [42]:

$$E_{\alpha }\left(t\right)={\sum }_{k=0}^{\infty }\frac{t^k}{\mathsf{\Gamma }(\alpha k+1)}$$

(6)

Definition 4. 

The Atangana–Baleanu fractional integral of a function  $y\left(\tau \right)\in H^1(0,c)$. $c>0$ is as follows [42,43]:

$${ }_0^{AB}{I}_1^{\alpha }y(t)=\frac{1-\alpha }{B(\alpha )}y(t)+\frac{\alpha }{B(\alpha )\mathsf{\Gamma }(\alpha )}{\int }_0^1 y(\tau )(t-\tau {)}^{\alpha -1}d\tau .$$

(7)

Definition 5. 

The Laplace transform of the Atangana–Baleanu Caputo (ABC) fractional derivative given in [42] is

$$\mathcal{L}\left[{}_0{}^{ABC}{D}_t^{\alpha }y\left(t\right)\right]=\frac{B\left(\alpha \right)}{1-\alpha }\frac{s^{\alpha }\mathcal{L}\left[y\left(t\right)\right]-s^{\alpha -1}y\left(0\right)}{s^{\alpha }+\frac{\alpha }{1-\alpha }}$$

(8)

3. Numerical Scheme for the ABC Fractional Derivative

The goal of this section is to investigate a novel scheme for the ABC-FD of the form introduced in [44]:

$$\begin{array}{c}{}_0{}^{ABC}{D}_t^{\alpha }y\left(t\right)=\mathrm{H}\left(t,y\left(t\right)\right)\\ y\left(0\right)=y_0\end{array}$$

(9)

By using the fundamental theorem of fractional calculus, the equation above can be transformed into a fractional integral equation:

$$x(t)-x(0)=\frac{\left(1-\alpha \right)}{B\left(\alpha \right)}\mathsf{{\rm E}}(t,x(t))+\frac{\alpha }{\mathsf{\Gamma }\left(\alpha \right)\times B\left(\alpha \right)}{\int }_0^t \mathsf{{\rm E}}(\tau ,x(\tau ))(t-\tau {)}^{\alpha -1}\mathrm{d}\tau .$$

(10)

$$y(t)-y(0)=\frac{\left(1-\alpha \right)}{B\left(\alpha \right)}\mathrm{H}(t,y(t))+\frac{\alpha }{\mathsf{\Gamma }\left(\alpha \right)\times B\left(\alpha \right)}{\int }_0^t \mathrm{H}(\tau ,y(\tau ))(t-\tau {)}^{\alpha -1}\mathrm{d}\tau .$$

(11)

$$z(t)-z(0)=\frac{\left(1-\alpha \right)}{B\left(\alpha \right)}\mathrm{P}(t,z(t))+\frac{\alpha }{\mathsf{\Gamma }\left(\alpha \right)\times B\left(\alpha \right)}{\int }_0^t \mathrm{P}(\tau ,z(\tau ))(t-\tau {)}^{\alpha -1}\mathrm{d}\tau .$$

(12)

where $t_{\mathrm{n}+1},n=0,1,2\dots$, and this is reformulated as

$$\begin{array}{rr}x\left(t_{n+1}\right)-x\left(0\right)& =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{E}\left(t_n,x\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}} \mathrm{E}(\tau ,x(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ & =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{E}\left(t_n,x\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n {\int }_{t_k}^{t_{k+1}} \mathrm{E}(\tau ,x(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau .\hfill \end{array}$$

(13)

$$\begin{array}{rr}y\left(t_{n+1}\right)-y\left(0\right)& =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{H}\left(t_n,y\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}} \mathrm{H}(\tau ,y(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ & =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{H}\left(t_n,y\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n {\int }_{t_k}^{t_{k+1}} \mathrm{H}(\tau ,y(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau .\hfill \end{array}$$

(14)

$$\begin{array}{rr}z\left(t_{n+1}\right)-z\left(0\right)& =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{P}\left(t_n,z\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_{n+1}} \mathrm{P}(\tau ,z(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ & =\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{P}\left(t_n,z\left(t_n\right)\right)+\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n {\int }_{t_k}^{t_{k+1}} \mathrm{P}(\tau ,z(\tau \left)\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}\mathrm{d}\tau .\hfill \end{array}$$

(15)

with $\left[t_k,t_{k+1}\right]$, the functions $\mathrm{E}\left(t_n,x\left(t_n\right)\right),\mathrm{H}\left(t_n,y\left(t_n\right)\right) and \mathrm{P}\left(t_n,z\left(t_n\right)\right),$the following can be approximated using two-step Lagrange polynomial interpolation:

$${\mathrm{P}}_{1k}\left(\tau \right)\simeq \frac{\mathrm{E}\left(t_k,x_k\right)}{h}\left(\tau -t_{k-1}\right)-\frac{\mathrm{E}\left(t_{k-1},x_{k-1}\right)}{h}\left(\tau -t_k\right),$$

(16)

$${\mathrm{P}}_{2k}\left(\tau \right)\simeq \frac{\mathrm{H}\left(t_k,y_k\right)}{h}\left(\tau -t_{k-1}\right)-\frac{\mathrm{H}\left(t_{k-1},y_{k-1}\right)}{h}\left(\tau -t_k\right),$$

(17)

$${\mathrm{P}}_{3k}(\tau )\simeq \frac{\mathrm{P}\left(t_k,z_k\right)}{h}\left(\tau -t_{k-1}\right)-\frac{\mathrm{P}\left(t_{k-1},z_{k-1}\right)}{h}\left(\tau -t_k\right).$$

(18)

The above approximation can therefore be included in (16)–(18) to produce the following:

$$\begin{array}{rr}{x}_{n+1}=& x_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{E}\left(t_n,x\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n \left(\begin{array}{c}\frac{\mathrm{E}\left(t_k,x_k\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_{k-1}\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \\ -\frac{\mathrm{E}\left(t_{k-1},x_{k-1}\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_k\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \end{array}\right).\hfill \end{array}$$

(19)

$$\begin{array}{rr}{y}_{n+1}=& y_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{H}\left(t_n,y\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n \left(\begin{array}{c}\frac{\mathrm{H}\left(t_k,y_k\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_{k-1}\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \\ -\frac{\mathrm{H}\left(t_{k-1},y_{k-1}\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_k\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \end{array}\right).\hfill \end{array}$$

(20)

$$\begin{array}{rr}{z}_{n+1}=& y_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{P}\left(t_n,z\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)\times \mathsf{\Gamma }\left(\alpha \right)}\sum _{k=0}^n \left(\begin{array}{c}\frac{\mathrm{P}\left(t_k,z_k\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_{k-1}\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \\ -\frac{\mathrm{P}\left(t_{k-1},z_{k-1}\right)}{h}{\int }_{t_k}^{t_{k+1}} \left(\tau -t_k\right){\left(t_{n+1}-\tau \right)}^{\alpha -1}d\tau \end{array}\right).\hfill \end{array}$$

(21)

When you solve the integrals on the right, you obtain the following numerical scheme:

$$\begin{array}{rr}{x}_{n+1}=& x_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{E}\left(t_n,x\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)}\sum _{k=0}^n \left(\frac{h^{\alpha }\mathrm{E}\left(t_k,x_k\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha }(n-k+2+\alpha )-(n-k{)}^{\alpha }(n-k+2+2\alpha )\right)\right.\hfill \\ & \left.-\frac{h^{\alpha }\mathrm{E}\left(t_{k-1},x_{k-1}\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha +1}-(n-k{)}^{\alpha }(n-k+1+\alpha )\right)\right)\hfill \end{array}$$

(22)

$$\begin{array}{rr}{y}_{n+1}=& y_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{H}\left(t_n,y\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)}\sum _{k=0}^n \left(\frac{h^{\alpha }\mathrm{H}\left(t_k,y_k\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha }(n-k+2+\alpha )-(n-k{)}^{\alpha }(n-k+2+2\alpha )\right)\right.\hfill \\ & \left.-\frac{h^{\alpha }\mathrm{H}\left(t_{k-1},y_{k-1}\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha +1}-(n-k{)}^{\alpha }(n-k+1+\alpha )\right)\right)\hfill \end{array}$$

(23)

$$\begin{array}{rr}{z}_{n+1}=& z_0+\frac{(1-\alpha )}{B\left(\alpha \right)}\mathrm{P}\left(t_n,z\left(t_n\right)\right)\hfill \\ & +\frac{\alpha }{B\left(\alpha \right)}\sum _{k=0}^n \left(\frac{h^{\alpha }\mathrm{P}\left(t_k,z_k\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha }(n-k+2+\alpha )-(n-k{)}^{\alpha }(n-k+2+2\alpha )\right)\right.\hfill \\ & \left.-\frac{h^{\alpha }\mathrm{P}\left(t_{k-1},z_{k-1}\right)}{\mathsf{\Gamma }(\alpha +2)}\left((n+1-k{)}^{\alpha +1}-(n-k{)}^{\alpha }(n-k+1+\alpha )\right)\right)\hfill \end{array}$$

(24)

4. Applications of the ABC-FD Scheme

This section examines the practicality of the innovative plan for the numerical solution of the fractional simple chaotic jerk system in Equation (3) by using the ABC-FD scheme.

Table 1. Solutions of Equation (3) where $\alpha =1$ and $t=0.1$.

$\mathit{h}$	$\mathit{x}$	$\mathit{y}$	$\mathit{z}$
$1/320$	0.090976942328261	0.081191356188869	0.177322835208675
$1/640$	0.090969016611603	0.081174043332743	0.177301836131932
$1/1280$	0.090965053816737	0.081165386973051	0.177291336693722
$1/2560$	0.090963072435248	0.081161058810380	0.177286086999779
$1/5120$	0.090962411977118	0.081159616092040	0.177284337105534
$1/\mathrm{10,240}$	0.090962081748498	0.081158894733348	0.177283462159113

displays numerical solutions obtained using the ABC-FD scheme and the fractional simple chaotic jerk system shown in Equation (3) when $\alpha =1$ and $t=0.1$.

Table 2. Solutions of Equation (3) where $\alpha =0.95$ and $t=1$.

$\mathit{h}$	$\mathit{x}$	$\mathit{y}$	$\mathit{z}$
$1/320$	0.070352357852901	−0.009929541563750	0.046646357032531
$1/640$	0.070372731184257	−0.009990601272302	0.046497044647704
$1/1280$	0.070382994076053	−0.010021089970340	0.070382994076053
$1/2560$	0.070388144603155	−0.010036324181601	0.046385167861274
$1/5120$	0.070390724614794	−0.010043938809839	0.046366530599472
$1/\mathrm{10,240}$	0.070392015699256	−0.010047745802063	0.046357212549094

exhibits numerical solutions obtained using the ABC-FD scheme and the fractional simple chaotic jerk system shown in Equation (3) when $\alpha =0.95$ and $t=1.$ It can be noted that when the step size $h$ decreases, the accuracy improves. Based on the convergence of the numerical data in Table 1 and Table 2, we can observe the numerical stability feature of the ABC-FD scheme.

In Figure 2, Figure 3 and Figure 4, we plot numerical solutions to Equation (3) with $t=1000$ obtained using the ABC-FD scheme for different values of $\alpha$; our approach simulates problem (3) with good precision. Figure 2 shows the chaotic attractors plot by using the ABC-FD new scheme of Equation (3) when $\alpha =1$ and $t=1000$. Figure 3 shows the chaotic attractors plot of Equation (3) when $\alpha =0.97$ and $t=1000$, and Figure 4 shows the chaotic attractors plot of Equation (3) when $\alpha =0.99$ and $t=1000$. In Figure 5, we show the circuital generated results for Equation (3) when $\alpha =0.99$ and $t=1000.$

In these figures, we display the projections of the fractional simple chaotic jerk system attractors shown in Equation (3) obtained using a novel scheme (ABC-FD); identifying chaotic behavior by using a novel scheme (ABC-FD) in the fractional simple chaotic jerk system helps researchers gain insights into the underlying dynamics of the system. This understanding can be crucial in various scientific disciplines, including physics, engineering, and mathematics. Researchers can use this knowledge to design control strategies and synchronization methods to stabilize or manipulate chaotic systems for specific applications.

The circuit simulation and numerical results were compared, signifying that they were both in good agreement with each other. The simulation results demonstrated a high level of concordance between the circuit-based simulations and the numerical simulations.

We have used the technique proposed by Wolf et al. [45] to compute the Lyapunov exponents of the system (3). The values of a and b were set to $a=3.846\times {10}^{-4}$ and $b$ $=0.7$, respectively. The three Lyapunov exponents are denoted as $L_1=0.21244$, $L_2=0,L_3=-1.22543$. The presence of a positive exponent value for $L_1$ indicates that the attractor may be classified as a weird attractor, exhibiting chaotic motion. The Kaplan–Yorke dimension, as described in reference [46], is a measure that quantifies the complexity of an attractor. It is mathematically stated by the following:

$$D_{KY}=j+\frac{1}{\left|L_{j+1}\right|}{\sum }_{i=1}^j L_i$$

(25)

where $j$ is the largest integer satisfying ${\sum }_{i=1}^j \ge 0$ and ${\sum }_{i=1}^{j+1} \le 0$. The Kaplan–Yorke dimension of system (3) for $a=3.846\times {10}^{-4}$ and $b=0.7$ is $D_{KY}$ $=2.1759$. The evidence suggests that the newly implemented system has a fractal dimension and has been empirically shown to exhibit chaotic behavior.

5. The New Iterative Laplace Method (NILM)

In this section, we investigate the new iterative method and new iterative Laplace method.

5.1. New Iterative Method

Consider the equation

$$y=\rho +\mathcal{H}\left(y\right)+\mathcal{J}\left(y\right),$$

(26)

where $\mathcal{H}$ (linear) and $\mathcal{J}$ (nonlinear) are functions of $y$, and $\rho$ is a function.

Assuming the solution of Equation (11) has the form

$$y={\sum }_{\mathcal{j}=0}^{\infty }{y}_{\mathcal{j}}$$

We write the linear term as follows.

$$\mathcal{H}\left({\sum }_{\mathcal{j}=0}^{\infty }{y}_{\mathcal{j}}\right)={\sum }_{\mathcal{j}=0}^{\infty }{\mathcal{H}(y}_{\mathcal{j}})$$

And we decompose the nonlinear term as

$$\mathcal{J}\left({\sum }_{\mathcal{j}=0}^{\infty }{y}_{\mathcal{j}}\right)=\mathcal{J}\left(y_0\right)+{\sum }_{\mathcal{j}=1}^{\infty }\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}}{\mathcal{J}(y}_{\mathcal{i}})-\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}-1}{\mathcal{J}(y}_{\mathcal{i}})\right)\right)$$

Hence, Equation (11) can be expressed as

$${\sum }_{\mathcal{j}=0}^{\infty }{y}_{\mathcal{j}}=\rho +{\sum }_{\mathcal{j}=0}^{\infty }{\mathcal{H}(y}_{\mathcal{j}})+\mathcal{J}\left(y_0\right)+{\sum }_{\mathcal{j}=1}^{\infty }\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}}{\mathcal{J}(y}_{\mathcal{i}})-\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}-1}{\mathcal{J}(y}_{\mathcal{i}})\right)\right).$$

Furthermore, the recurrence formula can be defined as:

$$y_0=\rho$$

$$y_1=\mathcal{H}\left(y_0\right)+\mathcal{J}\left(y_0\right)$$

$$y_{\mathcal{j}+1}=\mathcal{H}\left(y_{\mathcal{j}}\right)+\left(\mathcal{H}\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}}{y}_{\mathcal{j}}\left(\tau \right)\right)\right)-\left(\mathcal{J}\left({\sum }_{\mathcal{i}=0}^{\mathcal{j}-1}{y}_{\mathcal{i}}\left(\tau \right)\right)\right),m=\mathrm{1,2},\dots$$

5.2. New Iterative Laplace Method

Now, consider Equation (9) with (ABC-FD) being written as

$${}_0{}^{ABC}{D}_t^{\alpha }y\left(t\right)=\mathcal{H}y\left(t\right)+\mathcal{J}y\left(t\right)$$

(27)

applying Laplace transform to Equation (27), we obtain

$$\mathcal{L}\left[y\left(t\right)\right]=\frac{y_0}{\mathcal{o}}+\frac{1-\alpha }{B\left(\alpha \right)}\left(1+\frac{\alpha }{{\mathcal{o}}^{\alpha }(1-\alpha )}\right) \mathcal{L}\left[\mathcal{H}y\left(t\right)+\mathcal{J}y\left(t\right)\right]$$

(28)

Taking ${\mathcal{L}}^{-1}$, we obtain

$$y\left(t\right)=y_0+{\mathcal{L}}^{-1}\left[\frac{1-\alpha }{B\left(\alpha \right)}\left(1+\frac{\alpha }{{\mathcal{o}}^{\alpha }(1-\alpha )}\right) \mathcal{L}\left[\mathcal{H}y\left(t\right)+\mathcal{J}y\left(t\right)\right]\right]$$

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Now, we apply the new iterative method that we mentioned in the previous section.

6. Application of the New Iterative Laplace Method

Consider the fractional system in Equation (3) with initial conditions $x\left(0\right)=0.1,y\left(0\right)=0.1,z\left(0\right)=0.2$. Taking the Laplace transform and its inverse to Equation (1), we have:

$$\begin{array}{rr}& x\left(t\right)=0.1+{\mathcal{L}}^{-1}\left[\frac{1-\alpha }{B\left(\alpha \right)}\left(1+\frac{\alpha }{{\mathcal{o}}^{\alpha }(1-\alpha )}\right)\mathcal{L}\left[\frac{y}{C}\right]\right],\\ & y\left(t\right)=0.1+{\mathcal{L}}^{-1}\left[\frac{1-\alpha }{B\left(\alpha \right)}\left(1+\frac{\alpha }{{\mathcal{o}}^{\alpha }(1-\alpha )}\right)\mathcal{L}\left[-\frac{1}{L}\left[x+\beta z^{2}y-\beta y\right]\right]\right],\\ & z\left(t\right)=0.2+{\mathcal{L}}^{-1}\left[\frac{1-\alpha }{B\left(\alpha \right)}\left(1+\frac{\alpha }{{\mathcal{o}}^{\alpha }(1-\alpha )}\right)\mathcal{L}\left[-y-\mu z+yz\right]\right].\end{array}$$

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Now, we applied the new iterative method that was discussed in the previous section, and we obtain the following:

$$\begin{array}{c}{x}_0=0.1\\ x_1=-\frac{0.1\left((1-\alpha )+\frac{t^{\alpha }\alpha }{\mathsf{\Gamma }[1+\alpha ]}\right)}{B\left(\alpha \right)}\\ x_2=\frac{0.2\left(\frac{2t^{\alpha }\alpha (1-\alpha )}{\mathsf{\Gamma }[1+\alpha ]}+\frac{t^{2\alpha }{\alpha }^2+{(-1+\alpha )}^2\mathsf{\Gamma }[1+2\alpha ]}{\mathsf{\Gamma }[1+2\alpha ]}\right)}{{\left(B\left(\alpha \right)\right)}^2}\\ x_3=\begin{array}{c}\frac{(-0.1+0.100167a-0.2b)}{{\left(B\left(\alpha \right)\right)}^3}(-\frac{3t^{\alpha }(-1+\alpha )\alpha (1-\alpha )}{\mathsf{\Gamma }[1+\alpha ]}+\frac{3t^{2\alpha }{\alpha }^2(1-\alpha )}{\mathsf{\Gamma }[1+2\alpha ]}\hfill \\ +\frac{t^{3\alpha }{\alpha }^3+{(-1+\alpha )}^2\mathsf{\Gamma }[1+3\alpha ](1-\alpha )}{\mathsf{\Gamma }[1+3\alpha ])})\end{array}\end{array}$$

$$\begin{array}{c}{y}_0=0.1\\ y_1=-\frac{0.2\left((1-\alpha )+\frac{t^{\alpha }\alpha }{\mathsf{\Gamma }[1+\alpha ]}\right)}{B\left(\alpha \right)}\\ y_2=-\frac{(-0.1+0.100167a-0.2b)}{{\left(B\left(\alpha \right)\right)}^2}\left(\frac{2t^{\alpha }\alpha (1-\alpha )}{\mathsf{\Gamma }[1+\alpha ]}+\frac{t^{2\alpha }{\alpha }^2+{(-1+\alpha )}^2\mathsf{\Gamma }[1+2\alpha ]}{\mathsf{\Gamma }[1+2\alpha ]}\right)\\ \begin{array}{c}{y}_3=-\left(\right((0.3+a(-0.603003-0.3005b)+0.3b+0.6b^2)t^{\alpha }{\left(1 -\alpha \right)}^2\alpha \mathsf{\Gamma }[1+2\alpha ]\mathsf{\Gamma }[1+3\alpha ]+\mathsf{\Gamma }[1\\ +\alpha \left]\right(t^{2\alpha }(0.3 +b^2(0.6 -0.6\alpha )+b(0.3-0.3\alpha )+a(-0.603003+b(-0.3005+0.3005\alpha )\\ +0.603003\alpha )-0.3\alpha ){\alpha }^2\mathsf{\Gamma }[1+3\alpha ]+\mathsf{\Gamma }[1+2\alpha ]\left(\right(0.1 +a(-0.201001-0.100167b)\\ +0.1b+0.2b^2)t^{3\alpha }{\alpha }^3+((-0.1-0.1b-0.2b^2){\left(-1+\alpha \right)}^3+a(-0.201001+0.6003\alpha \\ -0.603003{\alpha }^2+0.201001{\alpha }^3+b(-0.100167+0.3005\alpha -0.3005{\alpha }^2+0.100167{\alpha }^3)\left)\right)\mathsf{\Gamma }[1\\ +3\alpha \left]\right)\left)\right))/{\left(ABC\left(\alpha \right)\right)}^3\mathsf{\Gamma }[1+\alpha \left]\mathsf{\Gamma }\right[1+2\alpha \left]\mathsf{\Gamma }\right[1+3\alpha \left]\right)\end{array}\end{array}$$

$$\begin{array}{c}{z}_0=0.2\\ z_1=\frac{(-0.1+0.100167a-0.2b)\left((1-\alpha )+\frac{t^{\alpha }\alpha }{\mathsf{\Gamma }[1+\alpha ]}\right)}{B\left(\alpha \right)}\\ \begin{array}{c}{z}_2=-\frac{1}{{\left(B(\alpha )\right)}^2\mathsf{\Gamma }[1+\alpha ]\mathsf{\Gamma }[1+2\alpha ]}(t^{\alpha }(-0.2+a(0.402002 +b(0.200334 -0.200334\alpha )-0.402002\alpha )\\ +b(-0.2+0.2\alpha )+b^2(-0.4+0.4\alpha )+0.2\alpha \left)\alpha \mathsf{\Gamma }\right[1+2\alpha ]+(-0.1+a(0.201001 \\ +0.100167b)-0.1b-0.2b^2)\mathsf{\Gamma }[1+\alpha ](t^{2\alpha }{\alpha }^2+{(1 -\alpha )}^2\mathsf{\Gamma }[1+2\alpha \left]\right))\end{array}\\ \begin{array}{c}{z}_3=-\left(\right((0.6 +0.302004a^2+0.3b+0.3b^2+0.6b^3+a(-0.307511-1.20601b\\ -0.3005b^2\left)\right)t^{\alpha }{\left(1-\alpha \right)}^2\alpha \mathsf{\Gamma }[1+\alpha ]\mathsf{\Gamma }[1+2\alpha ]\mathsf{\Gamma }[1+3\alpha ]+a{t}^{2\alpha }{\alpha }^2\mathsf{\Gamma }[1\\ +2\alpha \left]\right(-0.00200334t^{\alpha }\alpha \mathsf{\Gamma }[1+2\alpha ]+(-0.00200334+0.00200334\alpha )\mathsf{\Gamma }[1+3\alpha ])\\ +{\mathsf{\Gamma }\left[1+\alpha \right]}^2\left(t^{2\alpha }{\alpha }^2\right(0.6+b^3(0.6 -0.6\alpha )+a^2(0.302004 -0.302004\alpha )+b(0.3 -0.3\alpha )\\ +b^2(0.3 -0.3\alpha )-0.6\alpha +a(-0.305508+b^2(-0.3005+0.3005\alpha )+0.305508\alpha \\ +b(-1.206013+1.20601\alpha )\left)\right)\mathsf{\Gamma }[1+3\alpha ]+\mathsf{\Gamma }[1+2\alpha ]\left(\right(0.2 +0.100668a^2+0.1b+0.1b^2\\ +0.2b^3+a(-0.1005-0.402002b-0.100167b^2))t^{3\alpha }{\alpha }^3+(-0.100668a^2{\left(-1+\alpha \right)}^3\\ +(-0.2-0.1b-0.1b^2-0.2b^3){\left(-1+\alpha \right)}^3+a(-0.102504+0.307511\alpha -0.307511{\alpha }^2\\ +0.102504{\alpha }^3+0.100167b^2{\left(-1+\alpha \right)}^3+0.402002b{\left(-1+\alpha \right)}^3\left)\right)\mathsf{\Gamma }[1\\ +3\alpha \left]\right)\left)\right)/\left({\left(B\left(\alpha \right)\right)}^3{\mathsf{\Gamma }[1+\alpha ]}^2\mathsf{\Gamma }\right[1+2\alpha \left]\mathsf{\Gamma }\right[1+3\alpha \left]\right))\end{array}\end{array}$$

The fourth terms approximate the solution for system (1)

$$\begin{gathered} x={\sum }_{\mathcal{j}=0}^3{x}_{\mathcal{j}}, \\ y={\sum }_{\mathcal{j}=0}^3{y}_{\mathcal{j}}, \\ z={\sum }_{\mathcal{j}=0}^3{z}_{\mathcal{j}}. \end{gathered}$$

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7. Numerical Results and Discussions

In this section, we will compare our proposed methods with the Runge–Kutta 4th-order method.

In

Table 3. Comparison between the new iterative Laplace method ($\mathrm{N}\mathrm{I}\mathrm{L}\mathrm{M}$), the ABC-FD scheme, and the Runge–Kutta 4th-order method (RK4) for $x\left(t\right)$ in relation to the fractional model in Equation (3).

t	$\mathbf{N}\mathbf{I}\mathbf{L}\mathbf{M}$ (α = 1)	ABC-FD (α = 1)	RK4	NILM (α = 0.95)	ABC-FD (α = 0.95)
0	0.1	0.1	0.1	$0.09547000481551651$	0.1
0.1	$0.09096000642068867$	0.090962081748498	0.090961091064418	$0.08753324148633759$	0.089869861689729
0.2	$0.0836800513655094$	0.083697708763766	0.083696921816189	$0.08166530511016366$	0.082568219065243
0.3	$0.07792017335859427$	0.078003841100781	0.078003234241056	$0.07709021041099968$	0.077182566037544
0.4	$0.07344041092407529$	0.073696270254762	0.073695822106252	$0.07343423181045078$	0.073330615652345
0.5	$0.07000080258608454$	0.070608058099706	0.070607749243106	$0.07039902624870213$	0.070743324321391
0.6	$0.06736138686875408$	0.068587112593472	0.068586925294057	$0.06772091120788105$	0.069208376289489
0.7	$0.06528220229621598$	0.067494081299899	0.067493999271994	$0.06515662549654835$	0.068548994446406
0.8	$0.06352328739260225$	0.067200527940728	0.067200536145170	$0.062476639070642014$	0.068613188755336
0.9	$0.06184468068204502$	0.067587361436780	0.067587445907686	$0.0594614471074005$	0.069267344857624
1	$0.06000642068867628$	0.068543490599299	0.068543638296846	$0.0558992853432117$	0.070392015699256

,

Table 4. Comparison between the new iterative Laplace method (NILM), the ABC-FD scheme, and the Runge–Kutta 4th-order method (RK4) for $y\left(t\right)$ in relation to the fractional model in Equation (3).

t	$\mathbf{N}\mathbf{I}\mathbf{L}\mathbf{M}$ (α = 1)	ABC-FD (α = 1)	RK4	NILM (α = 0.95)	ABC-FD (α = 0.95)
0	0.1	0.1	0.1	$0.09056641672364649$	0.1
0.1	$0.08115515809130855$	0.081158894733348	0.081156730659191	$0.07237915836843764$	0.078697858340013
0.2	$0.06444203521310958$	0.064468469882468	0.064466554822516	$0.05732365186363223$	0.061441045771581
0.3	$0.04959273563721611$	0.049715435020836	0.049713744539749	$0.04410455234142804$	0.046778652926444
0.4	$0.03633936363544122$	0.036713809022460	0.036712322078024	$0.03221490164926509$	0.034214331286817
0.5	$0.024414023479597945$	0.025303570580615	0.025302269090408	$0.021278445959197333$	0.023434830115426
0.6	$0.013548819441499313$	0.015347290438630	0.015346158895808	$0.010974942566247245$	0.014214221188707
0.7	$0.003475855792958407$	0.006727134723188	0.006726159855136	$0.0010154055130991366$	0.006379687559002
0.8	−0.006072763194211767	−0.000657806050452	−0.000658635586089	−0.00886902358831887	−0.000205320463951
0.9	−0.015364933248198126	−0.006893901927439	−0.006894595814113	−0.01893301278653764	−0.005651002981482
1	−0.024668550097187616	−0.012055116925426	−0.012055683430509	−0.029420582762232722	−0.010047745802063

and

Table 5. Comparison between the new iterative Laplace method ($\mathrm{N}\mathrm{I}\mathrm{L}\mathrm{M}$), the ABC-FD scheme, and the Runge–Kutta 4th-order method (RK4) for $z\left(t\right)$ in relation to the fractional model in Equation (3).

t	$\mathbf{N}\mathbf{I}\mathbf{L}\mathbf{M}$ (α = 1)	ABC-FD (α = 1)	RK4	NILM (α = 0.95)	ABC-FD (α = 0.95)
0	0.2	0.2	0.2	$0.18862323634104103$	0.2
0.1	$0.1772787589345153$	0.177283462159113	0.177280837322658	$0.16632846175044896$	0.174285678175172
0.2	$0.15684904243314876$	0.156883877304611	0.156881515372682	$0.14740398860035317$	0.153093618621189
0.3	$0.13832341777814897$	0.138486158092275	0.138484022799073	$0.1302399568063941$	0.134632104456653
0.4	$0.12131445225176456$	0.121812504013073	0.121810563716953	$0.11417507942878388$	0.118257375299824
0.5	$0.10543471313624415$	0.106620815236158	0.106619042407072	$0.0987053733964305$	0.023434830115426
0.6	$0.09029676771383637$	0.092700733181998	0.092699103953248	$0.08339472588532076$	0.090206130934548
0.7	$0.07551318326678981$	0.079870166941685	0.079868660717431	$0.06784522557988668$	0.077984463211127
0.8	$0.060696527077353145$	0.067972243760504	0.067970842871361	$0.05168378633892073$	0.066697932709398
0.9	$0.04545936642777493$	0.056872629123477	0.056871318523386	$0.03455498311198596$	0.056195836458403
1	$0.02941426860030384$	0.046457168423638	0.046455935422015	$0.016116757196209835$	0.046357212549094

, we show that the numerical solution results of the new iterative Laplace method and the ABC-FD scheme of Equation (3) are in excellent agreement with the solutions of the Runge–Kutta 4th order method when $\alpha =1.$ As a result of our findings, we are confident that the methodologies offered are powerful mathematical instruments for solving equations. Also, we can utilize them to find analytical or approximative solutions to other problems.

This paper presents a novel methodology that utilizes the ABC-FD scheme and the NILM method to address the challenges of solving the symmetric chaotic jerk system. This approach offers a promising alternative to conventional techniques such as the RK4. The efficacy of the provided strategies is demonstrated by comparing them with RK4. Nevertheless, it is imperative to acknowledge that our endeavors are not conducted in a vacuum. In recent years, significant progress has been made in equation solving. Notable advancements include the development of optimal derivative-free one-point algorithms for the computation of multiple zeros of nonlinear equations [47], the introduction of a family of derivative-free optimal fourth-order methods for computing multiple roots [48], and the creation of derivative-free multiple-root finders that exhibit optimal fourth-order convergence [49]. Furthermore, a novel and cost-effective higher-order method inspired by the Traub–Steffensen approach has been developed for solving nonlinear systems [50]. These findings showcase the field’s dynamism, reflecting the collective efforts to provide accurate and efficient solutions to complex problems.

8. Conclusions

This research gives the numerical solution of a fractional symmetry chaotic jerk system using a new scheme for the fractional derivative (ABC-FD) and the new iterative Laplace method (NILM). We compare the solutions of these methods with the Runge–Kutta fourth-order (RK4) method. By using MATLAB software package tools, we were able to provide a numerical strategy that aided the ABC-FD method to compare the solutions, and using Mathematica software package tools, we were able to provide a numerical strategy that aided the iterative Laplace method to compare the solutions. The numerical results show that our techniques execute their fractional procedures in a way that meets expectations for how well they preserve their numerical stability. This is demonstrated by the fact that the numerical results obtained are accurate. Moreover, in the future, we will explore the application of ABC-FD and NILM methods in modeling biological processes, disease spread, and drug interactions for advancements in healthcare and pharmaceutical research. We can investigate using chaotic dynamics to model and predict stock prices, exchange rates, and investment strategies for improved financial modeling. We can apply the stable ABC-FD and NILM methods to enhance the design and analysis of control systems in aerospace, robotics, and automation and integrate these methods into machine learning algorithms to boost predictive capabilities and decision-making processes, advancing the field of artificial intelligence. Our methods can be applied to solve chaotic systems, and we expect the methods to succeed in hyperchaotic systems and extend them to applications in engineering and coding. We advised applying this strategy to novel fractional problems [51,52] and comparing numerical solutions with other methods [53,54].