Open AccessArticle

Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods

Mohamed Elbadri

Mohamed A. Abdoon

D. K. Almutairi

Dalal M. Almutairi

⁴

and

Mohammed Berir

^5,*

Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia

Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 1142, Riyadh 12373, Saudi Arabia

Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia

⁴

Department of Mathematics, College of Science and Humanities, Shaqra University, Al-Dawadmi 17472, Saudi Arabia

⁵

Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 61008, Saudi Arabia

Author to whom correspondence should be addressed.

Fractal Fract. 2024, 8(12), 709; https://doi.org/10.3390/fractalfract8120709

Submission received: 28 October 2024 / Revised: 22 November 2024 / Accepted: 26 November 2024 / Published: 29 November 2024

(This article belongs to the Special Issue Current Trends on Fractional-Order Systems: Bifurcations, Synchronization, and Chaos)

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Abstract

This study presents two methods: a novel numerical scheme that utilizes the Atangana–Baleanu–Caputo (ABC) derivative and the Laplace New Iterative Method (LNIM). Furthermore, some complex dynamic behavior of fractional-order Chen is observed. The NABC method illustrates chaotic systems. We used the LNIM method to find analytical solutions for fractional Chen systems. The method stands out for its user-friendliness and numerical stability. The proposed methods are effective and yield analytical solutions and detection of chaotic behavior. Simultaneously, this results in a more precise understanding of the system. As a result, we may apply the approach to different systems and achieve more accurate findings. Furthermore, it has been demonstrated to be effective in accurately identifying instances through the exhibition of attractor chaos. Future applications in science and engineering can utilize these two methods to find numerical simulations and solutions to a variety of models.

Keywords:

LNIM; simulation; dynamic behavior; high-precision; chaos

1. Introduction

Fractional derivatives, the part of mathematics that studies derivatives and integrals of non-integer order, also known as singular integrals [1], has already become the most growing and rapidly developed area of science and engineering, including physics and applied mathematics [2]. This ascent echoes in the set of significant works, which has boosted the development and utilization of fractional calculus in diverse aspects. Recently, it has attracted much attention due to its application in modeling systems with memory and genetic properties, which ordinary calculus cannot do. Fractional calculus has great prospects in the field of science and is used in physics, engineering, biology, finance, etc. For example, in control theory, it enhances the approximation performance of dynamic systems with long memory. In most viscoelastic materials, stress and strain are well characterized by fractional time derivatives. Biomedical use covers various applications, including disease propagation, because fractal models provide more flexible and accurate models compared to integer models. Due to its ability to provide a description of non-local phenomena, fractional calculus is of great importance in modern scientific and technological investigations [3]. Other similar works include Kumar et al.’s study of the chaotic behavior of fractional predator–prey dynamical systems [4], Toufik and Atangana’s novel numerical differentiation for fractional chaotic models [5], and Bingi et al.’s fractional ordered modeling and control of robotic manipulator systems [6]. These abovementioned works cross ecological models, numerical methods, and robotics as well as the fractional epidemiology of computer viruses [7], Oldham and Spanier on fractional calculus [8], Gorenflo and Mainardi on fractional calculus in continuum mechanics [9], and Samko, Kilbas, and Marichev on fractional integrals and derivatives [10]. Further exploration of the cryptic hidden attractors in complex dynamical systems [11] and the entropy-inspired analysis of analysis-song encryption and parameter estimation in new Chua circuits and other species of chaos [12] further develop the concept of the fractional calculus and propagate it to various and manifold domains of science and technology for invention and discovery. This demonstrates the versatility of fractional differential equations in that the concept has grown into a flexible tool that can model various physical occurrences with different advanced non-local and memory characteristics. Over the years, numerous techniques for FDE solutions have been advanced due to the emerging techniques in current computational methods. Such methods include numerical solutions, analytical methods, and mixed methods that are composed of the two.

FDEs are significant not only within the “classical” areas of physics and engineering incorporated to the improvement of the accuracy of the models of diffusion, control, and material dynamics but also in interdisciplinary sciences, including economics, biology, and management. For instance, in economics, FDEs simulate processes with long-range memory, which include stock-exchange rates or economic development. There, they assist in modeling such operational scenarios as population, gene regulation, the spread of epidemics, and much more when the future state of the system is determined by its past. Moreover, in the field of management sciences, FDEs are used when decisions are formulated with time delay, or there are some cumulative impacts. Due to the higher flexibility and solidity of FDEs, they are the essential tool to analyze and describe the existing and newly emerging essential characteristics of numerous scientific, technological, and social systems. Their growing applicability indicates the increased inflexibility of problem-solving in today’s world, where interdisciplinary solutions are needed.

The investigation of chaotic system modeling has emerged as a prominent trend, widely utilized across both natural and technological disciplines [11,12,13]. The integration of chaotic systems into electrical circuits results in a complex modeling process that accurately reflects real-world conditions, making prediction a particularly tough endeavor. It employs phase pictures and sophisticated algorithms to investigate the behaviors of chaotic systems, controlling the impact of model parameters, identifying the Lyapunov exponent, and monitoring chaotic and hyperchaotic behavior based on beginning circumstances [14,15,16]. The Laplace New Iterative Method (LNIM) [17] is an analytic approach used to solve many mathematical problems, especially differential equations. This method improves the standard Laplace transform methodology by including new iterative processes [18,19,20,21], providing for increased convergence and accuracy in discovering solutions. Many researchers have used this idea, which combines integrative transformations with one of the analytical methods [22,23,24]. The advantage of LNIM is that it does not need a difficult calculation of the Adomian polynomial or require any perturbation parameters.

By analyzing the impacts of fractional-order chaotic systems, this study field has acknowledged that fractional differential systems under the Atangana–Baleanu–Caputo (ABC) derivative provide answers to issues that result from non-integer derivatives [25,26]. This research area is focused on analyzing the effects of fractional-order chaotic systems. As a result of this issue, it is distinguished from other formulations that concentrate on characteristics other than Liouville–Caputo derivatives, bifurcations, and Lyapunov analysis [27,28,29,30,31,32]. Using the same derivatives, other works [33,34] continue to go further into a variety of fractional-order chaotic and hyperchaotic systems for further exploration. Furthermore, the investigation of amplitude control for dynamically symmetric systems broadens the extent of this significance [35] within the area of study. The focus of this research is on the new avenues of fractional calculus applications, which may disclose the hidden aspects of chaotic systems and offer new opportunities for the development of fractional calculus and its application in a variety of scientific domains. Performing more research on the undiscovered alternate formulations of derivatives and determining how they relate to chaotic dynamics might potentially result in major advancements in the future.

We discuss the fractional Chen system [36,37]:

\{\begin{matrix} {}_{0}^{A B C}D_{t}^{α} x (t) = a_{1} (y (t) - x (t)), \\ {}_{0}^{A B C}D_{t}^{α} y (t) = (a_{3} - a_{1}) x (t) - x (t) z (t) + a_{3} y (t), \\ {}_{0}^{A B C}D_{t}^{α} z (t) = x (t) y (t) - a_{2} z (t) . \end{matrix}

(1)

where

α \in (0, 1)

, with conditions

x (0) = - 5

y (0) = - 1

, and

z (0) = - 1

, where

t > 0

, and

{}_{0}^{A B C}D_{t}^{α}

is the ABC operator. The parameters are given as

a_{1} = 7.5

a_{2} = 1.0

, and

a_{3} = 5

The fractional Chen system is a system of differential equations that has remarkable benefits in the representation of complicated dynamic procedures that cannot be described with the use of ordinary differential equations. This system is especially useful for investigating various natural processes, for instance, disease dissemination, population growth, and climate change, and for predicting and preventing their effects. In addition, it assists in constructing complicated control systems by providing analysis of the complex dynamic behaviors of the systems and is therefore valuable for researchers and engineers who want to optimize the performance of different applications.

The contribution of this work is to introduce novel methodologies for addressing the fractional Chen system using a numerical approach utilizing the ABC operator (NABC) method and the Laplace New Iterative Method (LNIM). We have confirmed the NABC and the feasibility of the LNIM as efficient tools for numerical modeling and solving the fractional Chen system. The results demonstrate the possibility of improving system model analysis and attractor chaos detection in science and engineering. These two methods can give numerical simulations and model solutions for future study and engineering.

2. Mathematical Preliminaries

Definition 1.

The Atangana–Baleanu fractional integral of the function

y \in H^{1} (0, c)

, with

c > 0

, is as follows [38]:

{}_{0}^{A B}I_{t}^{α} H (t) = \frac{1 - α}{B (α)} H (t) + \frac{α}{B (α) Γ (α)} \int_{0}^{t} H (ρ) {(t - ρ)}^{α - 1} d ρ .

(2)

Definition 2.

The Atangana–Baleanu–Caputo (ABC) derivative of a function

H \in H^{1} (0, c)

, with

α \in (0, 1]

is defined as [38]:

{}_{0}^{A B C}D_{t}^{α} H (t) = \frac{B (α)}{1 - α} \int_{0}^{t} H^{'} (ρ) E_{α} (- \frac{α}{1 - α} {(t - ρ)}^{α}) d ρ, 0 < α < 1,

(3)

where

H^{1} (0, c), c > 0

is a space of square-integrable functions and is itself defined as:

H^{1} (0, c) = {H (t) \in L^{2} (0, c) ∣ H^{'} (t) \in L^{2} (0, c)}, B (α) = 1 - α + \frac{α}{Γ (α)} .

Definition 3.

Mittag–Leffler Function [38]

E_{α} (t) = \sum_{k = 0}^{\infty} \frac{t^{k}}{Γ (α k + 1)} .

(4)

Definition 4.

The AB Operator in RLI [38]

{}_{0}^{A B}D_{t}^{α} H (t) = \frac{B (α)}{1 - α} \frac{d}{d t} \int_{0}^{t} H (τ) E_{α} (\frac{- α}{1 - α} {(t - τ)}^{α}) d τ,

(5)

where B(α) satisfies the condition B(1) = B(0) = 1, 0 < α < 1.

Definition 5.

The Laplace transform of the Atangana–Baleanu–Caputo derivative is given by [38]

L [{}_{0}^{A B C}D_{t}^{α} H (t)] = \frac{s^{α} l [H (t)] B (α) - s^{α - 1} H (0) B (α)}{s^{α} (1 - α) + α}

(6)

Definition 6.

Lagrange’s Polynomial Interpolation [39]

P_{n} (x) = \sum_{i = 0}^{n} w (x_{i}) L_{i} (x),

(7)

L_{i} (x) = \prod_{\begin{matrix} j = 0 \\ j \neq i \end{matrix}}^{n} \frac{x - x_{j}}{x_{i} - x_{j}} .

(8)

3. Numerical Scheme Using ABC Operator

The purpose of this section is to study chaotic systems (1) in the sense of the ABC fractional derivative, which may be expressed as follows:

{}_{0}^{A B C}D_{t}^{α} H (t) = w (t, H (t)),

(9)

with initial condition

H (0) = H_{0}

It is possible to obtain a fractional integral equation from the equation that was presented earlier:

H (t) - H (0) = \frac{(1 - α) w (t, H (t))}{α \cdot A B C (α)} + Γ (α + 1) \cdot A B C (α) \int_{0}^{t} w (τ, H (τ)) {(τ - t)}^{α - 1} d τ,

(10)

where

n = 0, 1, 2, 3, \dots

, reformulated as

\begin{matrix} H (t_{n + 1}) - H (0) & = \frac{(1 - α) w (t_{n}, H (t_{n}))}{α \cdot A B C (α)} + A B C (α) \cdot Γ (α + 1) \int_{0}^{t_{n + 1}} w (τ, H (τ)) {(t_{n + 1} - τ)}^{α - 1} d τ \\ = \frac{(1 - α) w (t_{n}, H (t_{n}))}{A B C (α)} + A B C (α) \cdot Γ (α) \sum_{k = 0}^{n} \int_{t_{k}}^{t_{k + 1}} w (τ, H (τ)) {(t_{n + 1} - τ)}^{α - 1} d τ . \end{matrix}

(11)

The following can be approximated using two-step Lagrange polynomial interpolation:

\begin{matrix} P_{k} (τ) & = \frac{(τ - t_{k - 1}) w (t_{k}, H (t_{k})) - (τ - t_{k}) w (t_{k - 1}, H (t_{k - 1}))}{t_{k} - t_{k - 1}} \\ = \frac{w (t_{k}, H (t_{k})) (τ - t_{k - 1}) - w (t_{k - 1}, H (t_{k - 1})) (τ - t_{k})}{h} \\ \approx \frac{w (t_{k}, H_{k}) (τ - t_{k - 1}) - w (t_{k - 1}, H_{k - 1}) (τ - t_{k})}{h} . \end{matrix}

(12)

\begin{matrix} X_{1, n + 1} = H_{0} + \frac{A B C (α) \cdot x_{3} (t_{n}, H (t_{n}))}{α} + A B C (α) \cdot Γ (α + 1) \sum_{k = 0}^{n} \int_{t_{k}}^{t_{k + 1}} \\ ((τ - t_{k - 1}) {(t_{n + 1} - τ)}^{α - 1} - (τ - t_{k}) {(t_{n + 1} - τ)}^{α - 1}) d τ \end{matrix}

(13)

For simplicity:

\begin{matrix} A_{a, k, 1} & = \frac{h^{α + 1}}{α (α + 1)} [{(n + 1 - k)}^{α} (n - k + 2 + α) - (n - k) (n - k + 2 + 2 α)] \end{matrix}

(14)

\begin{matrix} A_{a, k, 2} & = \frac{h^{α + 1}}{α (α + 1)} [{(n + 1 - k)}^{α + 1} - (n - k) (n - k + 1 + α)] \end{matrix}

(15)

By combining Equations (11) and (12) and substituting into (10):

\begin{matrix} X_{1, n + 1} & = H_{0} + \sum_{j = 0}^{n} [\frac{(1 - α)}{α} \frac{A B C (α)}{Γ (1 + α)} h^{α} x_{3} (t_{j}, X_{1, j}) \\ + \frac{h \cdot x_{3} (t_{i - 1}, j - 1) ({(n - j + 1)}^{α + 1} + (j - n) (n - j + 1 + α))}{Γ (1 + α)}] \end{matrix}

(16)

3.1. Dynamics Behaviors

In this section, more specific numerical examples of the new chaotic behaviors are presented. Below, the system is started with

x (0) = - 5

y (0) = - 1

z (0) = - 1

for

t > 0

. There, parameters are defined as

a_{1} = 7.5

a_{2} = 1.0, a_{3} = 5

. The figures highlight different plots of the numbers correlated with simulation and time series of the given impulsive system with different parameters and fractional derivatives. Simulation of the system behaviors is depicted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22, which represent the system parameters as

[a_{1}, a_{2}, a_{3}]

and fractional order

α

. Figure 1, Figure 3, Figure 5, Figure 7, Figure 9, Figure 11, Figure 13, Figure 15, Figure 17 and Figure 19 represent the numerical simulations representing the system in terms of the chaotic attractors. All the system responses at different time instants are depicted through time series plots in Figure 2, Figure 4, Figure 6, Figure 8, Figure 10, Figure 12, Figure 14, Figure 16, Figure 18 and Figure 20. The simulations show that the system exhibits a wealth of dynamic behaviors and chaos, with a specific focus on the effect of various parameters and the order of fractional derivatives. Figure 21 and Figure 22 show fractional-order Lorenz systems focusing on the fact that such systems present numerous types of chaotic attractors. The presented simulation results coincide with other methods and, furthermore, indicate many new chaotic attractors for the integer order system.

3.2. Numerical Solutions

Using the ABC technique, the section offers a thorough examination of the variables x, y, and z at various time periods t, arranged in two tables. Each of these tables’ tabulated data comprise a numerical dataset that might represent several situations and numerical solutions for the system in question. These solutions provide the information needed to classify and analyze trends, fluctuations, and stability. They are also useful in showing the rate of change and increase of each variable over time. This is because it shows the data in an organized way, which makes it easier to comprehend how well the model performs and how changes in circumstances and assumptions affect it. These data are crucial for evaluating the system’s response and determining its behavior in response to variations in certain parameters.

Table 1 and Table 2 show the approximate solutions of the Chen system derived using the Atangana–Baleanu–Caputo (ABC) derivative, specifically with initial conditions

(x_{0}, y_{0}, z_{0}) = (- 5, - 1, - 1)

. These solutions illustrate the effect of altering

α

on the behavior of the system, exhibiting how changes in the fractional order

α

affect the system dynamics.

4. Laplace New Iterative Method (LNIM)

In this section, we present the LNIM methodology for solving fractional systems. Consider the following fractional differential system with generalized fractional derivatives.

{}_{0}^{A B C}D_{t}^{α} H_{j} (t) = L [H_{j} (t)] + N [H_{j} (t)]

(17)

Subject to the initial conditions

H_{j} (0) = g_{j}, j = 1, 2, 3, \dots, m .

(18)

where

{}_{0}^{A B C}D_{t}^{α}

is the ABC fractional derivative of the functions

H_{j}

, and j denotes the number of equations in the system.

L

and

N

are linear and nonlinear terms. Applying Laplace to Equation (25), we have

L [H_{j} (t)] = \frac{g_{j}}{s} + \frac{1}{B (α)} (1 - α + α s^{- α}) L [L [H_{j} (t)] + N [H_{j} (t)]]

(19)

Taking

L^{- 1}

, we obtain

H_{j} (t) = g_{j} + L^{- 1} [\frac{1}{B (α)} (1 - α + α s^{- α}) L [L [H_{j} (t)] + N [H_{j} (t)]]]

(20)

The new iterative method takes the solution of system (27) in the form

H_{j} = \sum_{r = 0}^{\infty} H_{j_{r}} .

(21)

Linear and nonlinear terms can be written as

L (\sum_{r = 0}^{\infty} H_{j_{r}}) = \sum_{r = 0}^{\infty} L (H_{j_{r}})

(22)

N (\sum_{r = 0}^{\infty} H_{j_{r}}) = N (H_{j_{0}}) + \sum_{r = 1}^{\infty} (\sum_{i = 0}^{r} N (H_{j_{i}}) - (\sum_{i = 0}^{r - 1} N (H_{j_{i}})))

(23)

Then, Equation (26) can be expressed as

\begin{matrix} \sum_{r = 0}^{\infty} H_{j r} & = g_{j} + L^{- 1} [\frac{1}{B (α)} (1 - α + α s^{- α}) L \sum_{r = 0}^{\infty} L (H_{j_{r}}) + N (H_{j_{0}}) \\ + \sum_{r = 1}^{\infty} (\sum_{i = 0}^{r} N (H_{j_{i}}) - (\sum_{i = 0}^{r - 1} N (H_{j_{i}})))] \end{matrix}

(24)

Moreover, the formula for recurrence can be defined as:

\begin{matrix} H_{j 0} & = g_{j} \\ H_{j 1} & = L^{- 1} [\frac{1}{B (α)} (1 - α + α s^{- α}) L (L [H_{j 0} (t)] + N [H_{j 0} (t)])], \end{matrix}

\begin{matrix} H_{j r + 1} & = L^{- 1} [\frac{1}{B (α)} (1 - α + α s^{- α}) L \\ (L (H_{j r}) + (N (\sum_{i = 0}^{r} L (H_{j i}) (t))) - (N (\sum_{i = 0}^{r - 1} L (H_{j i}) (t))))], \\ r & = 1, 2, \dots \end{matrix}

First, we obtain the approximate solution of the Chen system (1) using LNIM. We start with the initial conditions

x (0)

= −5,

y (0)

= −1, and

z (0)

= −1 as the first approximate solutions. Then, we derive the recurrence formula to obtain the components of the approximate solutions. Finally, we substitute these components into Equation (21) to obtain the approximate solutions x, y and, z. In the next section, we present the numerical results using the fifth-term solutions of the Chen system obtained through LNIM.

Numerical Analysis

The section provides a comprehensive analysis of the variables x, y, and z at different time points t categorized through three tables. Tabulated data in each of these tables serves as a numerical dataset that may signify different scenarios, and numerical solutions of the system under consideration. These solutions are helpful in depicting the rate of change and growth of every variable over time and give the necessary data to categorize and analyze trends, fluctuations, and stability. This is due to the fact that it presents the data in a structured manner, hence facilitating the understanding of model performance as well as the effect of changes in conditions and assumptions. The values are presented in the tables with high accuracy, and the changes in each of the variables at exactly defined intervals of time are characterized. Such information is important for the assessment of the system’s reaction and the identification of its behavior based on changes in certain parameters.

Table 3, Table 4 and Table 5 show the approximate solutions of the Chen system (1), derived using the Laplace New Iterative Method (LNIM) for various values of the Atangana fractional order, specifically

α = 0.99

α = 0.96

, and

α = 0.95

, respectively. With parameters

(a_{1}, a_{2}, a_{3}) = (7.5, 1, 5)

and initial conditions

(x_{0}, y_{0}, z_{0}) = (- 5, - 1, - 1)

. These solutions illustrate the effect of altering

α

on the behavior of the system, exhibiting how changes in the fractional order

α

affect the system dynamics.

5. Numerical Method Accuracy

In this section, we compare the accuracy of various numerical schemes to solve the system for

α = 1

. In Table 6 and Table 7, we compare the ABC method for different step sizes (

h = 0.01

h = 0.001

h = 0.0001

and

h = 0.00001

), the RK4 method for

h = 0.001

and the LNIM method for a 10-term iterative approach, providing insight into the convergence and accuracy of the methods.

Table 6 compares the accuracy of the ABC method for

α

= 1 on three different time-step sizes. It shows the difference in error quantity

Δ

produced between solutions from two successive step sizes. All errors of each variable x, y, and z decrease as the step size decreases.

Table 7 compares the errors between the LNIM, RK4, and ABC methods. The errors of LNIM and RK4 are smaller than those between LNIM and ABC, indicating that LNIM provides more accurate solutions. For instance, at

t = 0.10

, the error between LNIM and RK4 is much smaller than that between LNIM and ABC, especially for the x-component.

6. Conclusions

This work provides new techniques for solving the fractional Chen system using a numerical scheme under the ABC (ABC) operator method and the Laplace New Iterative Method (LNIM). In this work, we have confirmed the ABC and the feasibility of the LNIM as efficient tools for numerical modeling and solving the fractional Chen system. Applying these techniques, it is indeed possible to describe the nature of the system and its operation based on dynamics. Most important, the techniques are simple to implement. The results show the feasibility of an improved analysis of system models and identification of attractor chaos in further scientific and engineering practices. Therefore, the proposed methods can be used for other systems for more exact results. These two methodologies can provide the numerical simulations and the solutions of the model for future research and engineering uses.

Author Contributions

Methodology, Investigation, M.B.; Funding acquisition, Software, D.K.A. and D.M.A.; Resources, Methodology, Investigation M.A.A.; Writing—original draft, Methodology, Investigation, M.E.; writing—review and editing, D.K.A. and D.M.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Deanship of Postgraduate Studies and Scientific Research at Majmaah University.

Data Availability Statement

Data is contained within the article.

Acknowledgments

The author extends the appreciation to the Deanship of Postgraduate Studies and Scientific Research at Majmaah University for funding this research work.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

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Figure 1. Numerical simulation for [1] with

α = 1

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 1. Numerical simulation for [1] with

α = 1

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

$Fractalfract 08 00709 g001$

$Fractalfract 08 00709 g002$

Figure 2. Time series plots of [1] with

α = 1

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 2. Time series plots of [1] with

α = 1

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

$Fractalfract 08 00709 g002$

$Fractalfract 08 00709 g003$

Figure 3. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 3. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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$Fractalfract 08 00709 g004$

Figure 4. Time series plots of [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 4. Time series plots of [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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$Fractalfract 08 00709 g005$

Figure 5. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 5. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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$Fractalfract 08 00709 g006$

Figure 6. Time series plots of [1] with

α = 0.99

α = 0.99

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 6. Time series plots of [1] with

α = 0.99

α = 0.99

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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Figure 7. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 7. Numerical simulation for [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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Figure 8. Time series plots of [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 8. Time series plots of [1] with

α = 0.99

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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Figure 9. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 9. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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Figure 10. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 10. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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$Fractalfract 08 00709 g011$

Figure 11. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 11. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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Figure 12. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 12. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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Figure 13. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 13. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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Figure 14. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 14. Time series plots of [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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Figure 15. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 15. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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Figure 16. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

Figure 16. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [36, 3, 20]

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Figure 17. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 17. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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Figure 18. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

Figure 18. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [7.5, 1, 5]

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$Fractalfract 08 00709 g019$

Figure 19. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 19. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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$Fractalfract 08 00709 g020$

Figure 20. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

Figure 20. Time series plots of [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [5, 2.5, 1]

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$Fractalfract 08 00709 g021$

Figure 21. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [39, 9, 29]

Figure 21. Numerical simulation for [1] with

α = 0.97

[a_{1}, a_{2}, a_{3}] = [39, 9, 29]

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Figure 22. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [39, 9, 29]

Figure 22. Numerical simulation for [1] with

α = 0.98

[a_{1}, a_{2}, a_{3}] = [39, 9, 29]

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Table 1. Solutions of system (1) where

α = 1

, and

t = 2

Table 1. Solutions of system (1) where

α = 1

, and

t = 2

h	x	y	z
1/320	$1.970450059197159$	$1.892790486482604$	$3.948261033919270$
1/640	$1.968944380072037$	$1.885782995260466$	$3.946660700790122$
1/1280	$1.967426486957617$	$1.878498016746111$	$3.945652775510089$
1/2560	$1.967426486957617$	$1.878498016746111$	$3.945652775510089$
1/5120	$1.966966217862482$	$1.876593821952430$	$3.945478164578038$
1/10,240	$1.966966217862482$	$1.876273184100086$	$3.945451968729026$
RK4	$1.967502791702454$	$1.880792169956990$	$3.945065645235501$

Table 2. Solutions of system (1) where

α = 0.99

, and

t = 2

Table 2. Solutions of system (1) where

α = 0.99

, and

t = 2

h	x	y	z
1/320	$1.503544778425298$	$1.701090080546432$	$3.101376802618641$
1/640	$1.504939874429462$	$1.700864785232350$	$3.099571189331024$
1/1280	$1.505666756047958$	$1.700754564867528$	$3.098706583675867$
1/2560	$1.506037115610106$	$1.700699802140150$	$3.098283475022354$
1/5120	$1.506223965695771$	$1.700672473820942$	$3.098074173961615$
1/10,240	$1.506317758230647$	$1.700658693233841$	$3.097970083584368$

Table 3. Values of t, x, y, and z with

α = 0.99

for different time steps.

Table 3. Values of t, x, y, and z with

α = 0.99

for different time steps.

t	x	y	z
0.00	$- 4.717676534442966$	$- 0.9752209678479962$	$- 0.9442271146366138$
0.01	$- 4.4516449453286375$	$- 0.9503260955927028$	$- 0.892673688897514$
0.02	$- 4.2061742032715665$	$- 0.9266989056470135$	$- 0.8456419478085981$
0.03	$- 3.977525285438457$	$- 0.9042467985407454$	$- 0.8022734717801349$
0.04	$- 3.7640760080658806$	$- 0.8830231200601681$	$- 0.7621397724426929$
0.05	$- 3.5645968192518827$	$- 0.8630807798984207$	$- 0.724902729472965$
0.06	$- 3.378052860429984$	$- 0.8444599890421611$	$- 0.690275674669581$
0.07	$- 3.2035367110547006$	$- 0.8271851438218556$	$- 0.6580112058545047$
0.08	$- 3.0402366700827828$	$- 0.811262337864509$	$- 0.6278973582751224$
0.09	$- 2.887418800435286$	$- 0.796675101795547$	$- 0.5997584594245231$
0.10	$- 2.7444156491377165$	$- 0.7833769805980455$	$- 0.5734600950246401$

Table 4. Values of t, x, y, and z with

α = 0.96

for different time steps.

Table 4. Values of t, x, y, and z with

α = 0.96

for different time steps.

t	x	y	z
0.00	$- 4.038374865459166$	$- 0.9073880243827049$	$- 0.8165151728821144$
0.01	$- 3.8240536015073925$	$- 0.8849413480396292$	$- 0.7778133828078999$
0.02	$- 3.6337485089678943$	$- 0.865322663999055$	$- 0.7436855215701685$
0.03	$- 3.4580263753870932$	$- 0.8475933495266136$	$- 0.7125423617887194$
0.04	$- 3.2940337020361135$	$- 0.8314421526000831$	$- 0.684166714775769$
0.05	$- 3.1398188340444593$	$- 0.8165451849112016$	$- 0.6587569765220675$
0.06	$- 2.993699315521101$	$- 0.8024388648335294$	$- 0.6369156024729663$
0.07	$- 2.854037988620238$	$- 0.788412410800134$	$- 0.6197445486707193$
0.08	$- 2.7191241404623554$	$- 0.7733831247333017$	$- 0.6089980486496176$
0.09	$- 2.587091023192293$	$- 0.7557419223744537$	$- 0.6072867320872235$
0.10	$- 2.4558474477475563$	$- 0.7331608846647865$	$- 0.6183362807355592$

Table 5. Values of t, x, y, and z with

α = 0.95

for different time steps.

Table 5. Values of t, x, y, and z with

α = 0.95

for different time steps.

t	x	y	z
0.00	$- 3.854023176988972$	$- 0.8879344286716799$	$- 0.7834668804097433$
0.01	$- 3.6514332305839075$	$- 0.866947568052652$	$- 0.7477788098234303$
0.02	$- 3.4725724911627123$	$- 0.8491353747238412$	$- 0.7170613316589427$
0.03	$- 3.305786008536047$	$- 0.8332622203476312$	$- 0.6900217961781865$
0.04	$- 3.146943081899874$	$- 0.8187445495942969$	$- 0.6672397308913354$
0.05	$- 2.9926590527137993$	$- 0.8048254540149677$	$- 0.6501669441801001$
0.06	$- 2.8395061442853926$	$- 0.7903514232084065$	$- 0.6412497013458499$
0.07	$- 2.683697723552972$	$- 0.773556875091765$	$- 0.6441999313610214$
0.08	$- 2.520889434668927$	$- 0.7518107963632288$	$- 0.6643613303072801$
0.09	$- 2.3460167042991955$	$- 0.7213088687702536$	$- 0.7091663608281956$
0.10	$- 2.1531423587284078$	$- 0.6766996679911801$	$- 0.7886906802029681$

Table 6. A determination of the accuracy of the ABC method for

α = 1

Table 6. A determination of the accuracy of the ABC method for

α = 1

t	$Δ = \| {ABC}_{0.01} - {ABC}_{0.001} \|$			$Δ = \| {ABC}_{0.001} - {ABC}_{0.0001} \|$
t	$Δ x$	$Δ y$	$Δ z$	$Δ x$	$Δ y$	$Δ z$
0.10	1.39 × $10^{- 1}$	1.33 × $10^{- 2}$	2.58 × $10^{- 2}$	1.37 × $10^{- 2}$	1.30 × $10^{- 3}$	2.55 × $10^{- 3}$
0.15	1.00 × $10^{- 1}$	7.07 × $10^{- 3}$	1.91 × $10^{- 2}$	9.75 × $10^{- 3}$	6.72 × $10^{- 4}$	1.87 × $10^{- 3}$
0.20	7.07 × $10^{- 2}$	1.02 × $10^{- 3}$	1.50 × $10^{- 2}$	6.81 × $10^{- 3}$	7.06 × $10^{- 5}$	1.46 × $10^{- 3}$
0.25	0.10 × $10^{- 2}$	7.07 × $10^{- 3}$	1.91 × $10^{- 2}$	4.62 × $10^{- 3}$	4.92 × $10^{- 4}$	1.20 × $10^{- 3}$
0.30	3.13 × $10^{- 2}$	0.10 × $10^{- 3}$	1.10 × $10^{- 2}$	2.93 × $10^{- 3}$	0.10 × $10^{- 4}$	1.06 × $10^{- 3}$

Table 7. Error Comparison of the LNIM, RK4, and ABC methods with LNIM accuracy for

α

= 1.

Table 7. Error Comparison of the LNIM, RK4, and ABC methods with LNIM accuracy for

α

= 1.

t	$Δ = \| LNIM - RK 4 \|$			$Δ = \| LNIM - ABC \|$
t	$Δ x$	$Δ y$	$Δ z$	$Δ x$	$Δ y$	$Δ z$
0.10	1.48 × $10^{- 6}$	6.06 × $10^{- 6}$	6.38 × $10^{- 7}$	1.54 × $10^{- 4}$	8.36 × $10^{- 6}$	2.89 × $10^{- 5}$
0.15	1.13 × $10^{- 4}$	4.71 × $10^{- 4}$	6.65 × $10^{- 5}$	2.21 × $10^{- 4}$	4.64 × $10^{- 4}$	8.73 × $10^{- 5}$
0.20	2.40 × $10^{- 3}$	1.01 × $10^{- 2}$	1.73 × $10^{- 3}$	2.48 × $10^{- 3}$	1.01 × $10^{- 2}$	1.75 × $10^{- 3}$
0.25	2.53 × $10^{- 2}$	1.08 × $10^{- 1}$	2.11 × $10^{- 2}$	2.53 × $10^{- 2}$	1.08 × $10^{- 1}$	2.11 × $10^{- 2}$
0.30	1.71 × $10^{- 1}$	7.35 × $10^{- 1}$	1.60 × $10^{- 1}$	1.71 × $10^{- 1}$	7.35 × $10^{- 1}$	1.60 × $10^{- 1}$

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Elbadri, M.; Abdoon, M.A.; Almutairi, D.K.; Almutairi, D.M.; Berir, M. Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods. Fractal Fract. 2024, 8, 709. https://doi.org/10.3390/fractalfract8120709

AMA Style

Elbadri M, Abdoon MA, Almutairi DK, Almutairi DM, Berir M. Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods. Fractal and Fractional. 2024; 8(12):709. https://doi.org/10.3390/fractalfract8120709

Chicago/Turabian Style

Elbadri, Mohamed, Mohamed A. Abdoon, D. K. Almutairi, Dalal M. Almutairi, and Mohammed Berir. 2024. "Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods" Fractal and Fractional 8, no. 12: 709. https://doi.org/10.3390/fractalfract8120709

APA Style

Elbadri, M., Abdoon, M. A., Almutairi, D. K., Almutairi, D. M., & Berir, M. (2024). Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods. Fractal and Fractional, 8(12), 709. https://doi.org/10.3390/fractalfract8120709

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Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods

Abstract

1. Introduction

2. Mathematical Preliminaries

3. Numerical Scheme Using ABC Operator

3.1. Dynamics Behaviors

3.2. Numerical Solutions

4. Laplace New Iterative Method (LNIM)

Numerical Analysis

5. Numerical Method Accuracy

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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