Novel Dynamic Behaviors in Fractional Chaotic Systems: Numerical Simulations with Caputo Derivatives

Abstract

Over the last several years, there has been a considerable improvement in the possible methods for solving fractional-order chaotic systems; however, achieving high accuracy remains a challenge. This work proposes a new precise numerical technique for fractional-order chaotic systems. Through simulations, we obtain new types of complex and previously undiscussed dynamic behaviors.These phenomena, not recognized in prior numerical results or theoretical estimations, underscore the unique dynamics present in fractional systems. We also study the effects of the fractional parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ on the system’s behavior, comparing them to integer-order derivatives. It has been demonstrated via the findings that the suggested technique is consistent with conventional numerical methods for integer-order systems while simultaneously providing an even higher level of precision. It is possible to demonstrate the efficacy and precision of this technique through simulations, which demonstrates that this method is useful for the investigation of complicated chaotic models.

1. Introduction

Fractional calculus is a branch of mathematics involving derivatives and integrals of non-integer order. This has already turned into one of the most rapidly developing directions in applied mathematics, physics, and many other sciences and engineering disciplines. This ascent is evidenced by the collection of significant articles, which has helped in the growth and use of fractional calculus in a number of contexts and fields. This includes the research conducted by Lai et al. [1] on a novel chaotic system with multiple attractors; Petras [2] on the fractional-order Chua’s system; and Abd El-Maksoud et al. [3] on the FPGA realization of sound encryption using fractional-order chaotic systems, validating the adaptability and detection of dynamic features in chaotic systems through the utilization of the fractional calculus methodology.

Other studies that are comparable to this one include the fractional-ordered modeling and control of robotic manipulator systems by Bingi et al. [4], the novel numerical differentiation for fractional chaotic models by Toufik and Atangana [5], and the investigation of the chaotic characteristics of fractional predator–prey dynamical systems by Kumar et al. [6]. The works mentioned above cover a broad range of subjects, such as fractional epidemiology of computer viruses, robotics, numerical methods, and ecological models [7]. Other relevant works include fractional calculus by Oldham and Spanier [8] and fractional calculus in continuum mechanics by Gorenflo and Mainardi [9]. In addition, Samko, Kilbas, and Marichev’s major contributions to fractional integrals and derivatives [10] should be acknowledged. In addition to laying forth the theoretical foundation, these books offer a wide range of applications in other branches of science and mathematics.

Further investigations into hidden attractors in dynamical systems [11] and the entropy-driven analysis, parameter estimation, and signal encryption in New Chua’s circuits and other chaotic systems [12] expand on fractional calculus’s many applications in scientific research and technological advancements. The fact that several approaches have been recently created to solve the FDE is what makes it significant. The latest developments in computational approaches have enabled the use of these methodologies [13,14]. The FDE may also be used to tackle physics and engineering issues, and it can also account for multidisciplinary crossovers in domains like biology, economics, and management [15,16].

As a consequence of its broad usage in the natural and technical sciences, the study of modeling chaotic systems has emerged as one of the most actively researched issues [11,12,17]. This factor may be regarded as highly difficult to foresee because integrating chaotic systems into electrical circuits necessitates an elaborate modeling procedure and appropriate confrontation with real-life scenarios. Phase diagrams and complex algorithms are employed, along with the finding of Lyapunov and the monitoring of chaotic and even hyperchaotic behavior via starting conditions. These techniques are used to control the impacts of model parameters [18,19,20].

A variety of differential equations are solved using the $\rho$-Laplace technique and the new iterative method (NIM), introduced in [21], which is an extension of the Laplace transform and analytical approaches [22,23,24,25]. However, neither the calculation of Adomian polynomials nor the inclusion of perturbation parameters in the equation are necessary for the NIM to function. This is one of the benefits of the NIM.

This branch of research has recognized that fractional differential systems offer solutions to problems arising from non-integer derivatives through an analysis of the effects of fractional-order chaotic systems [26]. The analysis of the consequences of fractional-order chaotic systems is the main focus of this field of study. This problem sets it apart from previous formulations that focus on properties other than bifurcations, Lyapunov analysis, and Liouville–Caputo derivatives [27,28,29,30,31,32].

Other publications [33,34] continue to delve deeper for more study into a range of fractional-order chaotic and hyperchaotic systems using the same derivatives. Moreover, the field of research expands this relevance [35] with the examination of amplitude control for dynamically symmetric systems. This study focuses on the novel applications of fractional calculus that might reveal the inner workings of chaotic systems and provide fresh prospects for the advancement of fractional calculus and its use in several scientific fields. Further investigations into the as-yet-undiscovered alternative formulations of derivatives and their connections to chaotic dynamics might yield significant future developments. In this paper, we simulate the fractional new chaos system [36] as follows:

$$\left\{\begin{array}{cc}\hfill {\mathcal{D}}_0^{{\beta }_1}x\left(t\right)& =ax\left(t\right)-by\left(t\right)x\left(t\right),\hfill \\ \hfill {\mathcal{D}}_0^{{\beta }_2}y\left(t\right)& =-cx\left(t\right)-x\left(t\right)z\left(t\right),\hfill \\ \hfill {\mathcal{D}}_0^{{\beta }_3}z\left(t\right)& =x\left(t\right)y\left(t\right)z\left(t\right)-dz\left(t\right)+4,\hfill \end{array}\right.$$

(1)

subject to the following constraints:

$$\begin{array}{cccc}\hfill x\left(0\right)& =1,y\left(0\right)\hfill & \hfill =1,z\left(0\right)& =1,\hfill \end{array}$$

(2)

where $a=1$, $b=1$, $c=2$, and $d=1$.

The mathematical construction of a novel exact numerical approach for fractional-order chaotic systems is the contribution of this paper. This approach reveals the dynamic behavior that was previously unknown and advances our knowledge of the chaos in fractional systems. The purpose of this investigation is to examine the impact of the fractional parameters ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ to obtain a deeper comprehension of how the system behaves differently from the integer fractional-order model. The suggested solution produces improved accuracy, suggesting that it is stable, sound, and well-suited to apply current techniques for integer-order systems. The new method also offers up new avenues for the study of chaos theory because it is general and can be developed to evaluate other powerful chaotic systems.

2. Mathematical Preliminaries

Definition 1

([37]). A function $u\left(t\right)$ defined over the interval $(t_0,t)$ possesses a Riemann–Liouville fractional derivative (FD) of order β, which is formulated as follows:

$${}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (m-\beta )}}{\displaystyle \frac{d^m}{d{t}^m}}{\int }_{t_0}^t{\displaystyle \frac{u\left(\tau \right)}{{(t-\tau )}^{\beta -m+1}}}d\tau ,\hfill & if0\le m-1<\beta <m\hfill \\ {\displaystyle \frac{d^{m}u\left(t\right)}{d{t}^m}},\hfill & if\beta =mandm\in \mathbb{N}.\hfill \end{array}\right.$$

(3)

Definition 2

([37]). A function $u\left(t\right)\in C^m[t_0,t]$ about t on the interval $(t_0,t)$ has a Caputo FD of order β, which is given as follows:

$${}_{t_0}{}^{\mathit{C}}D_t^{\beta }u\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (m-\beta )}}{\int }_{t_0}^t{\displaystyle \frac{u^{\left(m\right)}\left(\tau \right)}{{(t-\tau )}^{\beta -m+1}}}d\tau ,\hfill & 0\le m-1<\beta <m\hfill \\ {\displaystyle \frac{d^{m}u\left(t\right)}{d{t}^m}},\hfill & \beta =m\in \mathbb{N}.\hfill \end{array}\right.$$

(4)

Definition 3

([38]). A function $u\left(t\right)$ defined on the interval $(t_0,t)$ possesses a Grünwald–Letnikov fractional derivative of order β, expressed as follows:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\beta }}}\sum _{j=0}^{⌊{\displaystyle \frac{t-t_0}{h}}⌋}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{j}}\right)u(t-jh),$$

(5)

where the coefficient

$${(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{j}}\right)={\displaystyle \frac{{(-1)}^j\Gamma (\beta +1)}{\Gamma (j+1)\Gamma (\beta -j+1)}}$$

is defined as

$${\omega }_j^{\left(\beta \right)}={(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{j}}\right)={\displaystyle \frac{{(-1)}^j\Gamma (\beta +1)}{\Gamma (j+1)\Gamma (\beta -j+1)}},j=0,1,\dots ,m.$$

(6)

Thus, the Grünwald–Letnikov derivative of order β can be represented as follows:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\beta }}}\sum _{j=0}^{⌊{\displaystyle \frac{t-t_0}{h}}⌋}{\omega }_j^{\left(\beta \right)}u(t-jh).$$

(7)

Utilizing Newton’s binomial theorem, we derive the following expression:

$${(1-z)}^{\beta }=\sum _{j=0}^m{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\beta }{j}}\right)z^j=\sum _{j=0}^m{\omega }_j^{\left(\beta \right)}{z}^j.$$

(8)

This leads to the following conclusion:

$$\sum _{j=0}^{\infty }{\omega }_j^{\left(\beta \right)}{z}^j={(1-z)}^{\beta }.$$

(9)

The expression ${(1-z)}^{\beta }$ is recognized as the generating function of the first order for the $\beta$-order Grünwald–Letnikov derivative over the interval $(t_0,t)$.

Finally, we have the following:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={\displaystyle \frac{1}{h^{\beta }}}\sum _{j=0}^{⌊{\displaystyle \frac{t-t_0}{h}}⌋}{\omega }_j^{\left(\beta \right)}u(t-jh)+o\left(h\right).$$

(10)

Theorem 1

([35]). Let f be a continuous function that is differentiable up to order $m-1$ on the interval $(t_0,t)$, and let us assume that $f^{\left(m\right)}\left(t\right)$ is integrable. Under these conditions, the Grünwald–Letnikov derivative can be formulated as follows:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)=\sum _{j=0}^{m-1}{\displaystyle \frac{u^{\left(j\right)}\left(t_0\right)}{\Gamma (1+j-\beta )}}{(t-t_0)}^{j-\beta }+{\displaystyle \frac{1}{\Gamma (m-\beta )}}{\int }_{t_0}^t{\displaystyle \frac{u^{\left(m\right)}\left(\tau \right)}{{(t-\tau )}^{\beta -m+1}}}d\tau ,$$

(11)

where $0\le m-1<\beta <m$.

If these conditions are satisfied, the following relationships hold among the Riemann–Liouville derivative, the Caputo derivative, and the Grünwald–Letnikov derivative:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right),$$

and

$${}_{t_0}{}^{\mathit{C}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)-\sum _{j=0}^{m-1}{\displaystyle \frac{u^{\left(j\right)}\left(t_0\right)}{\Gamma (1+j-\beta )}}{(t-t_0)}^{j-\beta }.$$

(12)

Furthermore, if $u^{\left(j\right)}\left(t_0\right)=0$ for all $j=0,1,\dots ,m-1$, then the following can be concluded:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{C}}D_t^{\beta }u\left(t\right).$$

Proof. 

For a detailed proof, please refer to [36]. □

From Theorem 1, it follows that if $u^{\left(j\right)}\left(t_0\right)=0$ for $j=0,1,\dots ,m-1$, then:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{C}D_t^{\beta }u\left(t\right).$$

If there exists at least one j such that $u^{\left(j\right)}\left(t_0\right)\ne 0$ for $j=0,1,\dots ,m-1$, we can use the Taylor series expansion. Let

$$\overline{u}\left(t\right)=u\left(t\right)-\sum _{j=0}^{m-1}{\displaystyle \frac{u^{\left(j\right)}\left(t_0\right)}{\Gamma (1+j-\beta )}}{(t-t_0)}^{j-\beta },$$

then we obtain:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{C}D_t^{\beta }\overline{u}\left(t\right).$$

For convenience, we denote:

$${}_{t_0}{}^{\mathit{GL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{RL}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\mathit{C}}D_t^{\beta }u\left(t\right)={}_{t_0}{}^{\beta }D_t^{\beta }u\left(t\right).$$

3. Numerical Approach

In [39,40], the authors described a recurrent technique that uses generating functions to solve fractional derivatives and integrals. To further improve accuracy, they also devised a construction technique for functions that may be applied to any p.

Definition 4.

The definition of a p-order polynomial function is as follows:

$$P_p\left(x\right)=\sum _{l=1}^p{\displaystyle \frac{1}{l}}{(1-x)}^l.$$

(13)

Theorem 2.

If $P_p\left(x\right)$ is a p-order polynomial function, then it can be expressed as follows:

$$P_p\left(x\right)=\sum _{l=0}^p{P}_l{x}^l,$$

(14)

where $P_l$ represents the solution to the following system of equations:

$$\left(\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& 2& 3& \cdots & (p+1)\\ 1& 2^2& 3^2& \cdots & {(p+1)}^2\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& 2^p& 3^p& \cdots & {(p+1)}^p\end{array}\right)\left(\begin{array}{c}{P}_0\\ P_1\\ P_2\\ ⋮\\ P_p\end{array}\right)=-\left(\begin{array}{c}0\\ 1\\ 2\\ ⋮\\ p\end{array}\right).$$

(15)

Proof. 

It can be observed from Equations (13) and (14) that

$$\sum _{l=0}^p{P}_l{x}^l=\sum _{l=1}^p{\displaystyle \frac{1}{l}}{(1-x)}^l.$$

(16)

Substituting $x=1$ into Equation (16), we find

$$\sum _{l=0}^p{P}_l=0.$$

(17)

Multiplying both sides of Equation (16) by x and calculating the first derivative, we obtain

$$\sum _{l=0}^p(l+1)P_l{x}^l=\sum _{l=1}^p{\displaystyle \frac{1}{l}}{(1-x)}^l-x\sum _{l=1}^p{(1-x)}^{l-1}.$$

(18)

Substituting $x=1$ in (18), we find

$$\sum _{l=0}^p(l+1)P_l=-1.$$

Multiplying both sides of Equation (18) by x and calculating the first derivative again, we obtain

$$\sum _{l=0}^p{(l+1)}^2{P}_l{x}^l=\sum _{l=1}^p{\displaystyle \frac{1}{l}}{(1-x)}^l-3x\sum _{l=1}^p{(1-x)}^{l-1}+x^2\sum _{l=2}^p{\displaystyle \frac{1}{l-1}}{(1-x)}^{l-2}.$$

(19)

By inserting $x=1$ into Equation (19), we obtain

$$\sum _{l=0}^p{(l+1)}^2{P}_l=-2.$$

By repeating this procedure, we find the following equations:

$$\begin{array}{cc}& P_0+P_1+P_2+\cdots +P_p=0,\hfill \\ & P_0+2P_1+3P_2+\cdots +(p+1)P_p=-1,\hfill \\ & P_0+2^2{P}_1+3^2{P}_2+\cdots +{(p+1)}^2{P}_p=-2,\hfill \\ & ⋮\hfill \\ & P_0+2^p{P}_1+3^p{P}_2+\cdots +{(p+1)}^p{P}_p=-p.\hfill \end{array}$$

This theorem establishes that Formula (15) is the matrix form of the equation. □

Definition 5.

The generating function $P_p^{\left(\beta \right)}\left(x\right)$ of the Grünwald–Letnikov derivative of order β is defined as follows:

$$P_p^{\left(\beta \right)}\left(x\right)={(g_0+g_{1}x+\cdots +g_p{x}^p)}^{\beta }.$$

(20)

Theorem 3.

If the generating function $P_p^{\left(\beta \right)}\left(x\right)$ can be expressed as

$$P_p^{\left(\beta \right)}\left(x\right)=\sum _{l=0}^{\infty }{g}_l^{(\beta ,p)}{x}^l,$$

(21)

then the coefficients $g_l^{(\beta ,p)}$ can be computed recursively using the following formula [41]:

$$\begin{array}{cc}\hfill g_0^{(\beta ,p)}& =P_0^{\beta },l=0,\hfill \\ \hfill g_l^{(\beta ,p)}& =-{\displaystyle \frac{1}{P_0}}\sum _{j=1}^p{P}_j\left(1-{\displaystyle \frac{1+\beta }{l}}\right)g_{l-j}^{(\beta ,p)},l=1,2,\cdots ,\hfill \\ \hfill g_l^{(\beta ,p)}& =0,l<0.\hfill \end{array}$$

(22)

Proof. 

For the proof, please refer to [41]. □

Corollary 1.

The p-order generating function $P_p^{\left(\beta \right)}\left(x\right)$ of the β-order Grünwald–Letnikov derivative might be represented as follows [39,40]:

$$P_p^{\left(\beta \right)}\left(x\right)=\sum _{j=0}^{\infty }{g}_j^{(\beta ,p)}{x}^j.$$

(23)

where

$$\begin{array}{cc}\hfill g_k^{(\beta ,p)}& =P_0,k=0,\hfill \\ \hfill g_k^{(\beta ,p)}& =-{\displaystyle \frac{1}{P_0}}\sum _{i=0}^{k-1}\left(1-{\displaystyle \frac{1+\beta }{k}}\right)P_i{g}_{k-i}^{(\beta ,p)},k=1,2,\cdots ,p-1,\hfill \\ \hfill g_k^{(\beta ,p)}& =-{\displaystyle \frac{1}{P_0}}\sum _{i=0}^p\left(1-{\displaystyle \frac{1+\beta }{k}}\right)P_i{g}_{k-i}^{(\beta ,p)},k=p,p+1,p+2,\cdots ,\hfill \\ \hfill g_k^{(\beta ,p)}& =0,k<0.\hfill \end{array}$$

(24)

With a p-order generating function $P_p^{\left(\beta \right)}\left(x\right)$, the $\beta$-order Grünwald–Letnikov fractional derivative is provided as follows:

$${}_{t_0}{}^{GL}D_t^{\beta }u\left(t\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\beta }}}\sum _{i=0}^{⌊{\displaystyle \frac{t-t_0}{h}}⌋}{g}_i^{(\beta ,p)}u(t-ih).$$

(25)

The Caputo FD of order $\beta$, associated with the p-order generating function $P_p^{\left(\beta \right)}\left(x\right)$, is approximated as described in (25).

$$D_t^{\beta }u\left(t\right)\simeq {\displaystyle \frac{1}{h^{\beta }}}\sum _{j=0}^m{g}_j^{(\beta ,p)}u(t-jh)-{\displaystyle \frac{1}{h^{\beta }}}\sum _{i=1}^m{g}_i^{(\beta ,p)}u(t_k-jh),$$

(26)

and ${lim}_{m\to \infty }{u}_m=D_t^{\beta }u\left(t\right)$.

As a result, the fractional-order Lorenz chaotic systems (1) are equipped with a highly accurate numerical technique provided by the following:

$$\left\{\begin{array}{cc}\hfill x\left(t_k\right)& =h^{{\beta }_1}\left[ay\right(t_{k-1}-bx\left(t_k\right)(y\left(t_{k-1}\right]+x\left(0\right)-\sum _{j=1}^m{g}_j^{({\beta }_1,P)}\left(x\left(t_{k-j}\right)-x\left(0\right)\right),\hfill \\ \hfill y\left(t_k\right)& =h^{{\beta }_2}[-c(x\left(t_k\right)-x\left(t_k\right)z\left(t_{k-1}\right)]+x\left(0\right)-\sum _{j=1}^m{g}_j^{({\beta }_1,P)}\left(y\left(t_{k-j}\right)-y\left(0\right)\right),\hfill \\ \hfill z\left(t_k\right)& =h^{{\beta }_3}\left(\right(x\left(t_k\right)y\left(t_k\right)z\left(t_{k-1}\right)-dz\left(t_{k-1}\right)+4+z\left(0\right)-\sum _{j=1}^m{g}_j^{({\beta }_3,P)}\left(z\left(t_{k-j}\right)-z\left(0\right)\right).\hfill \end{array}\right.$$

(27)

4. Numerical Simulations

In this section, we will look at several examples of simulations. It is demonstrated that there exist some novel chaotic behaviors. We take into consideration system (1) with: $x\left(0\right)=1$, $y\left(0\right)=1$, and $z\left(0\right)=1$. Additionally, we have the following parameters: $a=1$, $b=1$, $c=2$, and $d=1$.

The complex dynamic behaviors of the systems (1) are illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. In Figure 1, Figure 2 and Figure 3, x is shown in green, y in red, and z in blue. Figure 1 presents time series plots of the systems (1) under the left figure with $\beta =[1.1,1,1]$ and the right figure with $\beta =[1,1,1]$. Figure 2 presents time series plots of the systems (1) under the left figure with $\beta =[2.1,1.18,1.1]$, $a,b,c,d=[2,3,1.19,19]$, and the right figure with $\beta =[1.2,2.02,2.1]$, $a,b,c,d=[2,3,2,18]$. Figure 3 presents time series plots of the systems (1) under the left figure with $\beta =[2.1,1.18,0.95]$, $a,b,c,d=[2,10,2,15]$, and the right figure with $\beta =[2.12,1.18,1.1]$, $a,b,c,d=[2.12,10,1.9,30]$. Figure 4 shows numerical simulations for the systems (1) under varying parameters and different values of $\beta =[0.95,0.93,0.98]$, $a,b,c,d=[2,7,2,10]$. Figure 5 shows numerical simulations for the systems (1) under varying parameters and different values of $\beta =[0.95,1.03,1.08]$, $a,b,c,d=[3,10,2,20]$. Figure 6 presents numerical simulations for the systems (1) under varying parameters and different values of $\beta =[2.95,2.93,2.98]$, $a,b,c,d=[2,9,2,11]$. Figure 7 illustrates numerical simulations for the systems (1) under varying parameters and different values of $\beta =[1.5,1.3,1.8]$, $a,b,c,d=[2,9.3,2,12]$. Figure 8 shows numerical simulations for the systems (1) under varying parameters and different values of $\beta =[0.9,1.23,1.28]$, $a,b,c,d=[2,7,2,7]$. Figure 9 presents numerical simulations for the systems (1) under varying parameters and different values of $\beta =[1.95,1.93,1.98]$, $a,b,c,d=[2,7,2,10]$. Lastly, Figure 10 shows numerical simulations for the systems (1) under varying parameters and different values of $\beta =[1.5,1.9,0.98]$, $a,b,c,d=[2,7,2,9]$. The results obtained through integer order for the simulation are quite close to those from other methods. In this paper, we present new chaotic fractional attractors.

Figure 10, Figure 11 and Figure 12 illustrate the complex new chaotic attractors of the fractional new chaos system (1). The analysis reveals that even small changes in ${\beta }_1$ induce a move from periodic to chaotic oscillations, thus showing its role at the beginning of chaotic behavior. Changes in ${\beta }_2$ affect the attractor’s size and complexity, which in turn controls the amount and intensity of chaos. Changes in ${\beta }_3$, which in turn affect the attractor’s asymmetry and diffusion, determine how sensitive it is to initial conditions.

5. Numerical Solutions

In this section, we will look at numerical solutions with different values of $\beta$.

Table 1. Solutions of system (1) where ${\beta }_1={\beta }_2={\beta }_3=1$, and $t=2$.

h	x	y	z
1/320	$-2.825746803413559$	$0.064360251117179$	$0.901089025404170$
1/640	$-2.786482723259760$	$0.072295429991067$	$0.901502898770758$
1/1280	$-2.770809223476878$	$0.075444632799451$	$0.901681311260687$
1/2560	$-2.766893742675114$	$0.076229722662474$	$0.901727032838009$
1/5120	$-2.765388097718438$	$0.076531445188934$	$0.901744736049166$
1/10240	$-2.762979423485949$	$0.077013929698945$	$0.901773197044337$
RK4	$-2.763785675007464$	$0.074762088285368$	$0.901626569715117$

presents numerical solutions for system (1) for parameters ${\beta }_1={\beta }_2={\beta }_3=1$ and time $t=2$. h gives the step size used in the calculations. The values of variables x, y, z are given for each h. The last line, called RK4, is the solution using the fourth-order Runge–Kutta method for comparison; this helps in finding the stability and the accuracy of the numerical method.

Table 2. Solutions of system (1) where ${\beta }_1=0.98,{\beta }_2=1.1,{\beta }_3=2.18$, and $t=5$.

h	x	y	z
1/320	$2.391178408771527$	$0.260479415912504$	$0.684893827986470$
1/640	$0.885247889145796$	$-0.133988117437640$	$0.701999993868819$
1/1280	$-0.243615245928001$	$-0.337859803099012$	$0.769260636757267$
1/2560	$-0.787279194432375$	$-0.401390176035800$	$0.785989610869821$
1/5120	$-0.954713611001146$	$-0.415369231098121$	$0.787183101994207$
1/10240	$-1.034847310041085$	$-0.420999827668877$	$0.786931304352553$

presents numerical solutions for system (1) for parameters ${\beta }_1=0.98,{\beta }_2=1.1,{\beta }_3=2.18$, and $t=5$. h gives the step size used in the calculations. The values of variables x, y, and z are given for each h value. The last line, called RK4, is the solution using the fourth-order Runge–Kutta method for comparison; this helps in finding the stability and the accuracy of the numerical method.

In Table 1 and Table 2, the results are displayed for different step sizes h and the associated values of x, y, and z with different values of ${\beta }_1,{\beta }_2$, and ${\beta }_3$. The fourth-order Runge–Kutta technique (RK4) is utilized for the purpose of comparing the correctness of the results.

6. Conclusions

This work investigates a fractional chaotic system using a new, accurate numerical approach, revealing a number of new dynamic characteristics. Simulations yield novel forms of intricate and hitherto unexamined dynamic behaviors. These results highlight the distinctive dynamics inherent in fractional systems. We establish that the results for integer-order systems obtained using our method correlate almost perfectly with those attained using conventional methods. By investigating the role of fractional parameters ${\beta }_1$, ${\beta }_2$, ${\beta }_3$ in the new chaotic system, we found that the results for integer-order systems achieved using our method are in excellent agreement with those obtained using traditional approaches. This simulation result shows that the numerical approach is suitable for both the chaotic systems in this study and more complicated ones. In addition to the chaotic systems under investigation, this numerical technique is applicable for more complex chaotic systems. It shows that this method works for other models [42,43,44,45,46,47].