Analysis of Generalized Nonlinear Quadrature for Novel Fractional-Order Chaotic Systems Using Sinc Shape Function

Abstract

This paper introduces the generalized fractional differential quadrature method, which is based on the generalized Caputo type and is used for the first time to solve nonlinear fractional differential equations. One of the effective shape functions of this method is the Cardinal Sine shape function, which is used in combination with the fractional operator of the generalized Caputo kind to convert nonlinear fractional differential equations into a nonlinear algebraic system. The nonlinearity problem is then solved using an iterative approach. Numerical results for a variety of chaotic systems are introduced using the MATLAB program and compared with previous theoretical and numerical results to ensure their reliability, convergence, accuracy, and efficiency. The fractional parameters play an effective role in studying the proposed problems. The achieved solutions prove the viability of the presented method and demonstrate that this method is easy to implement, effective, highly accurate, and appropriate for studying fractional differential equations emerging in fields related to chaotic systems and generalized Caputo-type fractional problems in the future.

1. Introduction

Chaos theory is well-known for its interest in the behavior of complex computational systems. This is known as deterministic chaos, and it has piqued the interest of many researchers in recent years [1,2,3,4,5]. Chaotic behavior occurs in many natural systems and fields of study, including economics, meteorology, sociology, engineering, chemistry, ecology, and medicine, such as pandemic crisis management for COVID-19, and so on. A Lorenz oscillator is a common chaotic system. The behavior of a chaotic system can be examined by investigating its computational formula along with its graphical results, which include strange attractors, Poincare maps, and bifurcations. A system of ordinary differential equations or, in an extra complex form, a system of fractional differential equations with dynamic fraction [6] is typically used as the mathematical model.

The dynamics of a fractional-order Lorenz system with variable derivative order and parameters are studied using Lyapunov exponents (LEs), bifurcation diagrams, chaos diagrams, and phase diagrams. The results reveal that the fractional-order Lorenz system exhibits extensive dynamical behavior and could be used in applications. It is also discovered that the numerical algorithm and time step size have an effect on the minimum order. Finally, the fractional-order system is implemented on a digital signal processor (DSP) circuit. The phase diagrams produced by the DSP are consistent with those produced by simulation.

Several techniques for solving fractional differential equations have recently been introduced. These techniques are classified into two types: numerical and analytical. The techniques of Adomian decomposition [7], Haar wavelet [8], homotopy study [7,9], Taylor matrix [10], and homotopy perturbation [11,12,13] were used to solve the chaotic system differential equation. However, the convergence area of the results obtained seems to be quite small. Actually, the numerical solution of the fractional equation is complicated due to the fractional derivative feature of the differential operator of the spatial fractional derivative, particularly in high dimensional cases. For solving such fractional equations, numerical methodologies such as Finite volume [14], Galerkin [15], Collocation [16], and Finite difference [17] are used.

Over the last two decades, many scientists have focused on fractional-order chaotic dynamical systems. They demonstrated that chaotic attractors do exist in the fractional-order model with an order of less than 3. Sheu and Chen [18] discovered that the fractional-order Newton-Leipnik scheme has the lowest order of 2.82. Li and Peng [19] exposed the rich dynamical performance of the fractional-order Chen scheme in 2004, including limit cycles, fixed points, periodic, and chaotic motions.

Ivo Petras [6] presented a detailed review of Lorenz-type systems and also a novel organization of fractional-order Lorenz models for the first time. Many chaotic systems from the large Lorenz family are presented in 3-dimensional state space as special cases of the new general formula, along with their equilibriums, eigenvalues, and attractor systems [6]. Despite yielding successful results, the task of discovering more general chaotic differential equations remains interesting. Katugampola [20] suggested a generalized fractional derivative previously. There are two fractional parameters. The numerical solutions to the following fractional differential equations [21]:

$${}_{}{}^{c}D{}_{a_{+}}^{\gamma ,\lambda }w=f\left(t,w\right), w\left(a\right)=w_a, 0<\gamma \le 1, \lambda \in \left(0,1\right),$$

(1)

Fractional differential equations of Hadamard and Caputo result from the exceptional cases of $\lambda =0$ and $\lambda =1$, respectively. It outperforms standard derivatives in several ways and may have more interesting applications. For example, visual cryptography recommends chaos in fractional differential equations [22,23]. Chaos is more complex and difficult than traditional cases due to the fractional order. Utilizing two-parameter fractional chaotic equations can enhance the reliability of image encryption results. This derivative was previously proposed in quantum mechanics [24]. Furthermore, two-parameter models in control theory and diffusion problems can have degrees of freedom in control and fitting. This derivative is represented as a new branch of fractional calculus, along with its applications.

This paper’s main goal is to present a novel generalized fractional Differential Quadrature Method (GFDQM) for numerically solving initial value fractional differential equations. Some observations are made about the proposed algorithm’s convergence and stability. Furthermore, test problems are examined to demonstrate the performance, efficiency, and features of the proposed algorithm. To solve the fractional order differential equation, we can use GFDQM with a different shape function focused on the generalized Caputo definition fractional derivative. One of the effective shape functions that have been successfully used in initial value fractional problems is the cardinal sine shape function (SINC) [25,26]. In addition, we develop a MATLAB code for this method in order to obtain a numerical solution for a variety of chaotic systems. The derived numerical results are highly efficient and accurate when compared to previous analytical and numerical [27,28,29,30,31] research methods. Then, we present some parametric research to demonstrate the dependability of our method when fractional order derivatives are present. The numerical method, efficiency, and stability are all discussed in this paper. The primary priority is to provide these outcomes that are applicable to fractional modeling and real-world problem control.

2. Formulation of the Problem

The Lorenz oscillator is a three-dimensional chaotic flow dynamical system. The Lorenz attractor was named after Edward Norton Lorenz, who discovered it in 1963 from simplified equations of convection rolls appearing in atmospheric equations. It refers to the sensitive dependency on initial conditions and parameters in chaos theory. Small changes in the initial conditions or parameters of a dynamic system can cause enormous changes in the system’s long-term behavior. The expression relates to the theory that a butterfly’s wings can cause tiny changes in the atmosphere that can shift the route of a tornado or delay, accelerate, or even prevent the creation of a tornado in a specific region. The fluttering wing depicts a modest change in the system’s initial state, which sets off a sequence of events that leads to large-scale changes in events.

A variety of chaotic systems are introduced to investigate the accuracy and efficiency of GFDQM with the SINC shape function based on the generalized Caputo definition fractional derivative and a study of recent fractional-order Lorenz systems from a broad family that stems from the original Lorenz system is presented [32,33]:

2.1. Fractional-Order Lorenz System [34]

The Lorenz system can be written in fractional-order as follows:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_1\left(y\left(t\right)-x\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=x\left(t\right)\left(a_3-z\left(t\right)\right)-y\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=-a_2 z\left(t\right)+x\left(t\right)y\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(2)

where $D_0^{\gamma ,\lambda }$ denotes to the generalized Caputo-type fractional derivative operator [27,28,29,30,31]. The Prandtl and Rayleigh numbers are denoted by $a_{1 }$and $a_3$, respectively.

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=1\\ y\left(0\right)=1,\\ z\left(0\right)=1\end{array}\right.$$

(3)

2.2. Fractional-Order Chen System [5,35,36]

Chen discovered another simple three-dimensional independent system with a chaotic attractor that is not topologically similar to the Lorenz system.

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_1\left(y\left(t\right)-x\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=\left(a_3-a_1\right)x\left(t\right)-x\left(t\right)z\left(t\right)+a_{3}y\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=-a_2 z\left(t\right)+x\left(t\right)y\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(4)

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=-9\\ \mathrm{y}\left(0\right)=-5,\\ \mathrm{z}\left(0\right)=14\end{array}\right.$$

(5)

2.3. Fractional-Order Lu System [37]

It is referred to as a bridge between the systems of Lorenz and Chen and can be written as:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_1\left(y\left(t\right)-x\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=-x\left(t\right)z\left(t\right)+a_{3}y\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=-a_2 z\left(t\right)+x\left(t\right)y\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(6)

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=0.2\\ \mathrm{y}\left(0\right)=0.5,\\ \mathrm{z}\left(0\right)=0.3\end{array}\right.$$

(7)

2.4. Fractional-Order Yang System [38]

The algebraic form of the chaotic attractor is quite identical to Lorenz-type systems, yet both are not topologically equivalent. It has the following fractional-order derivative:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_1\left(y\left(t\right)-x\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=-x\left(t\right)z\left(t\right)+a_{3}x\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=x\left(t\right)y\left(t\right)-a_2 z\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(8)

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=0.1\\ \mathrm{y}\left(0\right)=0.1,\\ \mathrm{z}\left(0\right)=0.1\end{array}\right.$$

(9)

2.5. Fractional-Order Liu System [39]

Another system that is similar to the Lorenz chaotic system is expressed as follows:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_1\left(y\left(t\right)-x\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=a_{3}x\left(t\right)-k x\left(t\right)z\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=-a_2 z\left(t\right)+h x^2\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(10)

The system displays chaotic behavior for parameter $a_1=10, a_2=2.5, a_3=40,k=1,$ and $h=4$.

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=0.2\\ \mathrm{y}\left(0\right)=0,\\ \mathrm{z}\left(0\right)=0.5\end{array}\right.$$

(11)

2.6. Fractional-Order Shimizu–Morioka System [40]

It is a simple model with the solution exhibiting the Lorenz model’s behavior for high Rayleigh numbers:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=a_{1}y\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=x\left(t\right)\left[1-z\left(t\right)\right]-a_3 \mathrm{y}\left(t\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=-a_2 z\left(t\right)+x^2\left(t\right)\end{array}\right.\hspace{1em}{a}_1,a_2,a_3>0,$$

(12)

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=0.1\\ \mathrm{y}\left(0\right)=0.1,\\ \mathrm{z}\left(0\right)=0.2\end{array}\right.$$

(13)

2.7. Fractional-Order Burke-Shaw System [41]

Burke and Shaw developed the Burke-Shaw system from the Lorenz system [42]. This system has an algebraic structure similar to the Lorenz system but is topologically inequivalent to the generalized Lorenz-type system and can be described as follows:

$$\left\{\begin{array}{l}{}_{}{}^{C}D{}_0^{\gamma , \lambda }x\left(t\right)=-a\left(x\left(t\right)+y\left(t\right)\right)\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }y\left(t\right)=-k x\left(t\right)z\left(t\right)-\mathrm{y}\left(t\right),\\ {}_{}{}^{C}D{}_0^{\gamma , \lambda }z\left(t\right)=g \mathrm{x}\left(t\right)y\left(t\right)+d\end{array}\right.$$

(14)

The system displays chaotic behavior for parameter $a=k=g=10,$ and $d=13 \mathrm{or} 4.272$.

With the following initial condition:

$$\left\{\begin{array}{l}x\left(0\right)=0.1\\ \mathrm{y}\left(0\right)=0.1,\\ \mathrm{z}\left(0\right)=0.1\end{array}\right.$$

(15)

3. Method of Solution

The generalized fractional differential quadrature method (GFDQM) is applied to initial value fractional problems for the first time. This method has many shape functions which are used with the fractional operator of the generalized Caputo kind to convert fractional equations into an algebraic system. To begin, we define a fractional derivative; there are several definitions, the most recent of which is the generalized Caputo’s definition considered in this paper.

3.1. Generalized Caputo-Kind Fractional Derivative [27,28,29,30,31]

Caputo presented Caputo’s Fractional Derivative [43,44,45,46], an innovative definition of a fractional derivative depending based on the Riemann-Liouville Fractional Derivative [47]. According to the definition below, [48]:

The Riemann–Liouville fractional integral operator of order $0<\gamma$ is given by:

$$J^{\gamma }w(t)=\frac{1}{\Gamma (\gamma )}{\displaystyle \underset{a}{\overset{t}{\int }}{(t-x)}^{\gamma -1}. w(x) dx,}$$

(16)

If $\mathsf{\gamma }=0 \to J^{0}w\left(t\right)=w\left(t\right)$.

Assume$\gamma ϵR^{+}$, If $M$is a positive integer number, and $M-1<\gamma \le M$. The fractional derivative of a function$w\left(t\right)$of order $\gamma$ is depicted below in Riemann-Liouville fractional form, which is one of the most studied definitions:

$$D_a^{\gamma }w(t)=\frac{1}{\Gamma (M-\gamma )} \frac{d^M}{d{t}^M} {\displaystyle \underset{a}{\overset{t}{\int }}{(t-x)}^{M-\gamma -1} . w^M(x) dx,}$$

(17)

Generalized Caputo’s Fractional Derivative of order $\gamma$and operator $D_{a_{+}}^{\gamma ,\lambda }$ is written as:

$$D_{a_{+}}^{\gamma ,\lambda }w(t)=\frac{{\lambda }^{\gamma -M+1}}{\Gamma (M-\gamma )}{\displaystyle \underset{c}{\overset{t}{\int }}{x}^{\lambda -1}{(t^{\lambda }-x^{\lambda })}^{M-\gamma -1} {\left(x^{1-\lambda }\frac{d}{dx}\right)}^{M}w(x) dx,}$$

(18)

where $a\ge 0$ denotes to the integration lower limit.

$$D_{a_{+}}^{\gamma ,\lambda }{(t^{\lambda }-x^{\lambda })}^{\beta }={\rho }^{\gamma }\frac{\Gamma \left(\beta +1\right)}{\Gamma \left(\beta -\lambda +1\right)}{\left(t^{\lambda }-c^{\lambda }\right)}^{\beta -\gamma },$$

(19)

After that, we’ll go over the differential quadrature technique with the previously mentioned shape function (SINC):

3.2. Differential Quadrature Method Based on Cardinal Sine Shape Function (SINC-DQM)

The key idea in the differential quadrature application lies in the calculation of the weighting coefficients for the first order derivative based on Cardinal Sine function. This method leads to accurate solutions with fewer grid points comparing with finite difference and finite element methods. Suppose $w$ is unknown, and its $n^{\mathrm{th}}$ derivatives can be presented as a weighted linear sum of nodal values, $w_i$ as shown [25,26]:

$$S_j(x_i,h_x)=\frac{h_x \sin [\pi (x_i-x_j)/h_x]}{\pi (x_i-x_j)},\mathrm{where} h_x>0 \mathrm{is\; the\; mesh\; size}$$

(20)

$$w(x_i)={\displaystyle \sum _{j=-M}^M\frac{h_x \sin [\pi (x_i-x_j)/h_x]}{\pi (x_i-x_j)} w(x_j)}, (i=-M:M),$$

(21)

$$\frac{\partial w}{\partial x}\left|\begin{array}{c}\\ x=x_i\end{array}\right.={\displaystyle \sum _{j=-M}^M{G}_{ij}^{(1)}}w(x_j) , \frac{{\partial }^{2}w}{\partial x^2}\left|\begin{array}{c}\\ x=x_i\end{array}\right.={\displaystyle \sum _{j=-M}^M{G}_{ij}^{(2)}}w(x_j) , \left(i=-M:M\right),$$

(22)

where $w$ denotes to $\left(x, y, and z\right).$ $M$is grid points number. $G_{ij}^{\left(1\right)}\mathrm{and} G_{ij}^{\left(2\right)}$ can be determined by differentiating Equations (20) and (21) as:

$$G_{ij}^{\left(1\right)}=\left\{\left.\begin{array}{cc}\frac{{\left(-1\right)}^{i-j}}{h_x\left[i-j\right]}& at i\ne j\\ 0& at i=j\end{array}\right\}\right.,G_{ij}^{\left(2\right)}=\left\{\left.\begin{array}{cc}\frac{2{\left(-1\right)}^{i-j+1}}{{\left(h_x\left[i-j\right]\right)}^2}& at i\ne j\\ -\frac{{\pi }^2}{3h_x{}^2}& at i=j\end{array}\right\}\right.$$

(23)

The iterative method is used to overcome the nonlinear problem. The governing equation is first solved as a linear system. Then, we solve them as a nonlinear system iteratively until we achieve the required convergence, which is as follows:

$$\left|\frac{w_{m+1}}{w_m}\right|<1, \mathrm{where} \mathrm{m}=0, 1, 2, \dots ,$$

(24)

Finally, we use the SINC-DQM in the generalized Caputo’s fractional derivative to determine the weighting coefficients $G_{ij}^{\gamma ,\lambda }$ for $\gamma ϵ\left]0,1\right]$ and $\lambda >0$, as follows:

Generalized Caputo’s Fractional Derivative of order $\gamma ϵ\left]0,1\right]$ and $\lambda >0$is presented as:

$$\begin{array}{ll}{D}_{a_{+}}^{\gamma ,\lambda }w(t)& =\left\{\left.\begin{array}{l}=\frac{{\lambda }^{\gamma -M+1}}{\Gamma (M-\gamma )}{\displaystyle \underset{c}{\overset{t}{\int }}{x}^{\lambda -1}{(t^{\lambda }-x^{\lambda })}^{M-\gamma -1} {\left(x^{1-\lambda }\frac{d}{dx}\right)}^{M}w(x) dx}\\ ={\displaystyle \sum _{j=1}^N{G}_{ij}^{\gamma ,\lambda } \mathrm{w}(t_j},x), \end{array}\right\} at 0<\gamma \le 1,\right.\\ & ={\displaystyle \sum _{j=1}^N{G}_{ij}^{(1)} \mathrm{w}(t_j},x) at \gamma =\lambda =1\end{array}$$

(25)

So, the weighting coefficient $G_{ij}^{\gamma ,\lambda }$is calculated as:

$$G_{ij}^{\gamma ,\lambda }=A^{1-\gamma }{\lambda }^{\gamma }{ \mathrm{G}}_{ij}^{(1)}+\frac{{\lambda }^{\gamma }{G}_{1,j}^{(1)}}{\Gamma (2-\gamma )} {(t^{\lambda }-a^{\lambda })}^{1-\gamma }, A_{ij}=G_{ij}^{(1)}-G_{1j}^{(1)}$$

(26)

The Equation (26) can be proved as follows for $\lambda =1$:

$$\left\{\begin{array}{l}{w}^{\prime }(a)= dw(a),\hspace{1em}d=G_{1j}^{\left(1\right)}\\ \therefore J^{\gamma }{w}^{\prime }(a)=d J^{\gamma }w(a)=d \frac{w(a)}{\Gamma (\gamma )}{\displaystyle \underset{a}{\overset{x}{\int }}{(t-x)}^{\gamma -1} dx=}\frac{w(a)}{\Gamma (\gamma +1)}d {(t-a)}^{\gamma },\end{array}\right.$$

(27)

Consequently,

$$J_a^{1-\gamma }{w}^{\prime }(a)=\frac{w(a)}{\Gamma (2-\gamma )}d {(t-a)}^{1-\gamma },$$

(28)

Also,

$${\displaystyle \underset{a}{\overset{t}{\int }}w(t) dt=}{\displaystyle \sum _{j=1}^N(G_{ij}^1-G_{1j}^1)w(t_j},x), A_{ij}=G_{ij}^1-G_{1j}^1,$$

(29)

Then,

$$J^{1}w(t)={\displaystyle \underset{a}{\overset{t}{\int }}w(x) dx=A }w(t) \Rightarrow J^{2}w(t)= {\displaystyle \underset{a}{\overset{t}{\int }}{\displaystyle \underset{a}{\overset{t}{\int }}w()} dx={\displaystyle \underset{a}{\overset{t}{\int }}(t-x)w(x) dx=A^{2}w(t),} }$$

(30)

Further,

$$J^{\gamma }w(t)=A^{\gamma }w(t) \Rightarrow J^{1-\gamma }{w}^{\prime }(t)=A^{1-\gamma }{G}_{ij}^{1}w(t),$$

(31)

4. Numerical Results

In this section of the paper, the developed GFDQM is tested on a fractional-order Lorenz system problem, and the accuracy and efficiency of the proposed method are proved. SINC-DQM [25,26] and the concept of generalized Caputo are used to study fractional order differential equations in initial value fractional problems. We successfully completed our calculations by writing MATLAB code for this method. The principal objective of this paper is to discover the efficiency and accuracy of the developed technique by comparing the computed results to preceding numerical and analytical solutions [27,28,29,30,31].

4.1. Problem 1

After substituting the Equations (25) and (26) in the fractional-order Lorenz system (2):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right),$$

(32a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\left(a_3-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}\right),$$

(32b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right),}$$

(32c)

The initial condition (3) is also addressed by substituting in the governing Equation (32). We will now begin to demonstrate the obtained results to explain the accuracy and efficiency of GFDQM using SINC shape function with generalized Caputo sense, as follows:

Table 1. Computation of $x, y, \mathrm{and} z$ via SINC-DQM for fractional Chen system (Lorenz system) with time $\left(T=2\right)$ and fraction powers $\lambda =\gamma =1$ at various grid points$\left(N\right)$.$a_1=35,a_2=3, a_3=28$.

N	SINC-DQM			CPU (s)	N	Earlier Numerical Solutions [27,28,29,30,31]
	$\mathit{x}$	$\mathit{y}$	$\mathit{z}$			$\mathit{x}$	$\mathit{y}$	$\mathit{z}$
4	0.837919933	0.623902547	0.302407988	0.02	5	0.76932157	0.51235478	0.24658714
5	0.771720200	0.527485798	0.248194100	0.021	10	0.76547321	0.50805837	0.24436125
6	0.761378603	0.49832669	0.2415685802	0.022	20	0.76301723	0.50206750	0.24290222
7	0.760223666	0.497601296	0.2409074397	0.023	40	0.76241447	0.50063345	0.24253181
9	0.762203989	0.500122597	0.242400811	0.024	80	0.76226520	0.50028279	0.24243886
10	0.762216222	0.500169299	0.2424081878	0.025	160	0.76222806	0.50019611	0.24241560
11	0.76221575	0.500167396	0.242407838	0.025	320	0.76221880	0.50017456	0.24240979
12	0.76221573	0.500167394	0.2424078434	0.026	640	0.76221649	0.50016919	0.24240833
13	0.76221572	0.500167392	0.2424078432	0.028	1280	0.76221572	0.50016739	0.24240783

and

Table 2. Computation of $x, y, \mathrm{and} z$ via SINC-DQM for fractional Chen system (Lorenz system) with time $\left(T=5\right)$ and fraction powers $\lambda =1.2, \gamma =0.85$ at various grid points $\left(N\right)$.$a_1=35,a_2=3, a_3=28$.

N	SINC-DQM			CPU (s)	N	Earlier Numerical Solutions [27,28,29,30,31]
	x	y	z			x	y	z
4	1.98744231	1.95521788	1.77785472	0.08	5	1.87823591	1.91235487	1.77214577
5	1.87775412	1.94621547	1.76812478	0.082	10	1.87691249	1.90833765	1.76232887
6	1.87702358	1.93987952	1.76214785	0.083	20	1.87411505	1.93108732	1.75629505
7	1.87412004	1.93952410	1.75874532	0.084	40	1.87336221	1.93662856	1.75468966
9	1.87311223	1.9392005	1.75623587	0.086	80	1.87317222	1.93800680	1.75428450
10	1.87310922	1.93894152	1.75469874	0.087	160	1.87312481	1.93835118	1.75418327
11	1.87310905	1.93847456	1.75418742	0.088	320	1.87311299	1.93843729	1.75415800
12	1.87310903	1.93846621	1.75415001	0.089	640	1.87311004	1.93845883	1.75415169
13	1.87310902	1.93846581	1.75414950	0.09	1280	1.87310902	1.93846581	1.75414950

show the effect of using SINC-DQM on the computation of a fractional-order Lorenz system for different fractional power of $\left(\gamma ,\lambda \right)$ at various grid points$\left(N\right)$. In general, as the number of grid points increases, so does the accuracy with error $\le {10}^{-8}$and the performance time of approximately (0.028 s) at T = 2 s. Furthermore, increasing the number of grids with time leads to greater accuracy; for example, at time (t = 2), we use grid points (N = 13). Additionally, we use significantly fewer grids than in previous studies (N = 1280) [27,28,29,30,31].

Figure 1 depicts the dynamic behaviors of a fractional Lorenz system with fraction order of $\mathsf{\gamma }=0.993, \mathsf{\lambda }=1$ at initial conditions $\mathrm{x}\left(0\right)=1, \mathrm{y}\left(0\right)=1, \mathrm{z}\left(0\right)=1$ and ${\mathrm{a}}_1=35, {\mathrm{a}}_2=3,{ \mathrm{a}}_3=28$ to ensure the efficiency and accuracy of SINC-DQM. As a result, the numerical results show that the proposed technique is more accurate, efficient, and produce satisfactory results when compared to previous studies [27,28,29,30,31]. So, numerical simulations for a variety of chaotic systems are presented using the MATLAB program at various fraction orders $\left(\mathsf{\gamma }, \mathsf{\lambda }\right)$.

4.2. Problem 2

After substituting the Equations (25) and (26) in the fractional-order Chen system (4):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right),$$

(33a)

$$\begin{array}{l}{\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}=\\ \left(a_3-a_1\right){\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}+a_3{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}\end{array}$$

(33b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right),}$$

(33c)

The initial condition (5) is also addressed by substituting in the governing Equation (33). Figure 2 shows the dynamic behaviors of a fractional Chen system with fraction order of $\gamma =0.85, \lambda =1.2$ at initial conditions $x\left(0\right)=-9, y\left(0\right)=-5, z\left(0\right)=14$ and $a_1=35, a_2=3, a_3=28$.

4.3. Problem 3

After substituting the Equations (25) and (26) in the fractional-order Lu system (6):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right),$$

(34a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}=-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}+a_3{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}$$

(34b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right),}$$

(34c)

The initial condition (7) is also addressed by substituting in the governing Equation (34). Also, Figure 3 shows the dynamic behaviors of a fractional Lu system with fraction order of $\gamma =0.93, \lambda =1.2$ at initial conditions $x\left(0\right)=0.2, y\left(0\right)=0.5, z\left(0\right)=0.3$ and $a_1=36, a_2=3, a_3=20$.

4.4. Problem 4

After substituting the Equations (25) and (26) in the fractional-order Yang system (8):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right),$$

(35a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}=-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}+a_3{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}$$

(35b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right),}$$

(35c)

The initial condition (9) is also addressed by substituting in the governing Equation (35). Also, Figure 4 shows the dynamic behaviors of a fractional Yang system with fraction order of $\gamma =0.98, \lambda =0.95$ at initial conditions $x\left(0\right)=0.1, y\left(0\right)=0.1, z\left(0\right)=0.1$ and $a_1=10, a_2=8/3, a_3=16$.

4.5. Problem 5

After substituting the Equations (25) and (26) in the fractional-order Liu system (10):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right),$$

(36a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}=a_3{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}-k{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}$$

(36b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}=-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}+h{\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right)}^2,$$

(36c)

The initial condition (11) is also addressed by substituting in the governing Equation (36). In Figure 5 the dynamic behaviors of a fractional Liu system with fraction order of $\gamma =0.99, \lambda =0.8$at initial conditions $x\left(0\right)=0.2, y\left(0\right)=0.0, z\left(0\right)=0.5$ and $a_1=10, a_2=2.5, a_3=40,\mathrm{k}=1 \mathrm{and} \mathrm{h}=4$ are investigated.

4.6. Problem 6

After substituting the Equations (25) and (26) in the fractional-order Shimizu–Morioka system (12):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=a_1{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right),}$$

(37a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}={\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\left[1-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}\right]-a_3{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}$$

(37b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}=-a_2{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}+{\left({\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}\right)}^2,$$

(37c)

The initial condition (13) is also addressed by substituting in the governing Equation (37). In Figure 6 the dynamic behaviors of a fractional Shimizu–Morioka system with fraction order of $\gamma =0.98, \lambda =0.9$ at initial conditions $x\left(0\right)=0.1, y\left(0\right)=0.1, z\left(0\right)=0.2$ and $a_1=1, a_2=0.375, a_3=0.81$ are investigated.

4.7. Problem 7

After substituting the Equations (25) and (26) in the fractional-order Burke–Shaw system (14):

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{x}\left(t_j\right)}=-a\left[{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}\right],$$

(38a)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{y}\left(t_j\right)}=-k{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{z}\left(t_j\right)}-{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}$$

(38b)

$${\displaystyle \sum _{j=1}^L{G}_{ij}^{\gamma ,\lambda } \mathrm{z}\left(t_j\right)}=g{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{x}\left(t_j\right)}{\displaystyle \sum _{j=1}^N{\delta }_{ij}^{} \mathrm{y}\left(t_j\right)}+d$$

(38c)

The initial condition (15) is also addressed by substituting in the governing Equation (38). Finally, Figure 7 demonstrates the dynamic behaviors of a fractional Burke-Shaw system with fraction order of $\gamma =0.97, \lambda =1.1$ at initial conditions $x\left(0\right)=0.1, y\left(0\right)=0.1,z\left(0\right)=0.1$ and $\mathrm{a}=\mathrm{k}=\mathrm{g}=10,\mathrm{d}=13$.

5. Conclusions

We have successfully investigated a new numerical method for solving a variety of chaotic systems using nonlinear fractional differential equations in this work. The novel numerical method is called generalized fractional differential quadrature, and it is based on the Cardinal Sine function with a new generalized Caputo kind. The proposed problems are transformed into a nonlinear algebraic system using this method. The iterative method is then used to solve nonlinear problems. Furthermore, all computational results are generated using the MATLAB program. The numerical solutions compared with previous theoretical and numerical results to ensure their reliability, convergence, accuracy, and efficiency. The achieved solutions prove the viability of the presented method and demonstrate that this method is easy to implement, effective, highly accurate, and appropriate for studying fractional differential equations. Finally, we can conclude from our numerical solutions using the proposed method that the results achieved higher speed convergence via SINC-DQM than other techniques. SINC-DQM produces the best results when the grid points are N = 13 and the CPU time is 0.028 s and 0.09 s for T = 2 and T = 5 with error $\le {10}^{-8}$, respectively. In addition, the proposed technique has been used successfully to explain the dynamic behaviors of the fractional systems under consideration. Furthermore, the numerical results show that the solution is constantly dependent on the fractional derivative. As a result, the fractional parameters λ and γ play an effective role in studying the proposed problems. So, this method will be applied to more complex nonlinear equations. It can also be used to solve other field’s related to chaotic systems and generalized Caputo-type fractional problems in the future.