The dynamics of a new chaotic system through the Caputo–Fabrizio and Atanagan–Baleanu fractional operators

Abstract

The aim of this article is to analyze the dynamics of the new chaotic system in the sense of two fractional operators, that is, the Caputo–Fabrizio and the Atangana–Baleanu derivatives. Initially, we consider a new chaotic model and present some of the fundamental properties of the model. Then, we apply the Caputo–Fabrizio derivative and implement a numerical procedure to obtain their graphical results. Further, we consider the same model, apply the Atangana–Baleanu operator, and present their analysis. The Atangana–Baleanu model is used further to present a numerical approach for their solutions. We obtain and discuss the graphical results to each operator in details. Furthermore, we give a comparison of both the operators applied on the new chaotic model in the form of various graphical results by considering many values of the fractional-order parameter $\alpha$ . We show that at the integer case, both the models (in Caputo–Fabrizio sense and the Atangana–Baleanu sense) give the same results.

Introduction

Chaotic models are widely considered due to their applications in many branches of engineering and science and, especially, a rapid increase in the form of publications on chaotic models has been observed.^1–7 Particularly, chaotic models are considered for practical purposes in many areas, such as chaotic communication, ² image watermarking, ¹ and autonomous mobile robots. ³ Due to the applications of chaotic models, the researchers developed some new chaotic models and presented their analysis.^4,5 The literature shows clearly that most of the chaotic models have unstable equilibrium points,^6,7 and such system exhibits self-excited attractors. ⁷ Some other chaotic models that are studied by the researchers in order to identify their chaotic dynamics can be seen in Wei et al.^8–10 and the references therein.

The fractional-order modeling has gained a lot of attentions from the researchers once the new operators are defined. In these newly defined operators, the operator Caputo–Fabrizio and the Atangana–Baleanu derivative received more attention from the researchers, and various articles have been proposed on these, including the Caputo derivative.^11–18 There are many applications of the fractional models, such as the generalization of the model and the memory effects, which are usually attached to the real-life problems. The fractional-order modeling has the advantage of observing the dynamics of the real-life problems at any point of interests where such analysis for the integer-order models does not exist. The crossover behavior of the non-linear problems can only be handled by the newly introduced derivatives.

The chaotic systems studied in fractional calculus are numerous in literature.^19–24 A fractional chaotic model and its application to chaotic system have been discussed in the work of Atangana. ²⁴ A fractional model with no-indexed law and its application to chaos and statistics have been considered in the work of Atangana and Gómez-Aguilar. ²³ Owolabi and Atangana ²² considered a chaotic model in integer case and presented its dynamical analysis. Zhang et al. ²¹ considered a delayed chaotic model and analyzed its dynamics. A fractional chaotic model with no equilibrium has been discussed in the work of Mishra. ²⁰ Su et al. ¹⁹ presented a chaotic model and obtained a numerical scheme. Recently, a fractional logistic map and its application to dynamical analysis have been studied by Yuan et al. ²⁵ The real data application of fractional calculus to blood ethanol concentration calculation has been considered in Qureshi et al. ²⁶ The dynamics of chickenpox disease with field data is studied through fractional derivative in Qureshi and Yusuf. ²⁷ Application of the new derivatives to the partial differential equations is considered in Yusuf et al. ²⁸ A two-strain epidemic model with fractional derivatives has been proposed in Yusuf et al. ²⁹ The application of fractional operator known as Atangana–Baleanu to Korteweg–de Vries (KDV) equation is analyzed in Inc et al. ³⁰ The dynamics of chaotic attractors with fractional conformable derivative is proposed in Pérez et al. ³¹ A fractional-time wave equation with regular kernel is studied by Cuahutenango-Barro et al. ³² The authors studied a fractional Hunter–Saxton equation using Riemann–Liouville and Liouville–Caputo derivatives. ³³ Numerical solution of Fisher’s type equations of fractional nature with Atangana–Baleanu derivative is considered in Saad et al. ³⁴ The application of Feng’s first integral technique to the fractional modified Korteweg–de Vries (MKDV) equation and their analysis and solutions is presented in Yépez-Martnez et al. ³⁵

Motivated from the recent literature on the chaotic models, we consider a new chaotic model in two fractional operators, that is, the Caputo–Fabrizio derivative and the Atangana–Baleanu derivative, and present comparison results. Then, we present numerical approaches for both the derivatives and give various graphical results for the fractional-order parameter $\alpha$ . We discussed a brief overview of the recent literature on fractional models and on chaotic systems. Next, we present the basic mathematical background of fractional calculus that will be used later in the proposed model and its analysis. The basic model considered in “Model framework” section is used to formulate it in Caputo–Fabrizio derivative. A fractional model with Caputo–Fabrizio derivative, which is used further to present its numerical results by considering a numerical approach, is given in “Numerical approach for Caputo–Fabrizio derivative” section. A mathematical model with Atangana–Baleanu derivative is considered in “Model in Atangana–Baleanu derivative form” section, where the numerical scheme and its graphical results are presented. A comparison of Atangana–Baleanu and Caputo–Fabrizio model is considered in “Comparison of Atangana–Baleanu and Caputo–Fabrizio model” section, where we show the comparison by providing graphical illustrations. The main finding of the article is summarized in “Conclusion” section.

Basic concepts of Caputo–Fabrizio and Atangana–Baleanu derivative

This section presents the basic concepts regarding the two fractional operators, namely, Caputo–Fabrizio and Atangana–Baleanu derivatives as follows:

Definition 1

Let $z\in H^1(a,b)$ , where b is greater than a, and $\alpha \in [0,1]$ , then we can define the Caputo–Fabrizio derivative ³⁶ as follows

$$D_t^{\alpha }(z(t))=\frac{K(\alpha )}{1-\alpha }\underset{a}{\overset{t}{\int }}{z}^{\prime }(x)\exp [-\alpha \frac{t-x}{1-p}]\mathrm{dx}.$$

(1)

where the normal function is given by $K(\alpha )$ that holds $K(0)=K(1)=1$ . Further, if $z\notin H^1(a,b)$ , then we have

$$D_t^{\alpha }(z(t))=\frac{\alpha K(\alpha )}{1-\alpha }\underset{a}{\overset{t}{\int }}(z(t)-z(x))\exp [-\alpha \frac{t-x}{1-\alpha }]\mathrm{dx}$$

(2)

Remark 1

If $\varpi =((1-\alpha )/\alpha )\in [0,\infty ]$ , $\alpha =(1/(1+\varpi ))\in [0,1]$ , equation (2) is expressed as follows

$$D_t^{\alpha }(z(t))=\frac{K(\varpi )}{\varpi }\underset{a}{\overset{t}{\int }}{z}^{\prime }(x)\exp [-\frac{t-x}{\varpi }]\mathrm{dx},K(0)=K(\infty )=1$$

(3)

Moreover

$$\underset{\varpi \to 0}{lim}\frac{1}{\varpi }\exp [-\frac{t-x}{\varpi }]=\varphi (x-t)$$

(4)

The following integral associated with the derivative above is defined as follows ³⁷

Definition 2

Let $0<\alpha <1$ , then the integral of fractional order $\alpha$ is defined as

$$I_t^{\alpha }(z(t))=\frac{2(1-\alpha )}{(2-\alpha )K(\alpha )}g(t)+\frac{2\alpha }{(2-\alpha )K(\alpha )}\underset{0}{\overset{t}{\int }}z(s)\mathrm{ds},t\ge 0$$

(5)

Remark 2

In definition 2, the remainder of the integral of the function with $0<\alpha <1$ is a mean into z and its integral is of order 1. Therefore

$$\frac{2}{2K(\alpha )-\alpha K(\alpha )}=1$$

(6)

gives

$K(\alpha )=\frac{2}{2-\alpha }$ , $0<\alpha <1$

Losada and Nieto ³⁷ used the result given by equation (6) and presented the following definition

$$D_t^{\alpha }(z(t))=\frac{1}{1-\alpha }\underset{0}{\overset{t}{\int }}{z}^{\prime }(x)\exp [-\alpha \frac{t-x}{1-\alpha }]\mathrm{dx}$$

(7)

Next, we recall the fundamental concept of Atangana–Baleanu derivative ³⁸ and the related results.

Definition 3

Let $h\in F^1(a,b)$ , $b>a$ , $\alpha \in [0,1]$ , then the new derivative in the Caputo sense is presented as

$${{}_a{}^{\mathrm{AB}}D}_t^{\alpha }(h(t))=\frac{K(\alpha )}{1-\alpha }\underset{a}{\overset{t}{\int }}{h}^{\prime }(\chi )E_{\alpha }[-\alpha \frac{{(t-\chi )}^{\alpha }}{1-\alpha }]d\chi$$

Definition 4

Let $h\in F^1(a,b)$ , $b>a$ , $\alpha \in [0,1]$ , then, in the sense of Riemann–Liouville, the new derivative called the Atangana–Baleanu is shown by

$${{}_a{}^{\mathrm{ABR}}D}_t^{\alpha }(h(t))=\frac{K(\alpha )}{1-\alpha }\frac{d}{\mathrm{dt}}\underset{a}{\overset{t}{\int }}h(\chi )E_{\alpha }[-\alpha \frac{{(t-\chi )}^{\alpha }}{1-\alpha }]d\chi$$

Definition 5

The fractional integral with non-local kernel of Atangana–Baleanu fractional derivative is given by

$${{}_a{}^{\mathrm{AB}}I}_t^{\alpha }(h(t))=\frac{1-\alpha }{K(\alpha )}h(t)+\frac{\alpha }{K(\alpha )\mathrm{\Gamma }(\alpha )}\underset{a}{\overset{t}{\int }}h(y)(t-y{)}^{\alpha -1}\mathrm{dy}$$

Further, we present the following results.

Theorem 1

On $[a,b]$ , the following inequality holds for f when the function h is continuous on $[a,b]$ ³⁸

$$\begin{array}{c}\multicolumn{1}{c}{{\parallel }_a^{\mathrm{ABR}}{D}_t^{\alpha }(h(t))\parallel <\frac{B(\alpha )}{1-\alpha }\parallel h(x)\parallel ,\mathrm{where}\parallel h(x)\parallel }\\ \multicolumn{1}{c}{=\mathrm{ma}{x}_{a\le x\le b}|h(x)|}\end{array}$$

(8)

Theorem 2

Atangana–Baleanu and Atangana–Baleanu-R derivatives fulfill the Lipschitz conditions and are shown by ³⁸

$${\parallel }_a^{\mathrm{AB}}{D}_t^{\alpha }{h}_1(t){-}_a^{\mathrm{ABC}}{D}_t^{\alpha }{h}_2(t)\parallel <K\parallel h_1(t)-h_2(t)\parallel$$

(9)

and

$${\parallel }_a^{\mathrm{ABR}}{D}_t^{\alpha }{h}_1(t){-}_a^{\mathrm{ABR}}{D}_t^{\alpha }{h}_2(t)\parallel <K\parallel h_1(t)-h_2(t)\parallel$$

(10)

Theorem 3

For the given fractional differential equation ³⁸

$${{}_a{}^{\mathrm{AB}}D}_t^{\alpha }h(t)=s(t)$$

(11)

by applying the inverse Laplace transform and the convolution, we have a unique solution ³⁸

$$h(t)=\frac{1-\alpha }{\mathrm{AB}(\alpha )}s(t)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}\underset{a}{\overset{t}{\int }}s(\xi )(t-\xi {)}^{\alpha -1}d\xi$$

(12)

Model framework

Here, we consider a new chaotic model considered in Wang et al. ³⁹ and is given by

$$\begin{array}{c}\frac{d{x}_1}{\mathrm{dt}}=x_2(t)-{\alpha }_1\hfill \\ \frac{d{x}_2}{\mathrm{dt}}=-\beta x_1(t)x_3(t)+\varphi \hfill \\ \frac{d{x}_3}{\mathrm{dt}}=x_2(t)-x_3(t)+x_2(t)(x_2(t)-x_3(t))\hfill \end{array}$$

(13)

where the state variables are shown by $x_1(t)$ , $x_2(t)$ , and $x_3(t)$ and ${\alpha }_1$ , $\beta$ , and $\varphi$ are the positive parameters.

The fractional Caputo–Fabrizio derivative form of the system equation (13) will be

$$\begin{array}{l}{{}_0{}^{CF}D}_t^{\alpha }{x}_1=x_2(t)-{\alpha }_1\\ {{}_0{}^{CF}D}_t^{\alpha }{x}_2=-\beta x_1(t)x_3(t)+\varphi \\ {{}_0{}^{CF}D}_t^{\alpha }{x}_3=x_2(t)-x_3(t)+x_2(t)(x_2(t)-x_3(t))\end{array}$$

(14)

To find the equilibrium point of the model equation (14), we put the right-hand side of the model equation (14) equal to zero and obtain

$$P=(\frac{\varphi }{{\alpha }_1\beta },{\alpha }_1,{\alpha }_1)$$

The Jacobian matrix at the equilibrium point P of the model (14) is

$$J_P=\left(\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{-{\alpha }_1\beta }& 0\hfill & -\frac{\varphi }{{\alpha }_1}\hfill \\ \multicolumn{1}{c}{0}& {\alpha }_1+1\hfill & -{\alpha }_1-1\hfill \end{array}\right)$$

(15)

The characteristic equation for the Jacobian matrix $J_P$ is

$${\lambda }^3+({\alpha }_1+1){\lambda }^2+({\alpha }_1\beta +\frac{\varphi }{{\alpha }_1}+\varphi )\lambda +{\alpha }_1^2\beta +{\alpha }_1\beta =0$$

and the Routh–Hurwitz criteria can be easily satisfied

$$\begin{array}{c}\multicolumn{1}{c}{({\alpha }_1+1)({\alpha }_1\beta +\frac{\varphi }{{\alpha }_1}+\varphi )-({\alpha }_1^2\beta +{\alpha }_1\beta )}\\ \multicolumn{1}{c}{=\frac{{({\alpha }_1+1)}^2\varphi }{{\alpha }_1}>0}\end{array}$$

This shows that the fractional system (14) has one stable equilibrium point. Further, the system (14) is dissipative and can be easily obtained by taking the partial derivative of each variable of the model (14), and on adding, we get $\nabla V=-y(t)-1$ . Now considering the parameters values ${\alpha }_1=0.95$ , $\beta =1.45$ , and $\varphi =0.01$ , we obtain $P=(0.00725953,0.95,0.95)$ and the eigenvalues $-1.94226,-0.00387072\pm 1.176i$ . Next, we present the numerical scheme for the solution of the model (14) numerically in the following section.

Numerical approach for Caputo–Fabrizio derivative

The model given in Caputo–Fabrizio form (14) is used to obtain a numerical scheme by considering the scheme presented in Atangana and Owolabi. ⁴⁰ The obtained scheme is used further to obtain the graphical results for the model in Caputo–Fabrizio–order derivative (14) by taking different values of $\alpha$ . To do this, we first express the model given by (14) in the form of fractional Volterra type and then apply the fundamental theorem of integration. We start the numerical scheme by considering the first equation of (14) and using the fundamental theorem of integration on the first equation of the Caputo–Fabrizio model (14)

$$x_1(t)-x_1(0)=\frac{1-\alpha }{K(\alpha )}{\mathcal{H}}_1(t,x_1)+\frac{\alpha }{K(\alpha )}\underset{0}{\overset{t}{\int }}{\mathcal{H}}_1(\zeta ,x_1)d\zeta$$

(16)

For $t=t_{n+1}$ , where $n=0,1,2,\dots$ , we obtain

$$x_1(t_{n+1})-{x_1}_0=\frac{1-\alpha }{K(\alpha )}{\mathcal{H}}_1(t_n,{x_1}_n)+\frac{\alpha }{K(\alpha )}\underset{0}{\overset{t_{n+1}}{\int }}{\mathcal{H}}_1(t,x_1)\mathrm{dt}$$

(17)

Further, we obtain the following equation

$$\begin{array}{c}\multicolumn{1}{c}{{x_1}_{n+1}-{x_1}_n=\frac{1-\alpha }{K(\alpha )}\{{\mathcal{H}}_1(t_n,{x_1}_n)-{\mathcal{H}}_1(t_{n-1},{x_1}_{n-1})\}}\\ \multicolumn{1}{c}{+\frac{\alpha }{K(\alpha )}\underset{t_n}{\overset{t_{n+1}}{\int }}{\mathcal{H}}_1(t,x_1)\mathrm{dt}}\end{array}$$

(18)

We approximate the function ${\mathcal{H}}_1(t,x_1)$ by the interpolation polynomial in $[t_k,t_{(k+1)}]$ and have

$$P_k(t)\cong \frac{H(t_k,y_k)}{h}(t-t_{k-1})-\frac{H(t_{k-1},y_{k-1})}{h}(t-t_k)$$

(19)

where $h=t_n-t_{n-1}$ .

The approximations described above is used and we obtain the integral in equation (2)

$$\begin{array}{c}\multicolumn{1}{c}{\underset{t_n}{\overset{t_{n+1}}{\int }}{\mathcal{H}}_1(t,x_1)\mathrm{dt}=\underset{t_n}{\overset{t_{n+1}}{\int }}\frac{{\mathcal{H}}_1(t_n,{x_1}_n)}{h}(t-t_{n-1})}\\ \multicolumn{1}{c}{-\frac{\mathcal{H}(t_{n-1},{x_1}_{n-1})}{h}(t-t_n)\mathrm{dt}}\\ \multicolumn{1}{c}{=\frac{3h}{2}{\mathcal{H}}_1(t_n,{x_1}_n)-\frac{h}{2}{\mathcal{H}}_1(t_{n-1},{x_1}_{n-1})}\end{array}$$

(20)

Using equation (20) in equation (2), after some calculations, we have

$$\begin{array}{c}\multicolumn{1}{c}{{x_1}_{n+1}={x_1}_n+(\frac{1-\alpha }{K(\alpha )}+\frac{3h}{2K(\alpha )}){\mathcal{H}}_1(t_n,{x_1}_n)}\\ \multicolumn{1}{c}{-(\frac{1-\alpha }{K(\alpha )}+\frac{\alpha h}{2K(\alpha )}){\mathcal{H}}_1(t_{n-1},{x_1}_{n-1})}\end{array}$$

(21)

Using the procedure presented above in the same manner for the rest of the equations of the Caputo–Fabrizio model (14), we have

$$\begin{array}{c}\multicolumn{1}{c}{{x_2}_{n+1}={x_2}_0+(\frac{1-\alpha }{K(\alpha )}+\frac{3h}{2K(\alpha )}){\mathcal{H}}_2(t_n,{x_2}_n)}\\ \multicolumn{1}{c}{-(\frac{1-\alpha }{K(\alpha )}+\frac{\alpha h}{2K(\alpha )}){\mathcal{H}}_2(t_{n-1},{x_2}_{n-1})}\\ \multicolumn{1}{c}{{x_3}_{n+1}={x_3}_0+(\frac{1-\alpha }{K(\alpha )}+\frac{3h}{2K(\alpha )}){\mathcal{H}}_3(t_n,{x_3}_n)}\\ \multicolumn{1}{c}{-(\frac{1-\alpha }{K(\alpha )}+\frac{\alpha h}{2K(\alpha )}){\mathcal{H}}_3(t_{n-1},{x_3}_{n-1})}\end{array}$$

(22)

Using the procedure presented above for the solution of the Caputo–Fabrizio model (14), we obtain the graphical results in Figures 1–8 by taking $\alpha =1,0.95,0.9,0.85$ . Figures 1–4 represent the graphical results for the dynamics of the variables presented in Caputo–Fabrizio model (14), whereas Figures 5–8 show the chaotic attractor.

Figure 1.

Considering the model (14) when $\alpha =1$ : (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 2.

Considering the model (14) when $\alpha =0.95$ : (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 3.

Considering the model (14) when $\alpha =0.9$ : (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 4.

Considering the model (14) when $\alpha =0.85$ : (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 5.

Considering the model (14) when α = 1, shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 6.

Considering the model (14) when α = 0.95, shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 7.

Considering the model (14) when α = 0.9, shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 8.

Considering the model (14) when $\alpha =0.85$ , shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Model in Atangana–Baleanu derivative form

The model given by equation (13) can be written in fractional Atangana–Baleanu derivative form as

$$\begin{array}{l}{{}_0{}^{AB}D}_t^{\alpha }{x}_1=x_2(t)-{\alpha }_1,e^{i\theta }\\ {{}_0{}^{AB}D}_t^{\alpha }{x}_2=-\beta x_1(t)x_3(t)+\varphi \\ {{}_0{}^{AB}D}_t^{\alpha }{x}_3=x_2(t)-x_3(t)+x_2(t)(x_2(t)-x_3(t))\end{array}$$

(23)

Numerical procedure for Atangana–Baleanu derivative

Here, we give a procedure, similar to Khan et al. ¹⁴ , for the numerical solution of the model in the form of Atangana–Baleanu derivative equation (23) by considering the method explained in Toufik and Atangana, ⁴¹ which is effectively used for many problems that belong to the real-world life. We consider the same approach presented in Toufik and Atangana ⁴¹ and apply it on our model (23) as per the following: before we proceed to start the approach, first, we write the model in Atangana–Baleanu shape (equation (23)) in the form given below

$$\begin{array}{c}{{}_0{}^{\mathrm{AB}}D}_t^{\alpha }{x}_1={\mathcal{U}}_1(t,x_1,x_2,x_3)\hfill \\ \multicolumn{1}{c}{{{}_0{}^{\mathrm{AB}}D}_t^{\alpha }{x}_2={\mathcal{U}}_2(t,x_1,x_2,x_3)}\\ \multicolumn{1}{c}{{{}_0{}^{\mathrm{AB}}D}_t^{\alpha }{x}_3={\mathcal{U}}_3(t,x_1,x_2,x_3)}\end{array}$$

(24)

The model given by equation (23) can be expressed in the form after using the fundamental theorem of integration

$$\begin{array}{c}\multicolumn{1}{c}{x_1(t)-x_1(0)=\frac{(1-\alpha )}{\mathrm{AB}(\alpha )}{\mathcal{U}}_1(t,x_1)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}}\\ \multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}{\mathcal{U}}_1(\xi ,x_1)(t-\xi {)}^{\alpha -1}d\xi }\\ \multicolumn{1}{c}{x_2(t)-x_2(0)=\frac{(1-\alpha )}{\mathrm{AB}(\alpha )}{\mathcal{U}}_2(t,x_2)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}}\\ \multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}{\mathcal{U}}_2(\xi ,x_2)(t-\xi {)}^{\alpha -1}d\xi }\\ \multicolumn{1}{c}{x_3(t)-x_3(0)=\frac{(1-\alpha )}{\mathrm{AB}(\alpha )}{\mathcal{U}}_3(t,x_3)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}}\\ \multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}{\mathcal{U}}_3(\xi ,x_3)\times (t-\xi {)}^{\alpha -1}d\xi }\end{array}$$

(25)

We get from equation (25) for $t=t_{n+1}$ , $n=0,1,2,\dots$

$$\begin{array}{c}\multicolumn{1}{c}{x_1(t_{n+1})-x_1(t_0)=\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_1(t_n,x_1)}\\ \multicolumn{1}{c}{+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}\sum _{k=0}^n\underset{t_k}{\overset{t_{k+1}}{\int }}{\mathcal{U}}_1(\tau ,x_1)\times (t_{n+1}-\tau {)}^{\alpha -1}d\tau }\\ \multicolumn{1}{c}{x_2(t_{n+1})-x_2(t_0)=\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_2(t_n,x_2)}\\ \multicolumn{1}{c}{+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}\sum _{k=0}^n\underset{t_k}{\overset{t_{k+1}}{\int }}{\mathcal{U}}_2(\tau ,x_2)\times (t_{n+1}-\tau {)}^{\alpha -1}d\tau }\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{x_3(t_{n+1})-x_3(t_0)=\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_3(t_n,x_3)+\frac{\alpha }{\mathrm{AB}(\alpha )\mathrm{\Gamma }(\alpha )}\sum _{k=0}^n}\\ \multicolumn{1}{c}{\underset{t_k}{\overset{t_{k+1}}{\int }}{\mathcal{U}}_3(\tau ,x_3)\times (t_{n+1}-\tau {)}^{\alpha -1}d\tau }\end{array}$$

(26)

Using the integral given in equation (26), approximated using the two-point interpolation polynomial, we have the following iterative scheme for the model (23) after some simplifications. Using the same procedure for the remaining equations of Atangana–Baleanu model (23), we have the recursive formula as given by

$$\begin{array}{c}\multicolumn{1}{c}{x_1(t_{n+1})=x_1(t_0)+\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_1(t_n,x_1)+\frac{\alpha }{\mathrm{AB}(\alpha )}}\\ \multicolumn{1}{c}{\times \sum _{k=0}^n[\frac{h^{\alpha }{\mathcal{U}}_1(t_k,x_1)}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha }(n-k+2+\alpha )-{(n-k)}^{\alpha }(n-k+2+2\alpha ))-\frac{h^{\alpha }{\mathcal{U}}_1(t_{k-1},x_1)}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha +1}-{(n-k)}^{\alpha }(n-k+1+\alpha ))]}\end{array}$$

(27)

$$\begin{array}{c}\multicolumn{1}{c}{x_2(t_{n+1})=x_2(t_0)+\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_2(t_n,I)+\frac{\alpha }{\mathrm{AB}(\alpha )}}\\ \multicolumn{1}{c}{\times \sum _{k=0}^n[\frac{h^{\alpha }{\mathcal{U}}_2(t_k,x_{)}}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha }(n-k+2+\alpha )-{(n-k)}^{\alpha }(n-k+2+2\alpha ))-\frac{h^{\alpha }{\mathcal{U}}_2(t_{k-1},x_2)}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha +1}-{(n-k)}^{\alpha }(n-k+1+\alpha ))]}\end{array}$$

(28)

and

$$\begin{array}{c}\multicolumn{1}{c}{x_3(t_{n+1})=x_3(t_0)+\frac{1-\alpha }{\mathrm{AB}(\alpha )}{\mathcal{U}}_3(t_n,R)+\frac{\alpha }{\mathrm{AB}(\alpha )}}\\ \multicolumn{1}{c}{\times \sum _{k=0}^n\frac{h^{\alpha }{\mathcal{U}}_3(t_k,R)}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha }(n-k+2+\alpha )-{(n-k)}^{\alpha }(n-k+2+2\alpha ))-\frac{h^{\alpha }{\mathcal{U}}_3(t_{k-1},x_3)}{\mathrm{\Gamma }(\alpha +2)}((n+1-k{)}^{\alpha +1}-{(n-k)}^{\alpha }(n-k+1+\alpha ))}\end{array}$$

(29)

The numerical scheme presented for the model (23) is used further to obtain the numerical results by considering the fractional parameter $\alpha \in [0,1]$ . Figures 9–16 show the graphical results for the Atangana–Baleanu derivative model (23). Figures 9–12 show the dynamics of the model variables by using the fractional-order parameters values $\alpha =1,0.95,0.9,0.85$ and the graphical results given by Figures 13–16 with fractional-order parameter $\alpha =1,0.95,0.9,0.85$ show the chaotic attractor. The dashed line in Figures 9–16 shows the fractional-order parameter at the fractional order, whereas the bold line represents the integer case.

Figure 9.

Considering the Atangana–Baleanu model (23), with α = 1: (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 10.

Considering the Atangana–Baleanu model (23), with α = 0.95: (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 11.

Considering the Atangana–Baleanu model (23), with α = 0.90: (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 12.

Considering the Atangana–Baleanu model (23), with α = 0.85: (a) state variable x₁, (b) state variable x₂ and (c) state variable x₃.

Figure 13.

Considering the Atangana–Baleanu model (23), with $\alpha =1$ , where the model shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 14.

Considering the Atangana–Baleanu model (23), with α = 0.95, where the model shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 15.

Considering the Atangana–Baleanu model (23), with α = 0.9, where the model shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Figure 16.

Considering the Atangana–Baleanu model (23), with α = 0.85, where the model shows chaotic attractor: (a) Phase portraits (x₁-x₂ plane), (b) Phase portraits (x₁-x₃ plane), (c) Phase portraits (x₂-x₃ plane) and (d) 3D plot (x₁-x₂-x₃ plane).

Comparison of Atangana–Baleanu and Caputo–Fabrizio model

Here, we give the comparison of the fractional operators applied on the model (13) (see the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23)). We considered the numerical scheme presented for the Caputo–Fabrizio derivative model (14) and the Atangana–Baleanu model (23) and obtained the graphical results by taking the fractional-order parameter $\alpha =1,0.99,0.97$ , see Figures 17–25. In Figures 17–19, we took the fractional-order parameter $\alpha =1$ and presented the dynamics of the model (14) and (23). One can see that at integer case, both the models have the same results. Further, we took the fractional-order parameter $\alpha =0.99$ and presented the graphical results in Figures 20–22, where the bold lines show the integer cases for both model (14) (black line) and model (23) (blue line), and the dashed line (black) shows the Caputo–Fabrizio model in fractional values (14) and the dashed line (blue) shows the Atangana–Baleanu model (23) for the fractional values of the parameter $\alpha$ . One can see from Figures 20–22 the behavior of both the operators on the chaotic model (13). Further, Figures 23–25 show the dynamics of the model parameters and their comparison by using $\alpha =0.97$ .

Figure 17.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =1$ for $x_1$ .

Figure 18.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =1$ for $x_2$ .

Figure 19.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =1$ for $x_3$ .

Figure 20.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.99$ for $x_1$ .

Figure 21.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.99$ for $x_2$ .

Figure 22.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.99$ for $x_3$ .

Figure 23.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.97$ for $x_1$ .

Figure 24.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.97$ for $x_2$ .

Figure 25.

Considering the comparison graphs for the Caputo–Fabrizio model (14) and the Atangana–Baleanu model (23) when $\alpha =0.97$ for $x_3$ .

Conclusion

We presented the dynamics of the new chaotic model in two fractional operators, the Caputo–Fabrizio operator and the Atangana–Baleanu operator. We discussed some of the basic properties of the model (13) earlier. Initially, we considered the new chaotic model and showed that the model is a chaotic attractor for the specified values and then we applied the Caputo–Fabrizio derivative to the new chaotic model (see equation (14)) and then applied the Atangana–Baleanu derivative and obtained the model (see equation (23)). Then, we presented the numerical approach for the solution of Caputo–Fabrizio model and for the Atangana–Baleanu model and obtained the graphical results by considering various values of the fractional-order parameter. Further, we considered both the models, that is, in Caputo–Fabrizio and the Atangana–Baleanu derivative sense and presented the comparison plots. The comparison plots were obtained by using the value assigned to the fractional-order parameter $\alpha =1,0,0.99,\mathrm{and}0.97$ . One can see that both the schemes for the case of $\alpha =1$ give the same behavior for the models. Since the operators are different, the results obtained for the fractional values of the parameters are different (see Figures 17–25). We can observe from the graphical results the effectiveness of both the operators on a chaotic model and it can be applied for other chaotic problems confidently. The use of the two new fractional operators and their application to chaotic models are very rare in literature and so we hope that this article and its analysis will be more helpful for practical purpose when the mathematicians and scientist apply it for their proposed models. In future, this work can be extended into a new type of fractional and fractal combinations of operators in the sense of Caputo, Caputo–Fabrizio, and the Atangana–Baleanu derivatives.