# Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- ${\mathcal{D}}_{\alpha}(af+bg)=a{\mathcal{D}}_{\alpha}\left(f\right)+b{\mathcal{D}}_{\alpha}\left(g\right)$, for all $a,b\in \Re $.
- ${\mathcal{D}}_{\alpha}\left({t}^{p}\right)=p{t}^{p-\alpha}$, for all $p.$
- ${\mathcal{D}}_{\alpha}(\Xi )=0$, if $\Xi $ is a constant.
- ${\mathcal{D}}_{\alpha}\left(fg\right)=f{\mathcal{D}}_{\alpha}\left(g\right)+g{\mathcal{D}}_{\alpha}\left(f\right)$.
- ${\mathcal{D}}_{\alpha}\left(\frac{f}{g}\right)=\frac{g{\mathcal{D}}_{\alpha}\left(f\right)-f{\mathcal{D}}_{\alpha}\left(g\right)}{{g}^{2}}$.

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

## 3. Adams–Moulton Scheme for Fractional Conformable Derivatives

## 4. Application and Numerical Examples

- Rabinovich–Fabrikant attractor. The model of Rabinovich–Fabrikant [44] was initially designed as a physical model describing the stochasticity arising from the modulation instability in a non-equilibrium dissipative medium. The Rabinovich–Fabrikant system is described by the following equations:$$\begin{array}{c}\dot{x}=y\left(\right)open="("\; close=")">z-1+{x}^{2}+ax,\end{array}\dot{z}=-2z\left(\right)open="("\; close=")">b+xy,$$

- Conformable sense:$$\begin{array}{c}{}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{c\phantom{\rule{4pt}{0ex}}\beta}{\mathcal{D}}_{t}^{\alpha}x=\frac{1}{\Phi}\left(\right)open="["\; close="]">y\left(\right)open="("\; close=")">z-1+{x}^{2}+ax& ,\end{array}{}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{c\phantom{\rule{4pt}{0ex}}\beta}{\mathcal{D}}_{t}^{\alpha}y=\frac{1}{\Phi}\left(\right)open="["\; close="]">x\left(\right)open="("\; close=")">3z+1-{x}^{2}+ay& ,$$$$\begin{array}{c}{x}_{n+1}^{p}\left(t\right)={x}_{0}\left(t\right)+\frac{1}{\Gamma \left(\beta \right)}\sum _{j=0}^{n}{b}_{1,j,n+1}{f}_{1}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n},\end{array}{z}_{n+1}^{p}\left(t\right)={z}_{0}\left(t\right)+\frac{1}{\Gamma \left(\beta \right)}\sum _{j=0}^{n}{b}_{3,j,n+1}{f}_{3}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n},$$$$\begin{array}{c}{f}_{1}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n}:=\frac{1}{\Phi}\left(\right)open="["\; close="]">y\left(\right)open="("\; close=")">z-1+{x}^{2}& +ax\\ ,\end{array}$$
- β-conformable sense:$$\begin{array}{c}{}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{AC\phantom{\rule{4pt}{0ex}}\beta}{\mathcal{D}}_{t}^{\alpha}x=\frac{1}{\psi}\left(\right)open="["\; close="]">y\left(\right)open="("\; close=")">z-1+{x}^{2}+ax& ,\end{array}{}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{AC\phantom{\rule{4pt}{0ex}}\beta}{\mathcal{D}}_{t}^{\alpha}y=\frac{1}{\psi}\left(\right)open="["\; close="]">x\left(\right)open="("\; close=")">3z+1-{x}^{2}+ay& ,$$$$\begin{array}{c}{x}_{n+1}^{p}\left(t\right)={x}_{0}\left(t\right)+\frac{1}{\Gamma \left(\beta \right)}\sum _{j=0}^{n}{b}_{1,j,n+1}{g}_{1}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n},\end{array}{z}_{n+1}^{p}\left(t\right)={z}_{0}\left(t\right)+\frac{1}{\Gamma \left(\beta \right)}\sum _{j=0}^{n}{b}_{3,j,n+1}{g}_{3}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n},$$$$\begin{array}{c}{g}_{1}\left(\right)open="("\; close=")">{x}_{n},{y}_{n},{z}_{n},{t}_{n}:=\frac{1}{\psi}\left(\right)open="["\; close="]">y\left(\right)open="("\; close=")">z-1+{x}^{2}& +ax\\ ,\end{array}$$
- Observation. In the case when $\alpha \to 1$, we obtain the numerical solution of the Rabinovich–Fabrikant attractor in the Liouville–Caputo sense.

- Thomas’ cyclically symmetric attractor. Thomas in [45] proposed a mathematically three-dimensional cyclically symmetric attractor. This system is cyclically symmetric in the variables x, y, and z and considers a frictional damping b. The Thomas’ cyclically symmetric attractor is described by the following equations:$$\begin{array}{c}\dot{x}=sin\left(y\right)-bx,\\ \dot{y}=sin\left(z\right)-by,\\ \dot{z}=sin\left(x\right)-bz,\end{array}$$

- Observation. In the case when $\alpha \to 1$, we obtain the numerical solution of the Thomas’ cyclically symmetric attractor in the Liouville–Caputo sense.

- Newton–Leipnik attractor. The Newton–Leipnik system model was obtained by modifying Euler’s rigid body equations with the addition of a linear feedback in 1981. For this example, we consider a 3D system of fractional order nonlinear autonomous differential equations known as Newton–Leipnik attractor [47,48]:$$\begin{array}{c}\dot{x}=-ax+y+cyz,\\ \dot{y}=-x-ay+dxz,\\ \dot{z}=bz-dxy,\end{array}$$

- Observation. In the case when $\alpha \to 1$, we obtain the numerical solution of the Newton–Leipnik attractor in the Liouville–Caputo sense.

- If the distance between two points is larger than 0$$d\left(\right)open="("\; close=")">X,{X}_{d}$$
- The distance between two points is equal to 0, if and only if two points are overlapped$$\left(\right)open="("\; close=")">X,{X}_{d}$$

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numerical simulation for the Rabinovich–Fabrikant attractor (16) for $a=0.10$, $b=0.14$ with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$, $z\left(0\right)=0.5$.

**Figure 2.**Numerical simulation for the scheme given by Equation (18) for $a=0.10$, $b=0.14$ with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$, $z\left(0\right)=0.5$, for different particular cases of $\alpha $ and $\beta $. In (

**a**), $\alpha =1$, $\beta =0.998$. In (

**b**), $\alpha =1$, $\beta =0.997$. In (

**c**), $\alpha =0.996$, $\beta =1$. In (

**d**), $\alpha =0.988$, $\beta =1$, all values were arbitrarily chosen.

**Figure 3.**Numerical simulation for the Rabinovich–Fabrikant attractor (18) for $a=0.10$, $b=0.14$ with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$, $z\left(0\right)=0.5$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 4.**Numerical simulation for the scheme given by Equation (21) for $a=0.10$, $b=0.14$ with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$, $z\left(0\right)=0.5$, for different particular cases of $\alpha $ and $\beta $. In (

**a**) $\alpha =1$, $\beta =0.997$. In (

**b**) $\alpha =1$, $\beta =0.995$. In (

**c**) $\alpha =0.994$, $\beta =1$. In (

**d**) $\alpha =0.992$, $\beta =1$, all values were arbitrarily chosen.

**Figure 5.**Numerical simulation for the Rabinovich–Fabrikant attractor (21) for $a=0.10$, $b=0.14$ with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$, $z\left(0\right)=0.5$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 6.**Numerical simulation for the Thomas’ cyclically symmetric attractor (23) for $b=0.1998$, step size $h=1\times {10}^{-2}$, simulation time $t=150\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$$z\left(0\right)=1$.

**Figure 7.**Numerical simulation for the scheme given by Equation (24) for $b=0.1998$, step size $h=1\times {10}^{-2}$, simulation time $t=150\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$$z\left(0\right)=1$, for different particular cases of $\alpha $ and $\beta $. In (

**a**), $\alpha =1$, $\beta =0.996$. In (

**b**), $\alpha =1$, $\beta =0.993$. In (

**c**), $\alpha =0.995$, $\beta =1$. In (

**d**), $\alpha =0.955$, $\beta =1$, all values were arbitrarily chosen.

**Figure 8.**Numerical simulation for the Thomas’ cyclically symmetric attractor (24) for $b=0.1998$, step size $h=1\times {10}^{-2}$, simulation time $t=150\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$$z\left(0\right)=1$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 9.**Numerical simulation for the scheme given by Equation (25) for $b=0.1998$, step size $h=1\times {10}^{-2}$, simulation time $t=150\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$$z\left(0\right)=1$, for different particular cases of $\alpha $ and $\beta $. In (

**a**), $\alpha =1$, $\beta =0.994$. In (

**b**), $\alpha =1$, $\beta =0.992$. In (

**c**), $\alpha =0.994$, $\beta =1$. In (

**d**), $\alpha =0.985$, $\beta =1$, all values were arbitrarily chosen.

**Figure 10.**Numerical simulation for the Thomas’ cyclically symmetric attractor (25) for $b=0.1998$, step size $h=1\times {10}^{-2}$, simulation time $t=70\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=-1$, $y\left(0\right)=0$$z\left(0\right)=0.5$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 11.**Numerical simulation for the Newton–Leipnik attractor (26) for $a=0.4$, $b=0.175$, $c=10$, $d=5$, step size $h=1\times {10}^{-2}$, simulation time $t=400\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=0.349$, $y\left(0\right)=0$, $z\left(0\right)=-0.16$.

**Figure 12.**Numerical simulation for the scheme given by Equation (27) for $a=0.4$, $b=0.175$, $c=10$, $d=5$, step size $h=1\times {10}^{-2}$, simulation time $t=400\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=0.349$, $y\left(0\right)=0$, $z\left(0\right)=-0.16$, for different particular cases of $\alpha $ and $\beta $. In (

**a**), $\alpha =1$, $\beta =0.98$. In (

**b**), $\alpha =1$, $\beta =0.95$. In (

**c**), $\alpha =0.97$, $\beta =1$. In (

**d**), $\alpha =0.91$, $\beta =1$, all values were arbitrarily chosen.

**Figure 13.**Numerical simulation for the Newton–Leipnik attractor (27) for $a=0.4$, $b=0.175$, $c=10$, $d=5$, step size $h=1\times {10}^{-2}$, simulation time $t=400\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=0.349$, $y\left(0\right)=0$, $z\left(0\right)=-0.16$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 14.**Numerical simulation for the scheme given by Equation (28) for $a=0.4$, $b=0.175$, $c=10$, $d=5$, step size $h=1\times {10}^{-2}$, simulation time $t=400\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=0.349$, $y\left(0\right)=0$, $z\left(0\right)=-0.16$, for different particular cases of $\alpha $ and $\beta $. In (

**a**), $\alpha =1$, $\beta =0.97$. In (

**b**), $\alpha =1$, $\beta =0.94$. In (

**c**), $\alpha =0.999$, $\beta =1$. In (

**d**), $\alpha =0.9998$, $\beta =1$, all values were arbitrarily chosen.

**Figure 15.**Numerical simulation for the Newton–Leipnik attractor (28) for $a=0.4$, $b=0.175$, $c=10$, $d=5$, step size $h=1\times {10}^{-2}$, simulation time $t=400\phantom{\rule{3.33333pt}{0ex}}\left[s\right]$ and initial conditions $x\left(0\right)=0.349$, $y\left(0\right)=0$, $z\left(0\right)=-0.16$, for different particular cases of $\alpha $ and $\beta $, all values were arbitrarily chosen.

**Figure 16.**Bifurcation diagram of parameter a and portraits phase for the Newton–Leipnik system. In (

**a**), bifurcation diagram of parameter a. In (

**b**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $a=0.04$. In (

**c**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $a=0.775$. In (

**d**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $a=1.715$, all values were arbitrarily chosen.

**Figure 17.**Bifurcation diagram of parameter b and portraits phase for the Newton–Leipnik system. In (

**a**), bifurcation diagram of parameter b. In (

**b**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $b=0.07$. In (

**c**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $b=0.665$. In (

**d**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $b=1.31$, all values were arbitrarily chosen.

**Figure 18.**Bifurcation diagram of parameter c and portraits phase for the Newton–Leipnik system. In (

**a**), bifurcation diagram of parameter c. In (

**b**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $c=0.77$. In (

**c**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $c=0.775$. In (

**d**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $c=4.975$, all values were arbitrarily chosen.

**Figure 19.**Bifurcation diagram of parameter d and portraits phase for the Newton–Leipnik system. In (

**a**), bifurcation diagram of parameter d. In (

**b**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $d=0.18$. In (

**c**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $b=2.275$. In (

**d**), portrait phase of the Newton–Leipnik system with $\alpha =0.95$, $\beta =0.98$ and $d=5.245$, all values were arbitrarily chosen.

**Figure 20.**Visual projection distance of the different initial conditions along the time for the Newton–Leipnik system. In (

**a**), initial condition ${x}_{0}$. In (

**b**), initial condition ${y}_{0}$. In (

**c**), initial condition ${z}_{0}$.

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Pérez, J.E.S.; Gómez-Aguilar, J.F.; Baleanu, D.; Tchier, F.
Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors. *Entropy* **2018**, *20*, 384.
https://doi.org/10.3390/e20050384

**AMA Style**

Pérez JES, Gómez-Aguilar JF, Baleanu D, Tchier F.
Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors. *Entropy*. 2018; 20(5):384.
https://doi.org/10.3390/e20050384

**Chicago/Turabian Style**

Pérez, Jesús Emmanuel Solís, José Francisco Gómez-Aguilar, Dumitru Baleanu, and Fairouz Tchier.
2018. "Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors" *Entropy* 20, no. 5: 384.
https://doi.org/10.3390/e20050384