# Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated M-Derivative

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## Abstract

**:**

## 1. Introduction

- Three order non-local M-fractional derivative with a power law, exponential decay and Mittag–Leffler function.
- Numerical schemes based on Lagrange interpolation to predict chaotic behaviors of Rucklidge, Shimizu–Morioka and a hybrid strange attractor.
- Numerical analysis based on 0–1 test and sensitive dependence on initial conditions to support chaos existence in the above mentioned chaotic attractor.

## 2. Mathematical Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1.**

- 1.
- ${}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}(af+bg)\left(t\right)=a{}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}f\left(t\right)+b{}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}g\left(t\right)$.
- 2.
- ${}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}(f\xb7g)\left(t\right)=f\left(t\right){}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}g\left(t\right)+g\left(t\right){}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}f\left(t\right)$.
- 3.
- ${}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}\left(\frac{f}{g}\right)\left(t\right)\frac{g\left(t\right){}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}f\left(t\right)-f\left(t\right){}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}g\left(t\right)}{{\left[g\left(t\right)\right]}^{2}}$.
- 4.
- ${}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}\left(c\right)=0$, where $f\left(t\right)=c$ is a constant.
- 5.
- (Chain rule) If f is differentiable, then ${}_{\phantom{\rule{4pt}{0ex}}i}^{M}{\mathcal{D}}^{\alpha ,\beta}\left\{f\left(t\right)\right\}=\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}\frac{df\left(t\right)}{dt}$.

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

## 3. Rucklidge, Shimizu–Morioka and Hybrid Strange Attractors

#### 3.1. Rucklidge Attractor

#### 3.2. Shimizu–Morioka Attractor

#### 3.3. Strange Hybrid Attractor

## 4. Numerical Schemes and Simulations for Chaotic Attractors

#### 4.1. Truncated M-Fractional Rucklidge Attractor with Three Orders and Power-Law, Exponential Decay Law and Mittag–Leffler Memories

- Riemann–Liouville sense$$\begin{array}{cc}\hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-RL}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{x\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}f(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-RL}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{y\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}g(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-RL}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{z\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}h(x,y,z,t).\hfill \end{array}$$
- Caputo–Fabrizio sense$$\begin{array}{cc}\hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-CF}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{x\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}f(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-CF}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{y\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}g(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-CF}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{z\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}h(x,y,z,t).\hfill \end{array}$$
- Atangana–Baleanu sense$$\begin{array}{cc}\hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-AB}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{x\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}f(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-AB}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{y\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}g(x,y,z,t),\hfill \\ \hfill {}_{\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0}^{M-AB}{\mathcal{D}}_{t}^{\alpha ,\beta ,\gamma}\left\{z\right\}& =\frac{{t}^{1-\alpha}}{\Gamma (\beta +1)}h(x,y,z,t).\hfill \end{array}$$

#### 4.1.1. Numerical Scheme for Truncated M-Fractional Rucklidge Attractor in Liouville–Caputo Sense with Three Orders

#### 4.1.2. Numerical Simulations

**Case**

**1.**

**Case**

**2.**

- The Rucklidge attractor involving the M-derivative is shown in Figure 5b. This behavior is obtained when $\alpha =1$, $\beta \ne 1$ and $\gamma =1$.
- The Liouville–Caputo fractional Rucklidge attractor numerical approximation is displayed in Figure 5c with $\alpha =1$, $\beta =1$ and $\gamma \ne 1$.
- By choosing $\alpha \ne 1$, $\beta \ne 1$ and $\gamma =1$, the truncated M-Rucklidge attractor numerical approximation is showed in Figure 5d.

- Observation. When $\alpha ,\beta ,\gamma \to 1$, the classical Rucklidge attractor is recovered.

#### 4.1.3. Numerical Scheme for Truncated M-Fractional Rucklidge Attractor in Caputo–Fabrizio Sense with Three Orders

#### 4.1.4. Numerical Simulations

- Observation. When $\alpha ,\beta ,\gamma \to 1$, the classical Rucklidge attractor is obtained.

#### 4.1.5. Numerical Scheme for Truncated M-Fractional Rucklidge Attractor in Atangana–Baleanu Sense with Three Orders

#### 4.1.6. Numerical Simulations

- Observation. When $\alpha ,\beta ,\gamma \to 1$, the classical Rucklidge attractor is recovered.

#### 4.1.7. Chaos Analysis

#### 4.2. Truncated M-Fractional-Shimizu–Morioka Attractor with Three Orders and Mittag–Leffler Memory

- The Shimizu–Morioka attractor of Khalil’s type is recovered when $\alpha \ne 1$ and $\beta =1$, see Figure 10a,b.
- When $\alpha =1$ and $\beta \ne 1$, the M-derivative Shimizu–Morioka attractor is showed in Figure 10c,d.
- The behavior of the truncated M-Shimizu–Morioka attractor is obtained when $\alpha \ne 1$ and $\beta \ne 1$, see Figure 10e,f.

- Observation. When $\alpha ,\beta ,\gamma \to 1$, the classical Shimizu–Morioka attractor is obtained.

#### Chaos Analysis

#### 4.3. Truncated M-Fractional Strange Hybrid Attractor with Three Orders and Mittag–Leffler Memory

**Case****1**.- Choosing ${\alpha}_{1}=0.98$, ${\beta}_{1}=0.97$, ${\gamma}_{1}=0.96$ for the state x; ${\alpha}_{2}=0.96$, ${\beta}_{2}=0.97$, ${\gamma}_{2}=1$ for the state y; and ${\alpha}_{3}=1$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.97$ for the state z; Figure 13a shows interesting behavior for the truncated M-fractional strange hybrid attractor in Atangana–Baleanu sense.
**Case****2**.- Choosing ${\alpha}_{1}=0.97$, ${\beta}_{1}=0.96$, ${\gamma}_{1}=0.96$ for the first state x; ${\alpha}_{2}=0.95$, ${\beta}_{2}=0.98$, ${\gamma}_{2}=0.97$ for the second state y; and ${\alpha}_{3}=0.96$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.98$ for the third state z; Figure 13b shows interesting behavior for the truncated M-fractional strange hybrid attractor in Atangana–Baleanu sense.
**Case****3**.- Setting ${\alpha}_{1}=1$, ${\beta}_{1}=1$, ${\gamma}_{1}=0.96$ for the first state; ${\alpha}_{2}=0.96$, ${\beta}_{2}=1$, ${\gamma}_{2}=1$ for the second state; and ${\alpha}_{3}=0.96$, ${\beta}_{3}=0.98$, ${\gamma}_{3}=1$ for the last state; the truncated M-fractional strange hybrid attractor exhibits complex dynamics involving the Mittag–Leffler memory and the local M-derivative, see Figure 13c.
**Case****4**.- Considering for the state x the orders ${\alpha}_{1}=0.95$, ${\beta}_{1}=0.96$, ${\gamma}_{1}=1$; the values ${\alpha}_{2}=0.96$, ${\beta}_{2}=0.97$, ${\gamma}_{2}=0.98$ for the state y; and ${\alpha}_{3}=0.96$, ${\beta}_{3}=1$, ${\gamma}_{3}=1$ for z; Figure 13d depicts complex dynamics for the truncated M-fractional strange hybrid attractor in Atangana–Baleanu sense.
**Case****5**.- Next, to get the simulation shown in Figure 13e, in the state x was choosing ${\alpha}_{1}=0.95$, ${\beta}_{1}=1$, ${\gamma}_{1}=1$; for the state y${\alpha}_{2}=1$, ${\beta}_{2}=1$, ${\gamma}_{2}=0.97$; and the values ${\alpha}_{3}=0.96$, ${\beta}_{3}=0.95$, ${\gamma}_{3}=0.98$ in the state z.
**Case****6**.- The following parameters, randomly chosen, were set on the system (23): ${\alpha}_{1}=0.9819$, ${\beta}_{1}=0.8922$ and ${\gamma}_{1}=0.9521$ for the first state; ${\alpha}_{2}=0.8270$, ${\beta}_{2}=0.8801$ and ${\gamma}_{2}=1$ for the second; and ${\alpha}_{3}=1$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.9740$ for the third state. Here, the inner scroll still remains because the external dynamics depict an attractor strange-like with quasi-periodic orbits. It can be see in Figure 13f.

- Observation. When ${\alpha}_{i}$, ${\beta}_{i}$ and ${\gamma}_{i}\to 1$, the classical strange hybrid attractor is obtained.

#### Chaos Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical simulation of the Rucklidge attractor (8) for $a=6.7$, $k=2$; with initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$.

**Figure 2.**Numerical simulation of Shimizu–Morioka attractor (9) for $a=0.45$, $B=0.75$; with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=2$ and $z\left(0\right)=1$.

**Figure 3.**Numerical simulation for strange hybrid attractor (10) with initial conditions $x\left(0\right)=2$, $y\left(0\right)=0$ and $z\left(0\right)=0$.

**Figure 4.**Numerical simulation for truncated M-Rucklidge attractor (12) with initial conditions, $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$. Different α and β values were arbitrarily chosen.

**Figure 5.**Numerical simulation for truncated M-fractional Rucklidge attractor in Liouville–Caputo sense (13) with initial conditions, $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$. Different α, β and γ values were arbitrarily chosen.

**Figure 6.**Numerical simulations for truncated M-fractional Rucklidge attractor in Liouville–Caputo sense (13) with initial conditions, $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$. Different α, β and γ values were arbitrarily chosen.

**Figure 7.**Numerical simulations for truncated M-fractional Rucklidge attractor in Caputo–Fabrizio sense (14) with initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$. Different $\alpha $, $\beta $ and $\gamma $ values were arbitrarily chosen.

**Figure 8.**Numerical simulations for truncated M-fractional Rucklidge attractor in Atangana–Baleanu–Caputo sense (15) with initial conditions $x\left(0\right)=1$, $y\left(0\right)=0$ and $z\left(0\right)=4.5$. Different $\alpha $, $\beta $ and $\gamma $ values were arbitrarily chosen.

**Figure 10.**Numerical simulation for truncated M-Shimizu–Morioka attractor (21) with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=2$ and $z\left(0\right)=1$. Different $\alpha $, $\beta $ and $\gamma $ values were arbitrarily chosen.

**Figure 11.**Numerical simulations for truncated M-fractional Shimizu–Morioka attractor in Atangana–Baleanu–Caputo sense (21) with initial conditions $x\left(0\right)=-1$, $y\left(0\right)=2$, $z\left(0\right)=1$. Different $\alpha $, $\beta $ and $\gamma $ values were arbitrarily chosen.

**Figure 13.**Dynamic behaviors of the incommensurate truncated M-fractional strange hybrid attractor (23) for different orders with initial conditions $x\left(0\right)=2$, $y\left(0\right)=0$ and $z\left(0\right)=0$.

**Figure 14.**Variables p and q in the strange attractor (23) for different orders. (

**a**–

**c**) Case 1 with ${\alpha}_{1}=0.98$, ${\beta}_{1}=0.97$, ${\gamma}_{1}=0.96$, ${\alpha}_{2}=0.96$, ${\beta}_{2}=0.97$, ${\gamma}_{2}=1$, ${\alpha}_{3}=1$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.97$; (

**d**–

**f**) Case 2 with ${\alpha}_{1}=0.97$, ${\beta}_{1}=0.96$, ${\gamma}_{1}=0.96$, ${\alpha}_{2}=0.95$, ${\beta}_{2}=0.98$, ${\gamma}_{2}=0.97$, ${\alpha}_{3}=0.96$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.98$; (

**g**–

**i**) Case 3 with ${\alpha}_{1}=1$, ${\beta}_{1}=1$, ${\gamma}_{1}=0.96$, ${\alpha}_{2}=1$, ${\beta}_{2}=1$, ${\gamma}_{2}=1$, ${\alpha}_{3}=0.96$, ${\beta}_{3}=0.98$, ${\gamma}_{3}=1$, respectively.

**Figure 15.**Variables p and q in Strange attractor (23) ffor different orders. (

**a**–

**c**) Case 4 with ${\alpha}_{1}=0.95$, ${\beta}_{1}=0.96$, ${\gamma}_{1}=1$, ${\alpha}_{2}=0.96$, ${\beta}_{2}=0.97$, ${\gamma}_{2}=0.98$, ${\alpha}_{3}=0.96$, ${\beta}_{3}=1$, ${\gamma}_{3}=1$; (

**d**–

**f**) Case 5 with ${\alpha}_{1}=0.95$, ${\beta}_{1}=1$, ${\gamma}_{1}=1$, ${\alpha}_{2}=1$, ${\beta}_{2}=0.97$, ${\gamma}_{2}=0.96$, ${\alpha}_{3}=0.96$, ${\beta}_{3}=0.95$, ${\gamma}_{3}=0.98$; (

**g**–

**i**) Case 6 with ${\alpha}_{1}=0.9819$, ${\beta}_{1}=0.8922$, ${\gamma}_{1}=0.9521$, ${\alpha}_{2}=0.8270$, ${\beta}_{2}=0.8801$, ${\gamma}_{2}=1$, ${\alpha}_{3}=1$, ${\beta}_{3}=1$, ${\gamma}_{3}=0.9740$, respectively.

**Figure 16.**Differences between nearby starting trajectories for the state x in the incommensurate M-fractional Strange attractor.

**Figure 17.**Differences between nearby starting trajectories for the state y in the incommensurate M-fractional strange attractor.

**Figure 18.**Differences between nearby starting trajectories for the state z in the incommensurate M-fractional strange attractor.

Derivative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ |
---|---|---|---|

M–Liouville–Caputo | 0.9557 | 0.8915 | 0.9308 |

M–Caputo–Fabrizio | 0.9960 | 0.9796 | 0.9541 |

M–Atangana–Baleanu | 0.9926 | 0.9668 | 0.9358 |

Derivative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ |
---|---|---|---|

M–Atangana–Baleanu | 0.9676 | 0.9908 | 0.9900 |

Derivative | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{3}$ | Case |
---|---|---|---|---|

M–Atangana–Baleanu | 0.9962 | 0.9933 | 0.9120 | Case 1 |

M–Atangana–Baleanu | 0.9922 | 0.9942 | 0.9659 | Case 2 |

M–Atangana–Baleanu | 0.9897 | 0.9905 | 0.8573 | Case 3 |

M–Atangana–Baleanu | 0.9930 | 0.9773 | 0.9312 | Case 4 |

M–Atangana–Baleanu | 0.9946 | 0.9853 | 0.9442 | Case 5 |

M–Atangana–Baleanu | 0.9970 | 0.9901 | 0.9912 | Case 6 |

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**MDPI and ACS Style**

Solís-Pérez, J.E.; Gómez-Aguilar, J.F.
Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated *M*-Derivative. *Symmetry* **2020**, *12*, 626.
https://doi.org/10.3390/sym12040626

**AMA Style**

Solís-Pérez JE, Gómez-Aguilar JF.
Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated *M*-Derivative. *Symmetry*. 2020; 12(4):626.
https://doi.org/10.3390/sym12040626

**Chicago/Turabian Style**

Solís-Pérez, Jesús Emmanuel, and José Francisco Gómez-Aguilar.
2020. "Novel Fractional Operators with Three Orders and Power-Law, Exponential Decay and Mittag–Leffler Memories Involving the Truncated *M*-Derivative" *Symmetry* 12, no. 4: 626.
https://doi.org/10.3390/sym12040626