# On Two-Dimensional Fractional Chaotic Maps with Symmetries

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**The υth fractional sum of X can be defined as:

**Definition**

**2**

**.**The υ-Caputo-like difference, denoted by ${}^{C}{\Delta}_{a}^{\upsilon}X(t)$, of $X(t)$ is defined as:

**Theorem**

**1**

**.**The corresponding discrete integral equation of the delta fractional difference equation:

## 3. Fractional–Order Maps with Closed Curve Fixed Points

#### 3.1. Fractional–Order Map with Square-Shaped Fixed Points

#### Bifurcation Diagrams and Largest Lyapunov Exponents

#### 3.2. A New Fractional Map with Rectangle-Shaped Fixed Points

#### Bifurcation and Largest Lyapunov Exponents

#### 3.3. 0–1 Test

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical analysis of the fractional-order map (10) for $\nu =1$. (

**a**) Square-shaped fixed point of the fractional-order map are illustrated with black dots and the hidden chaotic attractor in red. (

**b**) Evolution states of the fractional-order map (10) of x and y. (

**c**) Bifurcation diagram of the fractional-order map (10) versus $\alpha $ for $\beta =2.4$. (

**d**) Bifurcation diagram of the fractional-order map (10) versus $\beta $ for $\alpha =0.2$.

**Figure 2.**(

**a**) Bifurcation diagram of the fractional-order map (10) versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.95$. (

**b**) Largest Lyapunov exponents corresponding to (

**a**). (

**c**) Bifurcation diagram of the fractional-order map (10) versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.9$. (

**d**) Largest Lyapunov exponents corresponding to (

**c**).

**Figure 3.**Bifurcation diagram for system (10) in the $x-{\upsilon}_{2}$ plane for $\alpha =0.2$ and $\beta =2.2$.

**Figure 4.**Coexisting chaotic attractor of the fractional-order map (16) for $\alpha =0.2$ and $\beta =2.2$.

**Figure 6.**(

**a**) Bifurcation diagram of the fractional-order map (16) versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.988$. (

**b**) Largest Lyapunov exponents corresponding to (

**a**). (

**c**) Bifurcation diagram of the fractional-order map (16) versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.995$. (

**d**) Largest Lyapunov exponents corresponding to (

**c**).

**Figure 7.**The 0–1 test method of the fractional-order map (10).

**Figure 8.**Transient state of the fractional-order map for $\upsilon =0.98$ (

**a**) when $n=570$, (

**b**) when $n=578$.

**Figure 9.**The 0–1 test of the fractional-order map (16). (

**a**) The asymptotic growth rate versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.998$. (

**b**) The asymptotic growth rate versus $\alpha $ for ${\upsilon}_{1}={\upsilon}_{2}=0.995$.

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**MDPI and ACS Style**

Hadjabi, F.; Ouannas, A.; Shawagfeh, N.; Khennaoui, A.-A.; Grassi, G.
On Two-Dimensional Fractional Chaotic Maps with Symmetries. *Symmetry* **2020**, *12*, 756.
https://doi.org/10.3390/sym12050756

**AMA Style**

Hadjabi F, Ouannas A, Shawagfeh N, Khennaoui A-A, Grassi G.
On Two-Dimensional Fractional Chaotic Maps with Symmetries. *Symmetry*. 2020; 12(5):756.
https://doi.org/10.3390/sym12050756

**Chicago/Turabian Style**

Hadjabi, Fatima, Adel Ouannas, Nabil Shawagfeh, Amina-Aicha Khennaoui, and Giuseppe Grassi.
2020. "On Two-Dimensional Fractional Chaotic Maps with Symmetries" *Symmetry* 12, no. 5: 756.
https://doi.org/10.3390/sym12050756