# Bifurcations, Hidden Chaos and Control in Fractional Maps

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

**Theorem**

**1.**

#### Stability of Fractional Order Maps

**Theorem**

**2.**

**Lemma**

**1.**

**Theorem**

**3.**

## 3. New Two and Three-Dimensional Fractional Maps

#### 3.1. Description of the New Two-Dimensional Fractional Map

#### 3.2. Bifurcation and 0-1 Test

#### 3.3. Description of the New Three-Dimensional Fractional Map

#### 3.4. Bifurcation and 0-1 Test

## 4. Chaos Control

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Strange attractor of the two-dimensional fractional map (10) for $\nu =0.999$ and $A=-0.83$.

**Figure 2.**Bifurcation diagrams of the two-dimensional fractional map (10) versus $\nu $ for $A=-0.83$.

**Figure 3.**The coexisting hidden attractors of the two-dimensional fractional map (10) for system parameter $A=-0.83$ and initial condition $(-0.32,-1.85)$ for red attractors and $(0.32,-1.85)$ for blue attractors, with fractional order varying: (

**a**) $\nu =0.9996$, (

**b**) $\nu =0.9989$, (

**c**) $\nu =0.9987$, (

**d**) $\nu =0.9984$.

**Figure 4.**The 0-1 test of the two-dimensional fractional map. (

**a**) Brownian like trajectories for both initial conditions with $\nu =0.9996$, (

**b**) bounded trajectories for both initial conditions with $\nu =0.9984$.

**Figure 5.**Strange attractor of the three-dimensional fractional map (17).

**Figure 7.**Coexisting hidden attractors of the three-dimensional fractional map (17) with initial condition $(-0.26,3.83,-2.22)$ for red attractors and $(-0.26,-3.83,-2.22)$ for blue attractors, (

**a**) for fractional order $\nu =0.993$, (

**b**) for fractional order $\nu =0.9984$.

**Figure 8.**The $p-q$ plots of the three-dimensional fractional map, (

**a**) bounded trajectories for $\nu =0.993$, (

**b**) Brownian-like trajectories for both initial conditions with $\nu =0.9948$.

**Figure 9.**(

**a**) Stabilization of state $x\left(n\right)$, (

**b**) stabilization of state $y\left(n\right)$, (

**c**) attractor of the controlled system (10) for $\nu =0.999$.

**Figure 10.**(

**a**) Stabilization of state $x\left(n\right)$, (

**b**) stabilization of state $y\left(n\right)$, (

**c**) stabilization of state $z\left(n\right)$, (

**d**) attractor of the controlled system (17) for $\nu =0.998$.

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**MDPI and ACS Style**

Ouannas, A.; Almatroud, O.A.; Khennaoui, A.A.; Alsawalha, M.M.; Baleanu, D.; Huynh, V.V.; Pham, V.-T.
Bifurcations, Hidden Chaos and Control in Fractional Maps. *Symmetry* **2020**, *12*, 879.
https://doi.org/10.3390/sym12060879

**AMA Style**

Ouannas A, Almatroud OA, Khennaoui AA, Alsawalha MM, Baleanu D, Huynh VV, Pham V-T.
Bifurcations, Hidden Chaos and Control in Fractional Maps. *Symmetry*. 2020; 12(6):879.
https://doi.org/10.3390/sym12060879

**Chicago/Turabian Style**

Ouannas, Adel, Othman Abdullah Almatroud, Amina Aicha Khennaoui, Mohammad Mossa Alsawalha, Dumitru Baleanu, Van Van Huynh, and Viet-Thanh Pham.
2020. "Bifurcations, Hidden Chaos and Control in Fractional Maps" *Symmetry* 12, no. 6: 879.
https://doi.org/10.3390/sym12060879