# A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Predictor–Corrector Scheme

#### 2.2. Stability of Fractional-Order Systems

**Definition**

**1.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. A New Three-Dimensional Fractional-Order Chaotic System

#### 3.1. Self-Excited Chaotic Attractor: Spiral Saddle Type of Equilibrium Points

**Proof.**

#### 3.2. Degenerate Case: Self-Excited Chaotic Attractor with Nonhyperbolic Equilibria

#### 3.3. Hidden Chaotic Attractor Localization in the Fractional-Order System without Equilibria

**Definition**

**2.**

#### 3.4. Coexistence of Hidden Attractors Regimes in the Fractional-Order System without Equilibria

#### 3.5. Mechanism of the Different Dynamics

## 4. Test 0–1 for Chaos

#### Detecting Chaos in the Proposed Fractional-Order System

## 5. Spectral Entropy Analysis

#### 5.1. Structural Complexity of the New Fractional-Order Chaotic System

#### 5.2. Design of a PRNG Using Hidden Attractors

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Self-excited attractor of the system (15) considering $a=-1$ and $q=0.93$. (

**a**) x–y plane; (

**b**) x–z plane; (

**c**) y–z plane.

**Figure 2.**Chaotic attractor of the system (15) with nonhyperbolic equilibrium points, $a=0.25$ and $q=0.99$. (

**a**) x–y plane; (

**b**) x–z plane; (

**c**) y–z plane.

**Figure 3.**(

**a**) Lyapunov exponents spectrum, and (

**b**) bifurcation diagram of the fractional-order nonhyperbolic system (15) when $a=1/4$.

**Figure 4.**Hidden attractor of the system (15) considering $a=0.35$, $q=0.97$, and initial conditions $\left(x\right(0),y(0),z(0\left)\right)=(1,1,1)$. (

**a**) x–y plane; (

**b**) x–z plane; (

**c**) y–z plane.

**Figure 5.**(

**a**) Lyapunov exponents spectrum, and (

**b**) bifurcation diagram of the fractional-order no-equilibrium system (15), when $a>1/4$.

**Figure 6.**Cross-section of the basins of attraction of the two coexisting attractors in the y–z plane at $x=0$ for the fractional-order chaotic system without equilibrium (15) when $a=0.35$ and $q=0.996$.

**Figure 7.**Coexistence of hidden chaotic and periodic attractors of the system (15) considering $a=0.35$ and $q=0.996$. (

**a**) x–y plane; (

**b**) x–z plane; (

**c**) y–z plane.

**Figure 8.**Hidden periodic attractor of the fractional-order system (15) with $a=0.35$, $q=0.996$, and initial conditions $\left(x\right(0),y(0),z(0\left)\right)=(0,75,-50)$. (

**a**) x–y plane; (

**b**) x–z plane; (

**c**) y–z plane.

**Figure 9.**Bi-dimensional map for the different dynamical behaviors of the fractional-order system (15) as a function of the parameter a and order q. The white region leads to a chaotic attractor, the black region evolves to periodic attractors, and the orange region converges to unbounded orbits. Self-excited, nonhyperbolic, and hidden chaotic attractors for $a<1/4$, $a=1/4$, and $a>1/4$, respectively.

**Figure 10.**Dynamics of the translation components $({p}_{1},{p}_{2})$ of the fractional-order system (15): (

**a**) Self-excited chaotic attractor ($q=0.93$, $a=-1$) with an asymptotic growth rate $K=0.9988$; (

**b**) hidden chaotic attractor ($q=0.97$, $a=0.35$), with $K=0.9985$.

**Figure 11.**Dynamics of the translation components $({p}_{1},{p}_{2})$ of the fractional-order system (15): (

**a**) Coexisting hidden chaotic attractor ($q=0.996$, $a=0.35$, $(x,y,z)=(1,1,1)$) with an asymptotic growth rate $K=0.9975$; (

**b**) coexisting hidden periodic attractor ($q=0.996$, $a=0.35$, $(x,y,z)=(0,75,-50)$ with $K=0.0364$.

**Table 1.**Equilibria, eigenvalues, and Lyapunov exponents of the fractional-order chaotic system (15).

New System | Parameters | FO | Equilibria | Eigenvalues | ${\mathit{x}}_{0},{\mathit{y}}_{0},{\mathit{z}}_{0}$ | LEs |
---|---|---|---|---|---|---|

Self-excited | $a=-1;$ | $q=0.93$ | (1,1.6180, −1.6180) | $2.3064,-0.3442\pm 0.9225i$ | $(1,1,1)$ | $L{E}_{1}=2.957$ |

$(-1,-0.6180,-0.6180)$ | $-1.0666,0.2243\pm 1.4304i$ | $L{E}_{2}=0.01$ | ||||

$(1,-0.6180,0.6180)$ | $-1.0666,0.2243\pm 1.4304i$ | $L{E}_{3}=-5.765$ | ||||

$(-1,1.6180,1.6180)$ | $2.3064,-0.3442\pm 0.9225i$ | |||||

Non-hyperbolic | $a=0.25;$ | $q=0.99$ | $(1,\frac{1}{2},-\frac{1}{2})$ | $0,-0.3750+0.5994i$ | $(1,1,1)$ | $L{E}_{1}=1.27$ |

$(-1,\frac{1}{2},\frac{1}{2})$ | $0,-0.3750+0.5994i$ | $L{E}_{2}=0.010$ | ||||

$L{E}_{3}=-1.72$ | ||||||

Hidden | $a=0.35;$ | $q=0.97$ | no-equilibria | $(1,1,1)$ | $L{E}_{1}=14.735$ | |

$L{E}_{2}=0.010$ | ||||||

$L{E}_{3}=-18.350$ | ||||||

Coexistence Chaotic | $a=0.35;$ | $q=0.996$ | no-equilibria | $(1,1,1)$ | $L{E}_{1}=11.066$ | |

$L{E}_{2}=0.080$ | ||||||

$L{E}_{3}=-13.161$ | ||||||

Coexistence Periodic | $a=0.35;$ | $q=0.996$ | no-equilibria | $(0,75,-50)$ | $L{E}_{1}=0$ | |

$L{E}_{2}=-3.695$ | ||||||

$L{E}_{3}=-3.705$ |

**Table 2.**Results of NIST statistical tests for the bit sequences based on the system (15) when it presents a hidden chaotic attractor.

Statistical Test | p-Value | Results |
---|---|---|

Frequency | 0.654721 | success |

Block Frequency | 0.420199 | success |

Cusum-Forward | 0.600222 | success |

Cusum-Reverse | 0.446686 | success |

Runs | 0.220773 | success |

Long Runs of Ones | 0.012522 | success |

Rank | 0.254592 | success |

Spectral DFT | 0.538167 | success |

Non-Overlapping Templates | 0.615839 | success |

Overlapping Templates | 0.102065 | success |

Universal | 0.830304 | success |

Approximate Entropy | 0.635119 | success |

Random Excursions | 0.407574 | success |

Random Excursions Variant | 0.444982 | success |

Linear Complexity | 0.634990 | success |

Serial | 0.301388 | success |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Munoz-Pacheco, J.M.; Zambrano-Serrano, E.; Volos, C.; Jafari, S.; Kengne, J.; Rajagopal, K.
A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors. *Entropy* **2018**, *20*, 564.
https://doi.org/10.3390/e20080564

**AMA Style**

Munoz-Pacheco JM, Zambrano-Serrano E, Volos C, Jafari S, Kengne J, Rajagopal K.
A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors. *Entropy*. 2018; 20(8):564.
https://doi.org/10.3390/e20080564

**Chicago/Turabian Style**

Munoz-Pacheco, Jesus M., Ernesto Zambrano-Serrano, Christos Volos, Sajad Jafari, Jacques Kengne, and Karthikeyan Rajagopal.
2018. "A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors" *Entropy* 20, no. 8: 564.
https://doi.org/10.3390/e20080564