# Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System

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## Abstract

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## 1. Introduction

## 2. Numerical Solution of the Fractional-Order Lorenz Hyperchaotic System

#### 2.1. Adomian Decomposition Method

#### 2.2. Fractional-Order Lorenz Hyperchaotic System

## 3. Dynamics Analysis of the Fractional-Order Lorenz Hyperchaotic System

#### 3.1. Bifurcation Analysis

**Figure 1.**Bifurcation and Lyapunov Characteristic Exponents (LCEs) with different q ($k=0.26$) (

**a**) Bifurcation diagram; (

**b**) Lyapunov characteristic exponents.

**Figure 2.**Bifurcation and LCEs with different k ($q=0.96$) (

**a**) Bifurcation diagram; (

**b**) Lyapunov characteristic exponents.

#### 3.2. Observation of Chaotic Attractors

**Figure 3.**Phase diagrams of fractional-order hyperchaotic Lorenz system ($k=0.26$) (

**a**) periodic orbits ($q=0.65$); (

**b**) hyperchaos ($q=0.72$); (

**c**) chaos ($q=0.89$); (

**d**) periodic orbits ($q=1.00$).

**Figure 4.**Phase diagrams of fractional-order hyperchaotic Lorenz system $(q=0.96)$ (

**a**) hyperchaos ($k=0.05$); (

**b**) chaos ($k=0.20$); (

**c**) periodic orbits ($k=0.25$); (

**d**) periodic orbits ($k=0.30$); (

**e**) chaos ($k=0.40$); (

**f**) quasi-periodic orbits ($k=0.50$); (

**g**) periodic orbits ($k=0.80$); (

**h**) periodic orbits ($k=1.00$).

#### 3.3. Spectral Entropy Complexity Analysis

**Figure 5.**Spectral entropy (SE) complexity results (

**a**) SE complexity versus fractional order q (k = 0.26); (

**b**) SE complexity versus fractional order k (q = 0.96); (

**c**) SE complexity in the $q-k$ parameter plane.

#### 3.4. C${}_{0}$ Complexity Analysis

**Figure 6.**C${}_{0}$ complexity results (

**a**) C${}_{0}$ complexity versus fractional order q (k = 0.26); (

**b**) C${}_{0}$ complexity versus fractional order k (q = 0.96); (

**c**) C${}_{0}$ complexity in the $q-k$ parameter plane.

## 4. Digital Circuit Implementation

#### 4.1. DSP Implementation

**Figure 10.**Phase diagrams of the fractional-order Lorenz hyperchaotic system by DSP $(k=0.26)$ (

**a**) periodic orbits ($q=0.65$); (

**b**) hyperchaos ($q=0.72$); (

**c**) chaos ($q=0.89$); (

**d**) periodic orbits ($q=1.00$).

**Figure 11.**Phase diagrams of the fractional-order Lorenz hyperchaotic system by DSP $(q=0.96)$ (

**a**) hyperchaos ($k=0.05$); (

**b**) chaos ($k=0.20$); (

**c**) quasi-periodic orbits ($k=0.50$); (

**d**) periodic orbits ($k=1.00$).

#### 4.2. Pseudo-Random Sequence Generator

p-Value | Proportion | Success | |
---|---|---|---|

Frequency | 0.366918 | 100% | √ |

B. Frequency | 0.002559 | 99% | √ |

C. Sums | 0.275709 | 99% | √ |

Runs | 0.048716 | 98% | √ |

Longest Run | 0.935716 | 99% | √ |

Rank | 0.883171 | 99% | √ |

FFT | 0.574903 | 100% | √ |

N.O.Temp. | 0.002203 | 97% | √ |

O.Temp. | 0.834308 | 100% | √ |

Universal | 0.759756 | 99% | √ |

App. Entropy | 0.759756 | 98% | √ |

R.Excur. | 0.005166 | 96.7% | √ |

R.Excur.V. | 0.105618 | 96.7% | √ |

Serial | 0.075719 | 99% | √ |

L. Complexity | 0.554420 | 99% | √ |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

He, S.; Sun, K.; Wang, H.
Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System. *Entropy* **2015**, *17*, 8299-8311.
https://doi.org/10.3390/e17127882

**AMA Style**

He S, Sun K, Wang H.
Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System. *Entropy*. 2015; 17(12):8299-8311.
https://doi.org/10.3390/e17127882

**Chicago/Turabian Style**

He, Shaobo, Kehui Sun, and Huihai Wang.
2015. "Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System" *Entropy* 17, no. 12: 8299-8311.
https://doi.org/10.3390/e17127882