# Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors

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## Abstract

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## 1. Introduction

## 2. Hidden Hyperchaotic Multi-Scroll Fractional-Order System

#### 2.1. Design of System Model

#### 2.2. Equilibrium Point Analysis

#### 2.3. Adomian Decomposition

## 3. Dynamics Analysis

#### 3.1. Phase Analysis

#### 3.2. Lyapunov Exponents Analysis

#### 3.3. Bifurcation Analysis

## 4. Hardware Implementation

## 5. Application in Medical Image Encryption

#### 5.1. Encryption Scheme Design

#### 5.2. Security Analysis

#### 5.2.1. Histogram

#### 5.2.2. Correlation

#### 5.2.3. Running Time

#### 5.2.4. Information Entropy

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Coexistence of attractors on the $x-z$ plane of the system. The red line represents the phase diagram of the system under initial conditions of $[0.2,0.4,0.8,0.6,0.1]$, and the blue line represents the phase diagram of the system under initial conditions of $[0.12,0.27,0.92,0.56,0.1]$.

**Figure 7.**Coexistence of attractors on the $x-y-z$ space of the system. The red line represents the phase diagram of the system under initial conditions of $[0.2,0.4,0.8,0.6,0.1]$, and the blue line represents the phase diagram of the system under initial conditions of $[0.12,0.27,0.92,0.56,0.1]$.

**Figure 12.**Bifurcation coexistence of system with respect to order q. The blue line represents the bifurcation diagram with initial values of $[0.12,0.27,0.92,0.56,0.1]$, while the red line represents the bifurcation diagram with initial conditions of $[0.12,0.27,0.92,0.56,-0.1]$.

**Figure 17.**Image encryption and decryption results and their histograms: (

**a**) Original image. (

**b**) Encrypted image. (

**c**) Decrypted image. (

**d**) Original image histogram. (

**e**) Encrypted Image histogram. (

**f**) Decrypted image histogram.

**Figure 18.**Correlation analysis between the original image and the encrypted image, (

**a**–

**d**) is the correlation map of the original image, and (

**e**–

**h**) is the correlation coefficient map of the encrypted image: (

**a**) Horizontal. (

**b**) Negative diagonal. (

**c**) Positive diagonal. (

**d**) Vertical. (

**e**) Horizontal. (

**f**) Negative diagonal. (

**g**) Positive diagonal. (

**h**) Vertical.

${\mathit{E}}_{0}$ | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{-1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{-2}$ |
---|---|---|---|---|

0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 |

0.001 | 0.0013 | 0.0013 | 0.0012 | 0.0012 |

−1.001 | −1.0013 | −1.0013 | −1.0012 | −1.0012 |

Resource | Utilization | Available | Utilization |
---|---|---|---|

LUT | 31,347 | 53,200 | 58.92 |

LUTRAM | 1413 | 17,400 | 8.12 |

FF | 37.452 | 106,440 | 35.18 |

DSP | 136 | 220 | 61.81 |

IO | 35 | 125 | 28 |

FF | 1 | 32 | 3.13 |

Direction | Horizontal | Diagonal | Vertical |
---|---|---|---|

Encrypted | 0.9735 | 0.9636 | 0.9622 |

Decrypted | 0.0323 | −0.0029 | −0.0031 |

Ref. [63] | 0.0770 | −0.0615 | −0.0724 |

Ref. [64] | −0.0893 | 0.0010 | 0.0034 |

Ref. [65] | −0.0289 | 0.0366 | 0.0146 |

Encryption Time (s) | Decrypted Time (s) |
---|---|

0.18832 | 0.18874 |

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## Share and Cite

**MDPI and ACS Style**

Yu, F.; Xu, S.; Lin, Y.; He, T.; Wu, C.; Lin, H.
Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors. *Mathematics* **2024**, *12*, 2227.
https://doi.org/10.3390/math12142227

**AMA Style**

Yu F, Xu S, Lin Y, He T, Wu C, Lin H.
Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors. *Mathematics*. 2024; 12(14):2227.
https://doi.org/10.3390/math12142227

**Chicago/Turabian Style**

Yu, Fei, Shuai Xu, Yue Lin, Ting He, Chaoran Wu, and Hairong Lin.
2024. "Design and Analysis of a Novel Fractional-Order System with Hidden Dynamics, Hyperchaotic Behavior and Multi-Scroll Attractors" *Mathematics* 12, no. 14: 2227.
https://doi.org/10.3390/math12142227