# Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication

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## Abstract

**:**

## 1. Introduction

- -
- The computation of the fractional order in the microcontroller needs memory allocation to compute the cumulative integration; this limitation in the memory of the atmega328p microcontroller was solved by using the numerical approximation method, Rung Kutta 4.
- -
- The noise produced by the electronics components, which negatively affects the quality of the transmitted and received data, was solved in two ways: two capacitors were added to the circuit to enhance the NRF24LO1 module’s performance, and a stochastic filter was implemented to filter the estimated input (message) in the slave part.
- -
- The optimal SBS-SMO parameters were chosen using HHO.

## 2. Fundamental on Fractional Calculus

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

## 3. The Transmitter System’s Configuration

#### The Inclusion of Private Information

**Remark**

**1.**

## 4. SBS-SMO Theory

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**2.**

**Theorem**

**1.**

## 5. The Receiver System’s Configuration

## 6. Selection of the Observer’s Optimal Parameters via HHO

## 7. Results and Discussion

#### 7.1. Retrieving the Secret Message

#### 7.2. Security Analysis and Check

#### 7.3. Experimental Validation

## 8. Conclusions

- The type of chaotic system utilised, such as Lorenz, Rossler, Chua’s, and Chen, among others.
- The dynamic parameters of the chaotic system.
- The order and initial conditions of the system, as well as the state in which the message was embedded.
- The mechanism of the synchronisation process, such as a controller or observer, and the specific type of controller or observer.
- The characteristics of the encrypted and decrypted data, such as text, images, voice, and video.

- Transmitting secure images instead of voice or signal data.
- Improving fractional Chua’s systems through real-time implementation of new 4D or 6D fractional chaotic systems.
- Replacing the synchronisation observer with advanced and more recent algorithms such as deep learning and machine learning approaches.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 13.**The original message (sinusoidal wave $m\left(t\right)=sin\left(t\right)$) and the recovered $\widehat{m}\left(t\right)$.

**Figure 14.**The original message ($m\left(t\right)=$ square(t)) and the recovered $\widehat{m}\left(t\right)$.

**Figure 15.**The original message ($m\left(t\right)=$ sawtooth(t)) and the recovered $\widehat{m}\left(t\right)$.

**Figure 16.**The original message ($m\left(t\right)=$ voice) and the recovered $\widehat{m}\left(t\right)$.

**Figure 17.**The original message $m=$ voice and the recovered $\widehat{m}\left(t\right)$ with initial conditions $x\left(0\right)={10}^{-6},\phantom{\rule{3.33333pt}{0ex}}y\left(0\right)=0,\phantom{\rule{3.33333pt}{0ex}}z\left(0\right)=0$, and $(\widehat{x}\left(0\right)=\widehat{y}\left(0\right)=\widehat{z}\left(0\right)=0)$. The order for both the transmitter and receiver is ${q}_{1}={q}_{2}={q}_{3}=0.9$.

**Figure 18.**The original message $m=voice$ and the recovered $\widehat{m}\left(t\right)$ with initial conditions $\left(x\right(0)=y(0)=z(0)=0)$ and $(\widehat{x}\left(0\right)=\widehat{y}\left(0\right)=\widehat{z}\left(0\right)=0)$. The orders for the transmitter are ${q}_{1}=0.905,{q}_{2}={q}_{3}=0.9$, whereas those for the receiver are set to $0.9$.

**Table 1.**Parameter values of the model given in Equation (4).

Parameter | ${\mathit{q}}_{1}={\mathit{q}}_{2}={\mathit{q}}_{3}$ | $\mathsf{\alpha}$ | $\mathsf{\beta}$ | $\mathsf{\gamma}$ | ${\mathsf{\delta}}_{1}$ | ${\mathsf{\delta}}_{2}$ | ${\mathsf{\delta}}_{3}$ | R | ${\mathit{i}}_{0}$ |
---|---|---|---|---|---|---|---|---|---|

Value | 0.9 | 1.5 | −1 | 0.0035 | 100 | 1 | 1 | 0.1 | 0.0005 |

Parameters | Values |
---|---|

Search agents | 15 |

Max iteration | 50 |

Parameters [μ_{1} μ_{2} μ_{3}] | $[185.3,80.1,145.08]$ |

Methods | Sin Message | Square Message | Sawtooth Message | Voice Message |
---|---|---|---|---|

Proposed | $7.5211\times {10}^{-5}$ | $2.1165\times {10}^{-4}$ | $2.0181\times {10}^{-5}$ | $6.2521\times {10}^{-4}$ |

[26] | $8.2397\times {10}^{-4}$ | $0.0024$ | $0.0022$ | $0.0069$ |

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**MDPI and ACS Style**

Nail, B.; Atoussi, M.A.; Saadi, S.; Tibermacine, I.E.; Napoli, C.
Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication. *Fractal Fract.* **2024**, *8*, 104.
https://doi.org/10.3390/fractalfract8020104

**AMA Style**

Nail B, Atoussi MA, Saadi S, Tibermacine IE, Napoli C.
Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication. *Fractal and Fractional*. 2024; 8(2):104.
https://doi.org/10.3390/fractalfract8020104

**Chicago/Turabian Style**

Nail, Bachir, Mahedi Abdelghani Atoussi, Slami Saadi, Imad Eddine Tibermacine, and Christian Napoli.
2024. "Real-Time Synchronisation of Multiple Fractional-Order Chaotic Systems: An Application Study in Secure Communication" *Fractal and Fractional* 8, no. 2: 104.
https://doi.org/10.3390/fractalfract8020104