# Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

^{2+}oscillatory biosignals is presented. This paper explains how those types of diagrams provide different types of information about the analyzed autonomous system and how they complement one another. Aside from robust control, which is the strategy for stabilization and synchronization used in this research study, some research studies found in the literature related to the stability and chaos synchronization of various types of chaotic systems are worth mentioning. For example, in papers such as that in [73], the synchronization patterns in Kuramoto oscillators are demonstrated, in which phase locked states with constant phase shifts between these oscillators are studied. The synchronization estimation for complex time series using cross sample entropy measure is then demonstrated in papers such as that in [74]. Other papers with interesting results include that in [75], in which a modified Chua’s circuit is used with a five segment piecewise linear Chua’s diode. This paper demonstrates that the attractors have small basins of attraction. Finally, in [76], it is shown that a tuned pendulum absorber can reduce vibration and, at the same time to harvest energy. Then, in [77], it is shown how Lagrangian descriptors can be implemented to characterize invariant tori of generic systems. Finally, in [78], the magnetic confinement of a neutral atom is presented. In this paper, a neutral atom inside a double-wire waveguide in the presence of two uniform bias fields is presented.

## 3. Definition of the Novel Chaotic System

#### 3.1. Definition of the Novel Chaotic System

#### 3.2. Bifurcation Analysis

## 4. Main Results

**Property**

**1.**

#### 4.1. Robust Stabilization of the Novel Chaotic System

**Theorem**

**1.**

**Proof.**

#### 4.2. Robust Synchronization of the Novel Chaotic System

**Theorem**

**2.**

**Proof.**

## 5. Numerical Experiments

#### 5.1. Experiment 1: Robust Stabilization of the Novel Chaotic System

#### 5.2. Experiment 2: Robust Synchronization of the Novel Chaotic System

#### 5.3. Experimental Results Analysis

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Phase portrait of the novel chaotic attractor (4).

**Figure 2.**Bifurcation diagrams of the novel chaotic system (4).

**Figure 7.**Input variable U of the novel chaotic system by the implementation of the proposed robust controller.

**Figure 10.**Evolution in time of the synchronized state variables of the drive and response novel chaotic system.

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## Share and Cite

**MDPI and ACS Style**

Azar, A.T.; Serrano, F.E.; Zhu, Q.; Bettayeb, M.; Fusco, G.; Na, J.; Zhang, W.; Kamal, N.A.
Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints. *Entropy* **2021**, *23*, 1110.
https://doi.org/10.3390/e23091110

**AMA Style**

Azar AT, Serrano FE, Zhu Q, Bettayeb M, Fusco G, Na J, Zhang W, Kamal NA.
Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints. *Entropy*. 2021; 23(9):1110.
https://doi.org/10.3390/e23091110

**Chicago/Turabian Style**

Azar, Ahmad Taher, Fernando E. Serrano, Quanmin Zhu, Maamar Bettayeb, Giuseppe Fusco, Jing Na, Weicun Zhang, and Nashwa Ahmad Kamal.
2021. "Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints" *Entropy* 23, no. 9: 1110.
https://doi.org/10.3390/e23091110