# A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems

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## Abstract

**:**

## 1. Introduction

**main**aim of this work is to design a new robust scheme by which two important types of chaotic synchronization (i.e., chaotic complete synchronization and chaotic phase synchronization) can be achieved based on a single-state feedback track synchronization control algorithm using the same controller. In addition, the robustness of the proposed controller is proved using a genetic algorithm. A new secure communication plan is also developed relying on the proposed robust single-state feedback track synchronization method as an application. The track control method is applied only to control chaotic systems [25]. To the best of our knowledge, this is the first time it has been applied in chaos synchronization and it is the first time two types of synchronization using the same controller have been achieved. In addition, phase synchronization is accomplished via a single-state feedback track controller that is less costly than feeding back all system states.

## 2. Track Synchronization Algorithm

- Construct the synchronization errors system ${w}_{i}={y}_{i}-{x}_{i}$.
- Construct the track synchronization control signals ${u}_{i}$.
- Construct the track synchronization control errors system ${e}_{i}={w}_{i}-{w}_{ir}$.
- Apply stability criteria to calculate the range of the controller such that the track errors system is asymptotically stable.
- If the reference values ${w}_{ir}=0$ then a complete synchronization will be reached while a phase synchronization is accomplished if ${w}_{ir}={y}_{i}\left(0\right)-{x}_{i}\left(0\right)$.

## 3. Implementation of the Novel Track Synchronization Controller

**Theorem**

**1.**

**Proof.**

#### Numerical Simulation of Tracking Synchronization of Two Identical Lorenz Systems

**Case 1. Complete synchronization**

**Case 2. Phase synchronization**

## 4. Robust Track Synchronization for Two Identical Lorenz Chaotic Systems

#### 4.1. Designing a Robust Tracking Synchronization Control Algorithm

**GA Track synchronization control procedure.**Select optimal gains ${k}_{1}$, ${k}_{2}$ and ${k}_{3}$ using GA algorithm.

- (1)
- Determine the dominant frequencies via FFT of the suggested measurable signal ${w}_{2}={y}_{2}-{x}_{2}.$
- (2)
- Design the fitness function as in (24).
- (3)
- The nonlinear constraints are set up as $Re\left({\lambda}_{i}\right)\le -\theta $, where $\left\{{\lambda}_{i}\left|det({\lambda}_{i}I-B)=0\right.\right\},$ $i=1$, 2, 3, and $\theta $ is a chosen positive real angle.
- (4)
- Utilize Matlab’s GA toolbox to search for the optimal gains $({k}_{1},\phantom{\rule{3.33333pt}{0ex}}{k}_{2},\phantom{\rule{3.33333pt}{0ex}}{k}_{3})$.

#### 4.2. Numerical Simulation for Robust Track Synchronization Control

**Remark**

**1.**

## 5. Secure Communication Application via Robust Track Synchronization Control Results

**(or more)**of the three state variables, with any pair of their products or with their product. For illustration purposes, after adding this to the state variable ${x}_{3}$ the encrypted message becomes $\Gamma \left(t\right)=\Theta +{x}_{3}$. After that, the chaotic carrying signal of the sender device is produced and the encrypted message is transferred to the recipient end. When the message is sent there is strong noise that will intercept the message as well as the signals of the transmitter system. Therefore, if the synchronization used in making this application is not robust, this could affect message extraction. At the receiver side, we construct the controller as proposed in Equation (20). As a result, the synchronization between the drive (master) (4) and the identical controlled (5) frameworks will occur after a certain time ${t}_{s}$ and the difference between the master system x and the slave system states y will be zero in a specific period ${t}_{c}$ that is a time period longer than the time period ${t}_{s}$ (${t}_{s}$ is the time period after which the synchronization process starts). For illustration purposes take ${t}_{c}=5$ s, recovering $\Gamma \left(t\right)$ via a straightforward mutation $\Theta =\Gamma \left(t\right)-{y}_{3}$ that starts at the receiver end. At the end, the sent data can be retrieved via an inverting operation ${r}^{*}\left(t\right)={\Phi}^{-1}({y}_{1},{y}_{2},{y}_{3},\Theta )$ because the nonlinear function $\Phi $ is invertible.

## 6. Conclusions

- Design of a single-state feedback track synchronization controller to accomplish two types of synchronization between two identical Lorenz chaotic systems.
- The complete and phase synchronizations of two identical Lorenz chaotic systems are achieved via the same controller, which has not been done before.
- The single-state feedback strategy is less costly and its realization is simple since it uses fewer sensors.
- The powerful GA algorithm has been combined with our proposed strategy to compute the suitable gains of the proposed synchronization controller.
- More complex chaotic systems will likely require more complex treatment of parameters to fit between systems to achieve a decent measure of synchronization. This can be seen from the condition in Theorem 1 required to measure the degree of synchronization.
- The proposed method has been successfully applied in a secure communication application.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Dynamics of master (x) and slave (y) Lorenz chaotic systems where ${x}_{0}=(10,\phantom{\rule{3.33333pt}{0ex}}5,\phantom{\rule{3.33333pt}{0ex}}-15)$, ${y}_{0}=(1,\phantom{\rule{3.33333pt}{0ex}}1,\phantom{\rule{3.33333pt}{0ex}}-1)$ and $(\sigma ,\phantom{\rule{3.33333pt}{0ex}}r,\phantom{\rule{3.33333pt}{0ex}}b)=(10,\phantom{\rule{3.33333pt}{0ex}}28,\phantom{\rule{3.33333pt}{0ex}}8/3)$.

**Figure 3.**Proposed tracking synchronization control signals: ${u}_{1},\phantom{\rule{3.33333pt}{0ex}}{u}_{2},\phantom{\rule{3.33333pt}{0ex}}{u}_{3}.$

**Figure 4.**(

**a**) Complete synchronization errors: ${w}_{i}={w}_{ir}=0,$ where $i=1,\phantom{\rule{3.33333pt}{0ex}}2,\phantom{\rule{3.33333pt}{0ex}}3$. (

**b**) Complete synchronization track control errors: ${e}_{1},\phantom{\rule{3.33333pt}{0ex}}{e}_{2},\phantom{\rule{3.33333pt}{0ex}}{e}_{3}$.

**Figure 5.**Phase synchronization errors and synchronization track control errors. (

**a**) Phase synchronization errors: ${w}_{1}={w}_{1r}={y}_{1}\left(0\right)-{x}_{1}\left(0\right)=-9,\phantom{\rule{3.33333pt}{0ex}}{w}_{2}={w}_{2r}={y}_{2}\left(0\right)-{x}_{2}\left(0\right)=-4,\phantom{\rule{3.33333pt}{0ex}}{w}_{3}={w}_{3r}={y}_{3}\left(0\right)-{x}_{3}\left(0\right)=14.$ (

**b**) Synchronization track control errors: ${e}_{1},\phantom{\rule{3.33333pt}{0ex}}{e}_{2},\phantom{\rule{3.33333pt}{0ex}}{e}_{3}$.

**Figure 6.**Phase differences of MPPS of models (4) and (5) are finite and can move in a chaotic style: (

**a**) ${\varphi}_{m}^{{x}_{1},{x}_{2}}-{\varphi}_{s}^{{y}_{1},{y}_{2}}$ against t, (

**b**) ${\varphi}_{m}^{{x}_{1},{x}_{3}}-{\varphi}_{s}^{{y}_{1},{y}_{3}}$ against t and (

**c**) the chaotic attractor of ${\varphi}_{m}^{{x}_{1},{x}_{2}}-{\varphi}_{s}^{{y}_{1},{y}_{2}}$ and ${\varphi}_{m}^{{x}_{1},{x}_{3}}-{\varphi}_{s}^{{y}_{1},{y}_{3}}.$

**Figure 10.**Synchronization robust track control errors where synchronization track controller gains are ${k}_{1}=-0.05;\phantom{\rule{3.33333pt}{0ex}}{k}_{2}=-200;\phantom{\rule{3.33333pt}{0ex}}{k}_{3}=-1.$

**Figure 11.**Synchronization robust track control errors where synchronization track controller gains are ${k}_{1}=-1;\phantom{\rule{3.33333pt}{0ex}}{k}_{2}=-100;\phantom{\rule{3.33333pt}{0ex}}{k}_{3}=-2.$

**Figure 13.**Secure communication simulation results. (

**a**) $r\left(t\right)$ is the authentic message. (

**b**) $\Gamma \left(t\right)$ is the transferred signal. (

**c**) ${r}^{*}\left(t\right)$ is the retrieved message. (

**d**) $r\left(t\right)-{r}^{*}\left(t\right)$ is the error signal.

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**MDPI and ACS Style**

Mahmoud, E.E.; Higazy, M.; Althagafi, O.A.
A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems. *Symmetry* **2020**, *12*, 1765.
https://doi.org/10.3390/sym12111765

**AMA Style**

Mahmoud EE, Higazy M, Althagafi OA.
A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems. *Symmetry*. 2020; 12(11):1765.
https://doi.org/10.3390/sym12111765

**Chicago/Turabian Style**

Mahmoud, Emad E., M. Higazy, and Ohood A. Althagafi.
2020. "A Novel Strategy for Complete and Phase Robust Synchronizations of Chaotic Nonlinear Systems" *Symmetry* 12, no. 11: 1765.
https://doi.org/10.3390/sym12111765