# Hybrid Synchronization Problem of a Class of Chaotic Systems by an Universal Control Method

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

**Notation**Throughout this paper, as usual, $\mathbb{R}$ stands for the real number, ${I}_{n}$ represents an $n\times n$ identity matrix, $\mathsf{\Lambda}=\{1,\cdots ,n\}$ is an index set, $\alpha =\mathrm{Diag}\{{\alpha}_{1},\cdots ,{\alpha}_{n}\}$ is a diagonal matrix, where ${\alpha}_{i}\ne 0$, and $u=K\left(e\right)\in {\mathbb{R}}^{n}$ is a controller, where $K(.)$ is a continuous vector function with $K\left(0\right)$ = 0.

## 2. Preliminaries and Problem Formation

**Definition**

**1.**

- If $\underset{t\to \infty}{lim}\parallel e\left(t\right)\parallel =0$ holds for the case: some ${\alpha}_{i}$ = 1, while the rest ${\alpha}_{j}$ = −1, $i\ne j\in \mathsf{\Lambda}$, then the system (6) and the system (7) are called to achieve the coexistence of synchronization and anti-synchronization, i.e., some variables: ${x}_{i}$, anti-synchronize the corresponding variables: ${y}_{i},i\in \mathsf{\Lambda}$, while the rest variables: ${x}_{i}$, synchronize the corresponding variables: ${y}_{j},j\in \mathsf{\Lambda}$, $i\ne j$.
- If $\underset{t\to \infty}{lim}\parallel e\left(t\right)\parallel =0$ holds simultaneously for the two cases: $\alpha ={I}_{n}$ and $\alpha =-{I}_{n}$, then the system (6) and the system (7) are called to achieve simultaneous synchronization and anti-synchronization, i.e., if the system (6) and the system (7) are anti-synchronized by the controller in this form: $u=K\left(e\right)$, then these two systems are also synchronized by the controller in this form: $u=K\left(e\right)$, vice versa.

## 3. Main Results

#### 3.1. The Existence of the Hybrid Synchronization Problem for a Class of Chaotic Systems

**Theorem**

**1.**

**Proof:**

**Remark**

**3.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

**Corollary**

**5.**

**Remark**

**4.**

**Corollary**

**6.**

**Theorem**

**2.**

**Proof:**

**Theorem**

**3.**

**Proof:**

#### 3.2. Solutions of the Hybrid Synchronization Problem for a Given Chaotic System

#### 3.3. The Implementation of the Hybrid Synchronization for the Given Chaotic Systems

**Theorem**

**4.**

## 4. Examples with Numerical Simulations

**Example**

**1.**

**Remark**

**5.**

**Example**

**2.**

**Remark**

**6.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, Z.; Guo, R.
Hybrid Synchronization Problem of a Class of Chaotic Systems by an Universal Control Method. *Symmetry* **2018**, *10*, 552.
https://doi.org/10.3390/sym10110552

**AMA Style**

Wang Z, Guo R.
Hybrid Synchronization Problem of a Class of Chaotic Systems by an Universal Control Method. *Symmetry*. 2018; 10(11):552.
https://doi.org/10.3390/sym10110552

**Chicago/Turabian Style**

Wang, Zuoxun, and Rongwei Guo.
2018. "Hybrid Synchronization Problem of a Class of Chaotic Systems by an Universal Control Method" *Symmetry* 10, no. 11: 552.
https://doi.org/10.3390/sym10110552