# Adaptive Coexistence of Synchronization and Anti-Synchronization for a Class of Switched Chaotic Systems

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## Abstract

**:**

## 1. Introduction

- Two necessary and sufficient conditions for the CSAS of the chaotic systems under a simple adaptive linear feedback controller are proposed.
- Based on the results above, two new algorithms are derived from searching the synchronization variables and anti-synchronization variables in chaotic systems.
- According to the results of the CSAS for chaotic systems, the problem of CSAS for the switched chaotic systems composed by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed which can realize CSAS of the switched chaotic systems under the arbitrary switching law.

## 2. Preliminaries

**Assumption A1.**

**Remark 1.**

**Definition 1**

**Remark 2.**

**Definition 2.**

**Remark 3.**

**Lemma 1**

**Lemma 2**

## 3. Main Results

#### 3.1. Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems

**Theorem 1.**

**Proof of Theorem 1.**

Algorithm 1: Distinguish the synchronous and anti-synchronous variables by analyzing the parity of the system expression. |

Step 1: Without loss of generality, we first select the variable ${x}_{1}$. If ${f}_{1}\left(x\right)={f}_{11}\left({x}_{1}\right)$ $+{f}_{12}({x}_{2};\cdots ;{x}_{n})$ is an odd function, or ${f}_{11}\left({x}_{1}\right)={\theta}_{1}{x}_{1}$, we can set ${E}_{1}={x}_{1}+{y}_{1}$, where ${\theta}_{1}$ is a real number; Step 2: If ${f}_{2}({x}_{2};\cdots ;{x}_{n})={\theta}_{2}{x}_{2}+{f}_{13}({x}_{3};\cdots ;{x}_{n})$, we should set ${E}_{2}={x}_{2}+{y}_{2}$, where ${\theta}_{2}$ is a real number. Else, if ${f}_{2}\left(x\right)={f}_{21}\left({x}_{2}\right)+{f}_{22}({x}_{1};{x}_{3};\cdots ;{x}_{n})$ is an odd function, or ${f}_{21}\left({x}_{2}\right)={\theta}_{3}{x}_{2}$, we can set ${E}_{2}={x}_{2}+{y}_{2}$, where ${\theta}_{3}$ is a real number. Then, we determine whether ${E}_{2}={x}_{2}+{y}_{2}$ is suitable or not according to the condition that the origin is an equilibrium point of the anti-synchronization error system; Step 3: When $i\le n$, we can set ${E}_{i}={y}_{i}+{x}_{i}$ or ${e}_{i}={y}_{i}-{x}_{i}$ by the similar procedure in Step 2. |

**Theorem 2.**

**Proof of Theorem 2.**

Algorithm 2: Distinguish the synchronous and anti-synchronous variables by decomposing the system. |

Step 1: $i=1.$ Check all elements of ${\beta}^{\left(i\right)}$ are the solutions of ${Q}^{\left(j\right)}$ or not, $2\le j\le n-1$. If one is yes, such as $\beta =diag[-1,1,\cdots ,-1]$ is a solution of ${Q}^{\left(j\right)}$, i.e., ${Q}^{\left(j\right)}=0$, then set ${w}_{m}=$ ${[{x}_{1},{x}_{3},\cdots ,{x}_{n}]}^{T}$, $z={x}_{2}$, thus the master system (1) is divided into the following two subsystems
$$\dot{x}=f\left(x\right)=\left(\begin{array}{c}{\dot{w}}_{m}\\ \dot{z}\end{array}\right)=\left(\begin{array}{c}M\left(z\right){w}_{m}\\ N({w}_{m},z)\end{array}\right).$$
Step 2: $i=2.$ Check all elements of ${\beta}^{\left(i\right)}$ are the solutions of ${Q}^{\left(j\right)}$ or not, $2\le j\le n-1$. If one is yes, such as $\beta =diag[-1,1,1,-1,\cdots ,-1]$ is a solution of ${Q}^{\left(j\right)}$, i.e., ${Q}^{\left(j\right)}=0$, then set ${w}_{m}={[{x}_{1},{x}_{4},\cdots ,{x}_{n}]}^{T}$, $z={[{x}_{2},{x}_{3}]}^{T}$, thus the master system (1) is divided into the following two subsystems
Step 3: This procedure goes on until $i=n-2$. |

**Theorem 3.**

**Proof of Theorem 3.**

**Remark 4.**

#### 3.2. Coexistence of Synchronization and Anti-Synchronization for Switched Chaotic Systems

**Theorem 4.**

**Proof of Theorem 4.**

## 4. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ren, L.; Peng, C.
Adaptive Coexistence of Synchronization and Anti-Synchronization for a Class of Switched Chaotic Systems. *Processes* **2023**, *11*, 530.
https://doi.org/10.3390/pr11020530

**AMA Style**

Ren L, Peng C.
Adaptive Coexistence of Synchronization and Anti-Synchronization for a Class of Switched Chaotic Systems. *Processes*. 2023; 11(2):530.
https://doi.org/10.3390/pr11020530

**Chicago/Turabian Style**

Ren, Ling, and Chenchen Peng.
2023. "Adaptive Coexistence of Synchronization and Anti-Synchronization for a Class of Switched Chaotic Systems" *Processes* 11, no. 2: 530.
https://doi.org/10.3390/pr11020530