# Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma 1**.

- (I)
- Asymptotically stable if and only if:$$\begin{array}{c}\hfill |arg({\lambda}_{i}\left(A\right)\left)\right|>\alpha \pi /2,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(i=1,2,\dots ,n),\end{array}$$
- (II)
- Stable if and only if:$$\begin{array}{c}\hfill |arg({\lambda}_{i}\left(A\right)\left)\right|\ge \alpha \pi /2,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(i=1,2,\dots ,n),\end{array}$$

## 3. The Scheme of Generalized Combination Complex Synchronization

#### 3.1. The Definition of Generalized Combination Complex Synchronization

**Remark 1**.

**Definition 1**.

**Remark 2**.

**Remark 3**.

#### 3.2. A General Method for Generalized Combination Complex Synchronization

**Theorem 1**.

**Proof**.

**Corollary 1**.

**Corollary 2**.

**Corollary 3**.

**Corollary 4**.

**Corollary 5**.

## 4. Numerical Simulations

#### 4.1. Synchronization among Three Fractional-Order Chaotic Complex Systems

#### 4.2. Synchronization between Two Fractional-Order Hyperchaotic Real Drive Systems and a Fractional-Order Chaotic Complex Response System

#### 4.3. Synchronization between Two Fractional-Order Chaotic Complex Drive Systems and a Fractional-Order Hyperchaotic Real Response System

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Jiang, C.; Liu, S.; Wang, D.
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems. *Entropy* **2015**, *17*, 5199-5217.
https://doi.org/10.3390/e17085199

**AMA Style**

Jiang C, Liu S, Wang D.
Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems. *Entropy*. 2015; 17(8):5199-5217.
https://doi.org/10.3390/e17085199

**Chicago/Turabian Style**

Jiang, Cuimei, Shutang Liu, and Da Wang.
2015. "Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems" *Entropy* 17, no. 8: 5199-5217.
https://doi.org/10.3390/e17085199