# Outer Synchronization between Fractional-Order Complex Networks: A Non-Fragile Observer-based Control Scheme

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Network Model

#### 2.1. Basic Concepts and Lemmas

**Lemma 2.2**

**Lemma 2.3**

**Lemma 2.4**

**Lemma 2.5**

**Lemma 2.6**

**(1)**- $\mathit{S}=\left[\begin{array}{cc}{\mathit{S}}_{11}& {\mathit{S}}_{12}\\ {\mathit{S}}_{21}& {\mathit{S}}_{22}\end{array}\right]<0$;
**(2)**- ${\mathit{S}}_{11}<0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathit{S}}_{22}-{\mathit{S}}_{12}^{T}{\mathit{S}}_{11}^{-1}{\mathit{S}}_{12}<0$;
**(3)**- ${\mathit{S}}_{22}<0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathit{S}}_{11}-{\mathit{S}}_{12}{\mathit{S}}_{22}^{-1}{\mathit{S}}_{12}^{T}<0$.

#### 2.2. Network Model

**Type 1:**$\Delta \mathit{L}\left(t\right)$ is with the norm-bounded additive form:

**Type 2:**$\Delta \mathit{L}\left(t\right)$ is with the norm-bounded multiplicative form:

## 3. Global Outer Synchronization Analysis

**Theorem 3.1.**

**Theorem 3.4.**

**Theorem 3.5.**

**Corollary 3.6.**

## 4. Numerical Simulations

#### 4.1. Outer Synchronization between Two FCNs with Nearest-Neighbor Network Topology

**Figure 1.**Chaotic behavior of the fractional-order jerk model (29). The fractional orders are: (

**a**) $\alpha =1$ and (

**b**) $\alpha =0.95$.

**Figure 2.**Synchronization errors between the FCNs (6) and (7), where each node is a chaotic fractional-order jerk model (29). (

**a**) The time evolutions of ${e}_{i1}\left(t\right)={x}_{i1}\left(t\right)-{\widehat{x}}_{i1}\left(t\right)$; (

**b**) the time evolutions of ${e}_{i2}\left(t\right)={x}_{i2}\left(t\right)-{\widehat{x}}_{i2}\left(t\right)$; (

**c**) the time evolutions of ${e}_{i3}\left(t\right)={x}_{i3}\left(t\right)-{\widehat{x}}_{i3}\left(t\right)\phantom{\rule{0.166667em}{0ex}}(i=1,2,\cdots ,10)$.

#### 4.2. Outer Synchronization between Two FCNs with Small-World Network Topology

**Figure 3.**Chaotic behavior of the fractional-order Duffing oscillator (34). The fractional orders are: (

**a**) $\alpha =1$ and (

**b**) $\alpha =0.98$.

**Figure 4.**Synchronization errors between the FCNs (6) and (7), where each node is a chaotic fractional-order Duffing oscillator (34). (

**a**) The time evolutions of ${e}_{i1}\left(t\right)={x}_{i1}\left(t\right)-{\widehat{x}}_{i1}\left(t\right)$; (

**b**) the time evolutions of ${e}_{i2}\left(t\right)={x}_{i2}\left(t\right)-{\widehat{x}}_{i2}\left(t\right)\phantom{\rule{0.166667em}{0ex}}(i=1,2,\cdots ,100)$.

## 5. Conclusions

## Acknowledgements

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Zhao, M.; Wang, J.
Outer Synchronization between Fractional-Order Complex Networks: A Non-Fragile Observer-based Control Scheme. *Entropy* **2013**, *15*, 1357-1374.
https://doi.org/10.3390/e15041357

**AMA Style**

Zhao M, Wang J.
Outer Synchronization between Fractional-Order Complex Networks: A Non-Fragile Observer-based Control Scheme. *Entropy*. 2013; 15(4):1357-1374.
https://doi.org/10.3390/e15041357

**Chicago/Turabian Style**

Zhao, Meichun, and Junwei Wang.
2013. "Outer Synchronization between Fractional-Order Complex Networks: A Non-Fragile Observer-based Control Scheme" *Entropy* 15, no. 4: 1357-1374.
https://doi.org/10.3390/e15041357