# Master–Slave Outer Synchronization in Different Inner–Outer Coupling Network Topologies

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries in Complex Dynamical Networks

## 3. Master Stability Function

## 4. MACM Chaotic System

## 5. Analysis of Master–Slave Inner–Outer Coupling Network Topologies

#### 5.1. Inner Topology of Ring, Star, and Small-World Networks in Master–Slave Configuration

#### 5.2. Outer Topology of Ring, Star, and Small-World Networks in Master–Slave Configuration

#### 5.3. A Big Network in Inner–Outer Network Coupling Topology $S-S$ in Master–Slave Configuration

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphic representation of ring network: (

**a**) inner ring topology in master–slave configuration and (

**b**) inner–outer ring coupling topology in master–slave configuration.

**Figure 2.**Graphic representation of star network: (

**a**) inner star topology in master–slave configuration and (

**b**) inner–outer star coupling topology in master–slave configuration.

**Figure 3.**Graphic representation of small-world network: (

**a**) inner small-world topology in master–slave configuration and (

**b**) inner–outer small-world coupling topology in master–slave configuration.

**Figure 4.**Phase planes of the MACM chaotic system (8) with $a=2$, $b=2$, $c=0.5$, and $d=10$: (

**a**) ${x}_{1}$ versus ${x}_{2}$ phase plane; (

**b**) ${x}_{1}$ versus ${x}_{3}$ phase plane; (

**c**) ${x}_{2}$ versus ${x}_{3}$ phase plane.

**Figure 5.**LEs for MACM chaotic system (8) with $b=2$ and $c=0.5$ for: (

**a**) $0\le a\le 10$ and (

**b**) $0\le b\le 10$ .

**Figure 6.**Maximum Lyapunov exponent ${\lambda}_{max}$ for $0\le {c}_{1}\le 3$ and different values of $\mathbf{\Gamma}$: (

**a**) inner ring topology, (

**b**) inner star topology, and (

**c**) inner small-world topology.

**Figure 7.**Maximum Lyapunov exponent ${\lambda}_{max}$ for ${c}_{1}=2$, $0\le {c}_{2}\le 5$ and ${\mathbf{\Gamma}}_{o1}=diag[1,0,0,0,0]$, ${\mathbf{\Gamma}}_{o2}=diag[1,1,0,0,0]$, ${\mathbf{\Gamma}}_{o3}=diag[1,1,1,0,0]$, ${\mathbf{\Gamma}}_{o4}=diag[1,1,1,1,0]$, ${\mathbf{\Gamma}}_{o5}=diag[1,1,1,1,1]$: (

**a**) $R-R$, (

**b**) $R-S$, (

**c**) $R-SW$, (

**d**) $S-R$, (

**e**) $S-S$, (

**f**) $S-SW$, (

**g**) $SW-R$, (

**h**) $SW-S$, and (

**i**) $SW-SW$.

**Figure 8.**Maximum Lyapunov exponent ${\lambda}_{max}$ for ${c}_{1}$ versus ${c}_{2}$ applying different inner–outer topologies with $\mathbf{\Gamma}=diag[1,0,1]$ and ${\mathbf{\Gamma}}_{o}=diag[1,0,0,0,0]$.

**Figure 9.**Inner–outer coupling network topology $S-S$ with $N=5$ and $M=100$. Temporal dynamics (

**a**) ${x}_{i1}$, (

**b**) ${x}_{i2}$, and (

**c**) ${x}_{i3}$, and errors between the master node and the other nodes in the network; (

**d**) ${x}_{11}-{x}_{i1}$, (

**e**) ${x}_{12}-{x}_{i2}$, and (

**f**) ${x}_{13}-{x}_{i3}$.

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**MDPI and ACS Style**

Arellano-Delgado, A.; López-Gutiérrez, R.M.; Murillo-Escobar, M.Á.; Posadas-Castillo, C.
Master–Slave Outer Synchronization in Different Inner–Outer Coupling Network Topologies. *Entropy* **2023**, *25*, 707.
https://doi.org/10.3390/e25050707

**AMA Style**

Arellano-Delgado A, López-Gutiérrez RM, Murillo-Escobar MÁ, Posadas-Castillo C.
Master–Slave Outer Synchronization in Different Inner–Outer Coupling Network Topologies. *Entropy*. 2023; 25(5):707.
https://doi.org/10.3390/e25050707

**Chicago/Turabian Style**

Arellano-Delgado, Adrian, Rosa Martha López-Gutiérrez, Miguel Ángel Murillo-Escobar, and Cornelio Posadas-Castillo.
2023. "Master–Slave Outer Synchronization in Different Inner–Outer Coupling Network Topologies" *Entropy* 25, no. 5: 707.
https://doi.org/10.3390/e25050707