# Outer Topology Network Synchronization Using Chaotic Nodes with Hidden Attractors

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## Abstract

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## 1. Introduction

## 2. Complex Dynamical Networks

## 3. Master Stability Function

## 4. Chaotic Node

## 5. Synchronization of Inner and Outer Coupling Topologies

#### 5.1. Inner Topologies of the Ring, Star, and Small-World Networks Synchronization

#### 5.2. Outer Topology of Ring, Star, and Small-World Networks Synchronization

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Phase planes of the chaotic system (9) with $a=1.35$ (

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

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

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

**Figure 5.**Phase planes of the chaotic system (9) with $a=1.4$ and initial conditions $({x}_{1},{x}_{2},{x}_{3})=(0,0.1,0)$ (blue), and $({x}_{1},{x}_{2},{x}_{3})=(0,-0.1,0)$ (red) (

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

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

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

**Figure 6.**The maximum Lyapunov exponent ${\lambda}_{max}$ with respect to ${c}_{1}$ for inner coupling network topologies R, S, and $SW$.

**Figure 7.**The maximum Lyapunov exponents ${\lambda}_{max}$ for the inner–outer coupling topologies R-R, R-S, R-$SW$, S-R, S-S, S-$SW$, $SW$-R, $SW$-S, and $SW$-$SW$ for ${c}_{1}=0.5$ and $0<{c}_{2}<2$.

**Figure 8.**The maximum Lyapunov exponent ${\lambda}_{max}$ for the inner–outer coupling topologies R-R, R-S, R-$SW$, S-R, S-S, S-$SW$, $SW$-R, $SW$-S, and $SW$-$SW$, considering a sweep of ${c}_{1}$ versus ${c}_{2}$.

**Figure 9.**Temporal dynamics of the inner–outer network coupling topology R-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 10.**Synchronization error of the inner–outer network coupling topology R-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 11.**Temporal dynamics of the inner–outer network coupling topology S-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 12.**Synchronization error of the inner–outer network coupling topology S-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 13.**Temporal dynamics of the inner–outer network coupling topology $SW$-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 14.**Synchronization error of the inner–outer network coupling topology $SW$-$SW$ for different values of ${c}_{1}$ and ${c}_{2}$, where the colors are to differentiate the states of each node.

**Figure 15.**The maximum Lyapunov exponent ${\lambda}_{max}$ for the inner–outer coupling topologies S-$SW$, and $SW$-$SW$ for $N=M=24$.

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**MDPI and ACS Style**

Villalobos-Aranda, C.A.; Arellano-Delgado, A.; Zambrano-Serrano, E.; Pliego-Jiménez, J.; Cruz-Hernández, C.
Outer Topology Network Synchronization Using Chaotic Nodes with Hidden Attractors. *Axioms* **2023**, *12*, 634.
https://doi.org/10.3390/axioms12070634

**AMA Style**

Villalobos-Aranda CA, Arellano-Delgado A, Zambrano-Serrano E, Pliego-Jiménez J, Cruz-Hernández C.
Outer Topology Network Synchronization Using Chaotic Nodes with Hidden Attractors. *Axioms*. 2023; 12(7):634.
https://doi.org/10.3390/axioms12070634

**Chicago/Turabian Style**

Villalobos-Aranda, Carlos Andrés, Adrian Arellano-Delgado, Ernesto Zambrano-Serrano, Javier Pliego-Jiménez, and César Cruz-Hernández.
2023. "Outer Topology Network Synchronization Using Chaotic Nodes with Hidden Attractors" *Axioms* 12, no. 7: 634.
https://doi.org/10.3390/axioms12070634