# Exploring the Role of Indirect Coupling in Complex Networks: The Emergence of Chaos and Entropy in Fractional Discrete Nodes

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Theorem**

**1**

**Remark**

**1.**

## 3. Fractional Order Network with Indirect Coupling

**Remark**

**2.**

## 4. Dynamic Analysis of the Fractional Order Network with Indirect Coupling

#### Bifurcation Diagrams and Maximal Lyapunov Exponent

## 5. Complexity Analysis via Spectral Entropy

## 6. FPGA Implementation of Complex Network with Two Fractional Order Chaotic Nodes

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Temporal dynamics of the states ${x}_{1}\left(n\right)$, ${x}_{2}\left(n\right)$, ${y}_{1}\left(n\right)$, and ${y}_{2}\left(n\right)$, with initial condition sets $X\left(0\right)$, $Y\left(0\right)$ and fractional order $v=0.9$, (

**a**) with coupling $\widehat{C}=[0,\phantom{\rule{0.277778em}{0ex}}0;\phantom{\rule{0.277778em}{0ex}}0,\phantom{\rule{0.277778em}{0ex}}0]$, (

**b**) with coupling $\widehat{C}=[0,\phantom{\rule{0.277778em}{0ex}}0;\phantom{\rule{0.277778em}{0ex}}0,\phantom{\rule{0.277778em}{0ex}}1]$ and $\Phi =[0,\phantom{\rule{0.277778em}{0ex}}0;\phantom{\rule{0.277778em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}0]$, respectively.

**Figure 3.**Phase portraits of fractionalized system (15) with different fractional orders: (

**a**) fractional order $v=1$, (

**b**) fractional order $v=0.99$, (

**c**) fractional order $v=0.9$.

**Figure 4.**Bifurcation diagrams where $\eta $ acts as a critical parameter with initial conditions sets $X\left(0\right)$, $Y\left(0\right)$. (

**a**) For $v=0.99$, (

**b**) for $v=0.9$, respectively.

**Figure 5.**Bifurcation diagrams where b acts as a critical parameter with initial condition sets $X\left(0\right)$, $Y\left(0\right)$. (

**a**) For $v=0.99$, (

**b**) for $v=0.9$, respectively.

**Figure 6.**Bifurcation diagram and maximal Lyapunov exponent of the state ${x}_{1}$ by varying the parameter v in the interval $v\in [0.01,1]$ with initial condition sets $X\left(0\right)$, $Y\left(0\right)$. (

**a**) Bifurcation diagrams, (

**b**) maximal Lyapunov exponent.

**Figure 7.**Spectral entropy b acts as a critical parameter with initial conditions sets $X\left(0\right)$, $Y\left(0\right)$. (

**a**) for $v=0.99$ (

**b**) for $v=0.9$, respectively., respectively.

**Figure 9.**FPGA implementation results. (

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

**b**) the phase plane ${x}_{1}$ versus ${y}_{1}$, (

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

Device Resources | Total | Used | Percent (%) |
---|---|---|---|

Total Slices | 13,300 | 10,588 | 79.6 |

Slice Registers | 106,400 | 26,778 | 25.2 |

Slices LUTs | 53,200 | 34,283 | 64.4 |

Block RAMs | 140 | 7 | 5 |

DSP48s | 220 | 22 | 10 |

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**MDPI and ACS Style**

Zambrano-Serrano, E.; Platas-Garza, M.A.; Posadas-Castillo, C.; Arellano-Delgado, A.; Cruz-Hernández, C.
Exploring the Role of Indirect Coupling in Complex Networks: The Emergence of Chaos and Entropy in Fractional Discrete Nodes. *Entropy* **2023**, *25*, 866.
https://doi.org/10.3390/e25060866

**AMA Style**

Zambrano-Serrano E, Platas-Garza MA, Posadas-Castillo C, Arellano-Delgado A, Cruz-Hernández C.
Exploring the Role of Indirect Coupling in Complex Networks: The Emergence of Chaos and Entropy in Fractional Discrete Nodes. *Entropy*. 2023; 25(6):866.
https://doi.org/10.3390/e25060866

**Chicago/Turabian Style**

Zambrano-Serrano, Ernesto, Miguel Angel Platas-Garza, Cornelio Posadas-Castillo, Adrian Arellano-Delgado, and César Cruz-Hernández.
2023. "Exploring the Role of Indirect Coupling in Complex Networks: The Emergence of Chaos and Entropy in Fractional Discrete Nodes" *Entropy* 25, no. 6: 866.
https://doi.org/10.3390/e25060866