# Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network

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## Abstract

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## 1. Introduction

## 2. Mathematical Model and Equilibrium Point Studies

#### 2.1. Mathematical Model

#### 2.2. The Equilibrium Points of the Small Heterogeneous Coupled Network

## 3. Numerical Simulation

#### 3.1. Firing Activity of the Small Heterogeneous Coupled Network

#### 3.2. Synchronization Behavior of the Small Heterogeneous Coupled Network

## 4. Circuit Simulation and Hardware Implementation

#### 4.1. Circuit Simulation

#### 4.2. Microcontroller Implementation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HR | Hindmarsh–Rose |

HNN | Hopfield neural network |

2D | Two dimensional |

## References

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**Figure 3.**Bifurcation diagram and the two largest Lyapunov exponents of the coupled network controlled by ${k}_{1}$, with initial states (0.1, 0, 0, 0, 0.1). (

**a**) Bifurcation diagram; (

**b**) Lyapunov exponents diagram.

**Figure 4.**Multiple periodic burstings with different spikes of a small heterogeneous coupled network controlled by ${k}_{1}$, and initial states (0.1, 0, 0, 0, 0.1). (

**a**) period-3 bursting with ${k}_{1}$ = 0.75; (

**b**) period-4 bursting with ${k}_{1}$ = 0.84; (

**c**) period-5 bursting with ${k}_{1}$ = 0.91; (

**d**) Period-6 bursting with ${k}_{1}$ = 0.99; (

**e**) period-7 bursting with ${k}_{1}$ = 1.083.

**Figure 5.**The firing patterns of the membrane potential in a small heterogeneous coupled network controlled by ${k}_{1}$, with initial states (0.1, 0, 0, 0, 0.1). (

**a**) Periodic spiking mode with ${k}_{1}$ = 0.63; (

**b**) chaotic spiking mode with ${k}_{1}$ = 0.71; (

**c**) stochastic bursting mode with ${k}_{1}$ = 1.178; (

**d**) chaotic bursting mode with ${k}_{1}$ = 1.39; (

**e**) periodic bursting mode with ${k}_{1}$ = 1.65.

**Figure 6.**The phase diagrams of the coupled network controlled by ${k}_{1}$, with initial states (0.1, 0, 0, 0, 0.1). (

**a**) Periodic spiking mode with ${k}_{1}$ = 0.63; (

**b**) chaotic spiking mode with ${k}_{1}$ = 0.71; (

**c**) stochastic bursting mode with ${k}_{1}$ = 1.178; (

**d**) chaotic bursting mode with ${k}_{1}$ = 1.39; (

**e**) periodic bursting mode with ${k}_{1}$ = 1.65.

**Figure 7.**Coexistence behavior controlled by initial value, and ${k}_{1}$ = 0.83. (

**a**) The coexisting attractor phase diagram of the network; (

**b**) the basin of attraction for the coexisting behavior.

**Figure 8.**The phase synchronization in a small heterogeneous coupled network controlled by ${k}_{1}$, with initial states (0.1, 0, 0, 0, 0.1). (

**a**) ${k}_{1}$ = 0.5; (

**b**) ${k}_{1}$ = 0.8; (

**c**) ${k}_{1}$ = 1.5; (

**d**) ${k}_{1}$ = 1.8; (

**e**) ${k}_{1}$ = 1.96; (

**f**) ${k}_{1}$ = 1.98.

**Figure 11.**Simulation results of the small heterogeneous coupled network. (

**a**). ${k}_{1}$ = 0.63; (

**b**). ${k}_{1}$ = 0.71; (

**c**). ${k}_{1}$ = 1.178; (

**d**). ${k}_{1}$ = 1.39; (

**e**). ${k}_{1}$ = 1.65.

**Figure 14.**Microcontroller implementation of the small heterogeneous coupled network. (

**a**). ${k}_{1}$ = 0.63; (

**b**). ${k}_{1}$ = 0.71; (

**c**). ${k}_{1}$ = 1.178; (

**d**). ${k}_{1}$ = 1.39; (

**e**). ${k}_{1}$ = 1.65.

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**MDPI and ACS Style**

Wang, M.; Peng, J.; He, S.; Zhang, X.; Iu, H.H.-C.
Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network. *Fractal Fract.* **2023**, *7*, 818.
https://doi.org/10.3390/fractalfract7110818

**AMA Style**

Wang M, Peng J, He S, Zhang X, Iu HH-C.
Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network. *Fractal and Fractional*. 2023; 7(11):818.
https://doi.org/10.3390/fractalfract7110818

**Chicago/Turabian Style**

Wang, Mengjiao, Jiwei Peng, Shaobo He, Xinan Zhang, and Herbert Ho-Ching Iu.
2023. "Phase Synchronization and Dynamic Behavior of a Novel Small Heterogeneous Coupled Network" *Fractal and Fractional* 7, no. 11: 818.
https://doi.org/10.3390/fractalfract7110818