# Kuramoto Model with Delay: The Role of the Frequency Distribution

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## Abstract

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## 1. Introduction

## 2. Model

## 3. Reduction of the Collective Dynamics

## 4. Studying the Role of the Coupling Delay

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The dynamics of the system in the asynchronous (

**a**,

**b**) and synchronous (

**c**,

**d**) regimes. The top panels show the time traces of 10 randomly chosen phases ${\theta}_{j}$, while the bottom panels show the time trace of the Kuramoto order parameter $\left|R\right|$. The coupling strength $K=1$ for (

**a**,

**b**) and $K=4$ for (

**c**,

**d**) The other parameters are $N=1000$, $n=1$, $\Omega =1$, $\Delta =1$, $\tau =0$.

**Figure 2.**The dependence of the Kuramoto order parameter on the coupling strength. Black circles: $\tau =0$, gray squares: $\tau =1.5$. The other parameters: $N=1000$, $n=1$, $\Omega =3$, $\Delta =1$. Solid lines indicate the results obtained by the simulation of the reduced system (17).

**Figure 3.**(

**a**) The Andronov-Hopf bifurcation curve for system (17) with $n=1$, $a=0$, $\Omega =3$ and $\Delta =0.1$. (

**b**) The same curve (black solid line) and its reappearing instances (black dashed lines). The gray thick line shows the synchronization border.

**Figure 4.**(

**a**) The synchronization borders of system (17) for $n=1$ (dotted line), $n=2$ (dash-dotted line) and $n=5$ (solid line). The mean frequency $\Omega =3$, the half-width $\Delta =\frac{n}{2}sin\frac{\pi}{2n}.$ (

**b**) Enlarged part of the panel (

**a**).

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Klinshov, V.V.; Zlobin, A.A.
Kuramoto Model with Delay: The Role of the Frequency Distribution. *Mathematics* **2023**, *11*, 2325.
https://doi.org/10.3390/math11102325

**AMA Style**

Klinshov VV, Zlobin AA.
Kuramoto Model with Delay: The Role of the Frequency Distribution. *Mathematics*. 2023; 11(10):2325.
https://doi.org/10.3390/math11102325

**Chicago/Turabian Style**

Klinshov, Vladimir V., and Alexander A. Zlobin.
2023. "Kuramoto Model with Delay: The Role of the Frequency Distribution" *Mathematics* 11, no. 10: 2325.
https://doi.org/10.3390/math11102325