Article Contents

2024, Volume 29, Issue 5: 2399-2418. Doi: 10.3934/dcdsb.2023183

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Synchronization of discrete time Kuramoto oscillators with delayed states

Hua Zhang^, and
Sisi Xiao

School of Science, Chongqing University of Technology, Chongqing, China

^*Corresponding author: Hua Zhang, zhanghwua@163.com
^*Corresponding author: Hua Zhang, zhanghwua@163.com

Received: October 2022

Revised: September 2023

Early access: October 2023

Published: May 2024

This work is supported by the Natural Science Foundation of China (Grant No.62262058), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grand No.KJQN201901122), and the Natural Science Foundation of Chongqing Science and Technology Bureau (Grant No. cstc2019jcyj-msxmX0020)

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Abstract

This paper investigates the synchronization problem of a kind of discrete-time Kuramoto oscillators based on delayed position states. For this kind of Kuramoto model, the oscillators can only obtain delayed phase and frequency states relative to each other at a series of discrete time instants. Some simple sufficient conditions on the time step, the time delay, the natural frequencies, the coupling strength, and the historical and initial values of the phases and frequencies are derived analytically to ensure the phase synchronization of homogeneous oscillators and the frequency synchronization of heterogeneous oscillators, respectively. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.

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Mathematics Subject Classification: 92B25, 39A12, 93C43, 34C15.

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Figure 1. The time evolution of the phases of $ N = 20 $ oscillators when $ \iota = 98 $

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Figure 2. The time evolution of the phases of $ N = 20 $ oscillators when $ \iota = 198 $

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Figure 3. The time responses of diameters $ D_{k}(\Theta) $ for $ \iota = 98 $ and $ \iota = 198 $

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Figure 4. The time response of the phases of the heterogeneous oscillators

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Figure 5. The time response of the frequency states of $ N = 17 $ heterogeneous oscillators

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