Emergence of synchronization in Kuramoto model with frustration under general network topology

Tingting Zhu

doi:10.3934/nhm.2022005

Article Contents

2022, Volume 17, Issue 2: 255-291. Doi: 10.3934/nhm.2022005

This issue Previous Article Stochastic two-scale convergence and Young measures Next Article

Emergence of synchronization in Kuramoto model with frustration under general network topology

Tingting Zhu

Key Laboratory of Applied Mathematics and Artificial Intelligence Mechanism, Hefei University, Hefei 230601, China

Received: July 31, 2021

Revised: November 30, 2021

Early access: February 2022

Published: April 2022

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Abstract

In this paper, we will study the emergent behavior of Kuramoto model with frustration on a general digraph containing a spanning tree. We provide a sufficient condition for the emergence of asymptotical synchronization if the initial data are confined in half circle. As lack of uniform coercivity in general digraph, we apply the node decomposition criteria in [25] to capture a clear hierarchical structure, which successfully yields the dissipation mechanism of phase diameter and an invariant set confined in quarter circle after some finite time. Then the dissipation of frequency diameter will be clear, which eventually leads to the synchronization.

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Mathematics Subject Classification: Primary: 34D06, 34C15; Secondary: 92B25, 70F99.

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Figure 1. The interaction network

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Figure 2. Frequency synchronization with $ K = 20,\alpha = 0.01 $

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Figure 3. Asymptotic behavior for $ K = 1 $ when $ \alpha = 0, 0.01, 0.1, 0.3 $

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Figure 4. Asymptotic behavior for $ \alpha = 0.1 $ when $ K = 0.8,1,1.2,1.4 $

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Figure 5. The four cases

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