On the generic complete synchronization of the discrete Kuramoto model

Woojoo Shim

doi:10.3934/krm.2020034

Article Contents

2020, Volume 13, Issue 5: 979-1005. Doi: 10.3934/krm.2020034

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On the generic complete synchronization of the discrete Kuramoto model

Woojoo Shim^,

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

^* Corresponding author: Woojoo Shim
^* Corresponding author: Woojoo Shim

Received: January 2020

Revised: May 2020

Published: August 2020

The work of Woojoo Shim is supported by the National Research Foundation of Korea (NRF-2017R1A5A1015-620)

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Abstract

We study the emergent behavior of discrete-time approximation of the finite-dimensional Kuramoto model. Compared to Zhang and Zhu's recent work in [38], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.

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Mathematics Subject Classification: Primary: 92B25, 39A28; Secondary: 37N25.

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