Uniform stability and mean-field limit for the augmented Kuramoto model

Seung-Yeal Ha; Jeongho Kim; Jinyeong Park; Xiongtao Zhang

doi:10.3934/nhm.2018013

Article Contents

2018, Volume 13, Issue 2: 297-322. Doi: 10.3934/nhm.2018013

This issue Previous Article Error bounds for Kalman filters on traffic networks Next Article Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring

Uniform stability and mean-field limit for the augmented Kuramoto model

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
2.
Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea
3.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
4.
Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea
5.
Center for Mathematical sciences, Huazhong University of Science and Technology, Wuhan, China

* Corresponding author: Jinyeong Park
* Corresponding author: Jinyeong Park

Received: June 30, 2017

Revised: February 28, 2018

Published: May 2018

The works of S.-Y. Ha and X. Zhang are supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03). The work of J. Kim is supported by the German Research Foundation (DFG), project number IRTG2235.

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Abstract

We present two uniform estimates on stability and mean-field limit for the "augmented Kuramoto model (AKM)" arising from the second-order lifting of the first-order Kuramoto model (KM) for synchronization. In particular, we address three issues such as synchronization estimate, uniform stability and mean-field limit which are valid uniformly in time for the AKM. The derived mean-field equation for the AKM corresponds to the dissipative Vlasov-McKean type equation. The kinetic Kuramoto equation for distributed natural frequencies is not compatible with the frequency variance functional approach for the complete synchronization. In contrast, the kinetic equation for the AKM has a similar structural similarity with the kinetic Cucker-Smale equation which admits the Lyapunov functional approach for the variance. We present sufficient frameworks leading to the uniform stability and mean-field limit for the AKM.

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Mathematics Subject Classification: 70F99, 92D25.

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