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

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

doi:10.3934/nhm.2018013

Networks and Heterogeneous Media

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

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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

Received: 01 July 2017 Revised: 01 March 2018
70F99, 92D25

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.
- The Kuramoto model,
- mean-field limit,
- synchronization,
- uniform stability
Citation: Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model[J]. Networks and Heterogeneous Media, 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013

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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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