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

SeungYeal Ha | Zendy; Jeongho Kim | Zendy; Jinyeong Park | Zendy; Xiongtao Zhang | Zendy

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

Author(s) -

SeungYeal Ha,

Jeongho Kim,

Jinyeong Park,

Xiongtao Zhang

Publication year - 2018

Publication title -

networks and heterogeneous media

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.732

H-Index - 34

eISSN - 1556-181X

pISSN - 1556-1801

DOI - 10.3934/nhm.2018013

Subject(s) - kuramoto model , mean field theory , limit (mathematics) , stability (learning theory) , dissipative system , mathematics , statistical physics , vlasov equation , mathematical analysis , field (mathematics) , synchronization (alternating current) , physics , computer science , pure mathematics , topology (electrical circuits) , combinatorics , quantum mechanics , machine learning , electron

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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