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

2018; American Institute of Mathematical Sciences; Volume: 13; Issue: 2 Linguagem: Inglês

10.3934/nhm.2018013

1556-181X

Seung‐Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang,

Ecosystem dynamics and resilience

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.