# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021013

## Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and Korea Institute for Advanced Study, Hoegiro 87, Seoul, 130-722, Republic of Korea 2 Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD 20742, USA 3 Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea

* Corresponding author

Dedicated to the celebration of the 80th birthday of Prof. Shuxing Chen

Received  June 2020 Revised  December 2020 Published  February 2021

Fund Project: The work of S. Y. Ha was supported by National Research Foundation of Korea(NRF-2020R1A2C3A01003881), and the work of Y. Zhang was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(NRF-2019R1A5A1028324)

We study phase concentration for the Kuramoto-Sakaguchi(K-S) equation with frustration via detailed estimates on the dynamics of order parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. When the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, we show that the amplitude order parameter has a positive lower bound uniformly in time, and we also show that the total mass concentrates on the translated phase order parameter by a frustration parameter asymptotically, whereas the mass in the region around the antipodal point decays to zero exponentially fast.

Citation: Seung-Yeal Ha, Javier Morales, Yinglong Zhang. Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021013
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