# American Institute of Mathematical Sciences

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March  2020, 25(3): 1059-1081. doi: 10.3934/dcdsb.2019208

## Emergent collective behaviors of stochastic kuramoto oscillators

 1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 00826, Korea (Republic of) 2 Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of) 3 DeustoTech, University of Deusto, and Facultad de Ingeniería, Universidad de Deusto, Avenida de las Universidades 24, Bilbao 48007, Basque Country, Spain 4 Department of Mathematical Sciences, Seoul National University, Seoul 00826, Korea (Republic of) 5 Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei Province, China

* Corresponding author: Dongnam Ko

Received  September 2018 Revised  May 2019 Published  September 2019

Fund Project: The work of S.-Y. Ha is supported by the NRF grant (2017R1A2B2001864). The work of D. Ko is supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (ERC-2015-AdG-694126-DyCon) and partially supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE-2016-0035. The work of X. Zhang is supported by the National Natural Science Foundation of China (Grant No. 11801194).

We study the collective dynamics of Kuramoto ensemble under uncertain coupling strength. For a finite ensemble, we can model the dynamics of the Kuramoto ensemble by the stochastic Kuramoto system with multiplicative noise. In contrast, for an infinite ensemble, the dynamics is effectively described by the Kuramoto-Sakaguchi-Fokker-Planck(KS-FP) equation with state dependent degenerate diffusion. We present emergent synchronization estimates for the stochastic and kinetic models, which yield the stability of the phase-locked state for identical Kuramoto ensemble with the same natural frequencies. We also provide a brief explanation on the mean-field limit between two models.

Citation: Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
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