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Exponential Asymptotic Stability of the Kuramoto System with Periodic Natural Frequencies and Constant Inertia

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Abstract

We consider the temporal asymptotic behavior of the all-to-all coupled Kuramoto model with inertia and time-periodic natural frequencies. Due to the inertial effect, there are three cases of the dynamical ensemble with respect to the coupling strength; large coupling, near boundary, and small coupling. For each case, we present the asymptotic behavior of the solution to the inertial Kuramoto model with periodic natural frequencies: the solutions commonly consist of a macroscopic phase, a mean-centered-periodic solution, and an exponential decay term. The macroscopic phase is a drift-type term determined by initial data and natural frequencies, and the mean-centered-periodic solution is a standing wave independent of initial data. We provide sufficient conditions for the existence of a mean-centered-periodic solution with a time-periodic phase difference between nodes for each case and its exponential stability. We also provide several simulations to confirm our mathematical results.

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The authors declare that the data supporting the findings of this study are available within the article.

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Acknowledgements

S.-H. Choi is partially supported by Korea Electric Power Corporation(Grant Number: R18XA02). S.-H. Choi and H. Seo are partially supported by NRF of Korea (No. 2017R1E1A1A03070692).

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Authors and Affiliations

  1. Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, South Korea

    Sun-Ho Choi

  2. Department of Mathematics, Kunsan National University, 558, Daehak-ro, Gunsan-si, Jeollabuk-do, 54150, Republic of Korea

    Hyowon Seo

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  1. Sun-Ho Choi
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Correspondence to Hyowon Seo.

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Communicated by Ram Ramaswamy.

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Choi, SH., Seo, H. Exponential Asymptotic Stability of the Kuramoto System with Periodic Natural Frequencies and Constant Inertia. J Nonlinear Sci 33, 15 (2023). https://doi.org/10.1007/s00332-022-09870-1

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