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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.

References

  • Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137 (2005)

    Article  Google Scholar 

  • Apaydin, N.M., Kaya, Y., Şafak, E., Alçik, H.: Vibration characteristics of a suspension bridge under traffic and no traffic conditions. Earthq. Eng. Struct. Dyn. 41, 1717–1723 (2012)

    Article  Google Scholar 

  • Cantero, D., McGetrick, P., Kim, C.W., OBrien, E.: Experimental monitoring of bridge frequency evolution during the passage of vehicles with different suspension properties. Eng. Struct. 187, 209–219 (2019)

    Article  Google Scholar 

  • Chi, D., Choi, S.-H., Ha, S.-Y.: Emergent behaviors of a holonomic particle system on a sphere. J. Math. Phys. 55, 052703 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, S.-H., Ha, S.-Y.: Complete entrainment of Lohe oscillators under attractive and repulsive couplings. SIAM J. Appl. Dyn. Syst. 13, 1417–1441 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, S.-H., Ha, S.-Y.: Emergent behaviors of quantum Lohe oscillators with all-to-all coupling. J. Nonlinear Sci. 25, 1257–1283 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, S.-H., Seo, H.: Synchronization in a power balance system with inertia and nonlinear derivatives. SIAM J. Appl. Math. 81, 1202–1225 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, S.-H., Seo, H.: Asymptotic behavior of the Kuramoto system with periodic natural frequency. J. Differ. Equ. 308, 160–187 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, Y.-P., Ha, S.-Y., Yun, S.-B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D Nonlinear Phenom. 240, 32–44 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, Y., Ha, S.-Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D Nonlinear Phenom. 241, 735–754 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Choi, S.-H., Cho, J., Ha, S.-Y.: Practical quantum synchronization for the Schrödinger–Lohe system. J. Phys. A Math. Theor. 49, 205203 (2016)

    Article  MATH  Google Scholar 

  • Ebling, F.J.P., Barrett, P.: The regulation of seasonal changes in food intake and body weight. J. Neuroendocrinol. 20, 827–833 (2008)

    Article  Google Scholar 

  • Femat, R., Solis-Perales, G.: On the chaos synchronization phenomena. Phys. Lett. A 262, 50–60 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Filatrella, G., Nielsen, A.H., Pedersen, N.F.: Analysis of a power grid using a Kuramoto-like model. Eur. Phys. J. B 61, 485–491 (2008)

    Article  Google Scholar 

  • Flaherty, J.E., Hoppensteadt, F.C.: Frequency entrainment of a forced van der Pol oscillator. Stud. Appl. Math. 58, 5–15 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  • Fujino, Y., Pacheco, B.M., Nakamura, S.I., Warnitchai, P.: Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge. Earthq. Eng. Struct. Dyn. 22, 741–758 (1993)

    Article  Google Scholar 

  • Glass, L.: Synchronization and rhythmic processes in physiology. Nature 410, 277–284 (2001)

    Article  Google Scholar 

  • Gupta, S., Campa, A., Ruffo, S.: Kuramoto model of synchronization: equilibrium and nonequilibrium aspects. J. Stat. Mech. Theory Exp. 2014, R08001 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Ha, S.-Y., Ha, T., Kim, J.-H.: On the complete synchronization of the Kuramoto phase model. Physica D Nonlinear Phenom. 239, 1692–1700 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Harvey, A., Koopman, S.J.: Forecasting hourly electricity demand using time-varying splines. J. Am. Stat. Assoc. 88, 1228–1236 (1993)

    Article  Google Scholar 

  • Hoppensteadt, F.C.: An Introduction to the Mathematics of Neurons: Modeling in the Frequency Domain. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  • Lohe, M.A.: Non-Abelian Kuramoto model and synchronization. J. Phys. A Math. Theor. 42, 395101 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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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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