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

Sun Ho Choi, Hyowon Seo

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

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.

Original languageEnglish
Article number15
JournalJournal of Nonlinear Science
Volume33
Issue number1
DOIs
Publication statusPublished - Feb 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Exponential asymptotic stability
  • Inertia
  • Kuramoto model
  • Mean-centered-periodic solution
  • Time-periodic natural frequency

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