On the relaxation dynamics of the Kuramoto oscillators with small inertia

Young Pil Choi; Seung Yeal Ha; Se Eun Noh

doi:10.1063/1.4809943

On the relaxation dynamics of the Kuramoto oscillators with small inertia

Young Pil Choi, Seung Yeal Ha, Se Eun Noh

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. ["Asymptotic formation and orbital stability of phase locked states for the Kuramoto model," Physica D241, 735-754 (2012)10.1016/j.physd.2011.11.011; Choi et al. "Complete synchronization of Kuramoto oscillators with finite inertia," Physica D240, 32-44 (2011)]10.1016/j.physd.2010.08.004.

Original language	English
Article number	072701
Journal	Journal of Mathematical Physics
Volume	54
Issue number	7
DOIs	https://doi.org/10.1063/1.4809943
Publication status	Published - 2013 Jul 1

Bibliographical note

Funding Information:
The work of Y.-P. Choi is partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496). The work of S.-Y. Ha is partially supported by NRF-2011-0015388, and the work of S. E. Noh is supported by 2012 Research Fund of Myongji University.

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.4809943

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title = "On the relaxation dynamics of the Kuramoto oscillators with small inertia",

abstract = "For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. [{"}Asymptotic formation and orbital stability of phase locked states for the Kuramoto model,{"} Physica D241, 735-754 (2012)10.1016/j.physd.2011.11.011; Choi et al. {"}Complete synchronization of Kuramoto oscillators with finite inertia,{"} Physica D240, 32-44 (2011)]10.1016/j.physd.2010.08.004.",

author = "Choi, {Young Pil} and Ha, {Seung Yeal} and Noh, {Se Eun}",

note = "Funding Information: The work of Y.-P. Choi is partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496). The work of S.-Y. Ha is partially supported by NRF-2011-0015388, and the work of S. E. Noh is supported by 2012 Research Fund of Myongji University.",

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PY - 2013/7/1

Y1 - 2013/7/1

N2 - For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. ["Asymptotic formation and orbital stability of phase locked states for the Kuramoto model," Physica D241, 735-754 (2012)10.1016/j.physd.2011.11.011; Choi et al. "Complete synchronization of Kuramoto oscillators with finite inertia," Physica D240, 32-44 (2011)]10.1016/j.physd.2010.08.004.

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On the relaxation dynamics of the Kuramoto oscillators with small inertia

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