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On the Finiteness of Collisions and Phase-Locked States for the Kuramoto Model

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Abstract

Synchronization phenomenon is ubiquitous in our complex systems, and many phenomenological models have been proposed and studied analytically and numerically. Among them, the Kuramoto model serves as a prototype model for the phase synchronization of weakly coupled oscillators. In this paper, we study the finiteness of collisions (crossings) among Kuramoto oscillators in the relaxation process toward the phase-locked states and the total number of phase-locked states with positive (Kuramoto) order parameters. For identical oscillators, it is well known that collisions between distinct oscillators cannot occur in finite-time, and we show that there are only a finite number of phase-locked states with positive order parameters. However, for non-identical oscillators, oscillators with different natural frequencies can cross each other in their relaxation process, and estimating the total number of phase-locked states is a nontrivial matter. We show that, for the non-identical case, asymptotic phase-locking is equivalent to the finiteness of collisions, and the total number of phase-locked states with positive order parameters is bounded above by \(2^N\), where N is the number of oscillators.

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Acknowledgments

The work of S.-Y. Ha was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of H. K. Kim was supported by the National Research Foundation of Korea (NRF2015R1D1A1A01056696).

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Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, South Korea

    Seung-Yeal Ha

  2. School of Mathematics, Korea Institute for Advanced Studies, Seoul, 130-722, South Korea

    Hwa Kil Kim

  3. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, South Korea

    Seung-Yeon Ryoo

Authors
  1. Seung-Yeal Ha
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  2. Hwa Kil Kim
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Correspondence to Seung-Yeal Ha.

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The original online version of this article was revised: "In this article the author’s name Seung-Yeon Ryoo was incorrectly written as Sang Woo Ryoo.

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Ha, SY., Kim, H.K. & Ryoo, SY. On the Finiteness of Collisions and Phase-Locked States for the Kuramoto Model. J Stat Phys 163, 1394–1424 (2016). https://doi.org/10.1007/s10955-016-1528-6

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  • DOI: https://doi.org/10.1007/s10955-016-1528-6

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