Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Remarks on the nonlinear stability of the Kuramoto model with inertia


Authors: Young-Pil Choi, Seung-Yeal Ha and Se Eun Noh
Journal: Quart. Appl. Math. 73 (2015), 391-399
MSC (2000): Primary 92D25, 74A25, 76N10
DOI: https://doi.org/10.1090/qam/1383
Published electronically: March 31, 2015
MathSciNet review: 3357501
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short note, we present an a priori nonlinear stability estimate for the Kuramoto model with finite inertia in $ \ell ^{\infty }$-norm under some a priori condition on the size of the phase diameter. As a direct corollary of our nonlinear stability estimate, we show that phase-locked states obtained in Choi, Ha, and Yun (2011) are orbital-stable in $ \ell ^{\infty }$-norm, which means that the perturbed phase-locked state approaches the phase-shift of the given phase-locked state. The phase-shift is explicitly determined by the averages of initial phase and frequency distribution and the strength of inertia $ m$.


References [Enhancements On Off] (What's this?)

  • [1] J. A. Acebrón, L .L. Bonilla, C. J. P. Pérez Vicente, F. Ritort, R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137-185.
  • [2] J. A. Acebrón, R. Spigler, Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators, Phys. Rev. Lett. 81 (1998), 2229-2332.
  • [3] D. Aeyels, J. Rogge, Existence of partial entrainment and stability of phase locking behavior of coupled oscillator, Prog. Theor. Phys. 112 (2004), 921-942.
  • [4] Young-Pil Choi, Seung-Yeal Ha, and Seok-Bae Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D 240 (2011), no. 1, 32-44. MR 2740100 (2012b:34093), https://doi.org/10.1016/j.physd.2010.08.004
  • [5] Young-Pil Choi, Seung-Yeal Ha, Sungeun Jung, and Yongduck Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), no. 7, 735-754. MR 2897541, https://doi.org/10.1016/j.physd.2011.11.011
  • [6] B. C. Daniels, S. T. Dissanayake, B. R. Trees, Synchronization of coupled rotators: Josephson junction ladders and the locally coupled Kuramoto model, Phys. Rev. E. 67 (2003), 026216.
  • [7] B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol. 29 (1991), no. 6, 571-585. MR 1118757, https://doi.org/10.1007/BF00164052
  • [8] Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators (Kyoto Univ., Kyoto, 1975), Lecture Notes in Phys., 39, Springer, Berlin, 1975, pp. 420-422. MR 0676492 (58 #32705)
  • [9] Renato E. Mirollo and Steven H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Phys. D 205 (2005), no. 1-4, 249-266. MR 2167156 (2006d:34080), https://doi.org/10.1016/j.physd.2005.01.017
  • [10] K. Park, M. Y. Choi, Synchronization in networks of superconducting wires, Phys. Rev. B 56 (1997), 387-394.
  • [11] Steven H. Strogatz and Renato E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys. 63 (1991), no. 3-4, 613-635. MR 1115806 (92h:34077), https://doi.org/10.1007/BF01029202
  • [12] S. H. Strogatz, R. E. Mirollo, Splay states in globally coupled Josephson arrays: Analytical prediction of Floquet multipliers, Phys. Rev. E 47 (1993), 220-227.
  • [13] James W. Swift, Steven H. Strogatz, and Kurt Wiesenfeld, Averaging of globally coupled oscillators, Phys. D 55 (1992), no. 3-4, 239-250. MR 1156731 (92m:34085), https://doi.org/10.1016/0167-2789(92)90057-T
  • [14] K. Wiesenfeld, R. Colet, S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model, Phys. Rev. E 57 (1988), 1563-1569.
  • [15] K. Wiesenfeld, R. Colet, S. H. Strogatz, Synchronization transitions in a disordered Josephson series arrays, Phys. Rev. Lett. 76 (1996), 404-407.
  • [16] S. Watanabe and J. W. Swift, Stability of periodic solutions in series arrays of Josephson junctions with internal capacitance, J. Nonlinear Sci. 7 (1997), no. 6, 503-536. MR 1474640 (98f:34051), https://doi.org/10.1007/s003329900038
  • [17] S. Watanabe, S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74 (1994), 197-253.
  • [18] K. Wiesenfeld, J. W. Swift, Averaged equations for Josephson junction series arrays, Phys. Rev. E 51 (1995), 1020-1025.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 92D25, 74A25, 76N10

Retrieve articles in all journals with MSC (2000): 92D25, 74A25, 76N10


Additional Information

Young-Pil Choi
Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, England
Email: young-pil.choi@imperial.ac.uk

Seung-Yeal Ha
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Email: syha@snu.ac.kr

Se Eun Noh
Affiliation: Department of Mathematics, Myongji University, Yongin 449-728, Republic of Korea
Email: senoh@mju.ac.kr

DOI: https://doi.org/10.1090/qam/1383
Received by editor(s): September 13, 2013
Published electronically: March 31, 2015
Additional Notes: The first author was supported by the Basic Science Research Program through the NRF funded by the Ministry of Education, Science and Technology (2012R1A6A3A03039496).
The second author was supported by NRF grant 2011-0015388.
The third author was partially supported by the 2012 Research Fund of Myongji University.
Article copyright: © Copyright 2015 Brown University

American Mathematical Society