Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators
Authors:
Seung-Yeal Ha and Moon-Jin Kang
Journal:
Quart. Appl. Math. 71 (2013), 707-728
MSC (2010):
Primary 92D25, 74A25, 76N10
DOI:
https://doi.org/10.1090/S0033-569X-2013-01302-0
Published electronically:
August 29, 2013
MathSciNet review:
3136992
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Abstract: We present asymptotic relaxation estimates to bi-cluster configurations for the ensemble of Kuramoto oscillators with two different natural frequencies which have been observed in numerical simulations. We provide a set of initial configurations with a positive Lebesgue measure in $\mathbb {T}^N$ leading to bi-(point) cluster configurations consisting of linear combinations of two Dirac measures in super-threshold and threshold-coupling regimes. In a super-threshold regime where the coupling strength is larger than the difference of two natural frequencies, we use the $\ell _1$-contraction property of the Kuramoto model to derive exponential convergence toward bi-cluster configurations. The exact location of bi-cluster configurations is explicitly computable using the coupling strength, the difference of natural frequencies, and the total phase. In contrast, for the threshold-coupling regime where the coupling strength is exactly equal to the difference of natural frequencies, the mixed ensemble of Kuramoto oscillators undergoes two dynamic phases. First, the initial configuration evolves to the segregated phase (two segregated subconfigurations consisting of the same natural frequency) in a finite time. After this segregation phase, each subconfiguration relaxes to the asymptotic phase algebraically slowly. Our analytical results provide a rigorous framework for the observed numerical simulations.
References
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- S. Watanabe, S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74, 197-253 (1994).
- Arthur T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin-New York, 1980. MR 572965
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References
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- D. Aeyels, J. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys. 112, 921-941 (2004).
- N. J. Balmforth and R. Sassi, A shocking display of synchrony, Phys. D 143 (2000), no. 1-4, 21–55. Bifurcations, patterns and symmetry. MR 1783383 (2001g:82007), DOI https://doi.org/10.1016/S0167-2789%2800%2900095-6
- E. Barreto, B. Hunt, E. Ott, and P. So, Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators, Phys. Rev. E (3) 77 (2008), no. 3, 036107, 7. MR 2495429 (2010d:37047), DOI https://doi.org/10.1103/PhysRevE.77.036107
- J. Buck, E. Buck, Biology of synchronous flashing of fireflies, Nature 211, 562 (1966).
- Y.-P. Choi, S.-Y. Ha, and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Phys. D 240 (2011), no. 1, 32–44. MR 2740100 (2012b:34093), DOI https://doi.org/10.1016/j.physd.2010.08.004
- Y.-P. Choi, S.-Y. Ha, S.-E. Jung, Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Phys. D 241 (2012), 735–754, DOI 10.101b/j.physd.2011.11.011.
- N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control 54 (2009), no. 2, 353–357. MR 2491964 (2010d:34097), DOI https://doi.org/10.1109/TAC.2008.2007884
- F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Phys. D 234 (2007), no. 2, 81–89. MR 2371859 (2008m:34089), DOI https://doi.org/10.1016/j.physd.2007.06.025
- F. Dorfler, F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Applied Dynamical Systems.
- G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol. 22 (1985), no. 1, 1–9. MR 802731 (86m:92010), DOI https://doi.org/10.1007/BF00276542
- S.-Y. Ha, T. Ha, and J.-H. Kim, On the complete synchronization of the Kuramoto phase model, Phys. D 239 (2010), no. 17, 1692–1700. MR 2684624 (2011m:34144), DOI https://doi.org/10.1016/j.physd.2010.05.003
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- A. Jadbabaie, N. Motee, M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators in: Proceedings of the American Control Conference, Boston, Massachusetts, 2004.
- Y. Kawamura, H. Nakao, K. Arai, H. Kori, and Y. Kuramoto, Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case, Chaos 20 (2010), no. 4, 043110, 8. MR 2791147 (2011m:34103), DOI https://doi.org/10.1063/1.3491346
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432 (87e:92054)
- Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, (Kyoto Univ., Kyoto, 1975) Springer, Berlin, 1975, pp. 420–422. Lecture Notes in Phys., 39. MR 0676492 (58 \#32705)
- R. E. Mirollo and S. 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), DOI https://doi.org/10.1016/j.physd.2005.01.017
- R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci. 17 (2007), no. 4, 309–347. MR 2335124 (2009a:37049a), DOI https://doi.org/10.1007/s00332-006-0806-x
- E. Montbrió, J. Kurths, B. Blasius, Synchronization of two interacting populations of oscillators, Phys. Review E 70, 056125 (2004).
- E. Montbrió and B. Blasius, Using nonisochronicity to control synchronization in ensembles of nonidentical oscillators, Chaos 13 (2003), no. 1, 291–308. MR 1964974 (2004b:34095), DOI https://doi.org/10.1063/1.1525170
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- C. S. Peskin, Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences New York University, New York, 1975. Notes based on a course given at New York University during the year 1973/74. MR 0414135 (54 \#2239)
- A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization, Cambridge Nonlinear Science Series, vol. 12, Cambridge University Press, Cambridge, 2001. A universal concept in nonlinear sciences. MR 1869044 (2002m:37001)
- J. H. Sheeba, V. K. Chandrasekar, A. Stefanovska, and P. V. E. McClintock, Asymmetry-induced effects in coupled phase-oscillator ensembles: routes to synchronization, Phys. Rev. E (3) 79 (2009), no. 4, 046210, 9. MR 2551228 (2010i:34091), DOI https://doi.org/10.1103/PhysRevE.79.046210
- J. H. Sheeba, V. K. Chandrasekar, A. Stefanovska, P. V. E. McClintock, Routes to synchrony between asymmetrically interacting oscillator ensembles, Phys. Review E. 78, 025201(R) (2008).
- S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Phys. D 143 (2000), no. 1-4, 1–20. Bifurcations, patterns and symmetry. MR 1783382 (2001g:82008), DOI https://doi.org/10.1016/S0167-2789%2800%2900094-4
- S. H. Strogatz and R. 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), DOI https://doi.org/10.1007/BF01029202
- S. H. Strogatz, R. E. Mirollo, and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett. 68 (1992), no. 18, 2730–2733. MR 1160290 (93d:92004), DOI https://doi.org/10.1103/PhysRevLett.68.2730
- J. L. van Hemmen, W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys. 72, 145-166 (1993).
- S. Watanabe, S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D 74, 197-253 (1994).
- A. T. Winfree, The geometry of biological time, Biomathematics, vol. 8, Springer-Verlag, Berlin, 1980. MR 572965 (82c:92003)
- A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol. 16, 15-42 (1967).
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Moon-Jin Kang
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
Email:
hiofte@snu.ac.kr
Keywords:
The Kuramoto model,
bi-cluster configurations,
$\ell _1$-contraction,
natural frequency
Received by editor(s):
November 21, 2011
Published electronically:
August 29, 2013
Additional Notes:
The work of S.-Y. Ha is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (2011-0015388), and the work of M.-J. Kang is supported by Hi Seoul Science/Humanities Fellowship funded by Seoul Scholarship Foundation
Article copyright:
© Copyright 2013
Brown University