Emergence of partial locking states from the ensemble of Winfree oscillators
Authors:
Seung-Yeal Ha, Dongnam Ko, Jinyeong Park and Sang Woo Ryoo
Journal:
Quart. Appl. Math. 75 (2017), 39-68
MSC (2010):
Primary 70F99, 92B25
DOI:
https://doi.org/10.1090/qam/1448
Published electronically:
August 12, 2016
MathSciNet review:
3580095
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Abstract: We study the emergence of partial locking states for a subsystem whose dynamics is governed by the Winfree model. The Winfree model is the first mathematical model for synchronization. Thanks to the lack of conservation laws except for the number of oscillators, it exhibits diverse asymptotic nonlinear patterns such as partial and complete phase locking, partial and complete oscillator death, and incoherent states. In this paper, we present two sufficient frameworks for a majority sub-ensemble to evolve to the phase-locked state asymptotically. Our sufficient frameworks are characterized in terms of the mass ratio of the subsystem compared to the total system, ratio of the coupling strength to the natural frequencies, and the phase diameter of the subsystem. We also provide several numerical simulations and compare their results to the analytical results.
References
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- L. Perea, G. Gómez, and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics 32 (2009), 526–536.
- D. Dane Quinn, Richard H. Rand, and Steven H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E (3) 75 (2007), no. 3, 036218, 10. MR 2358543, DOI https://doi.org/10.1103/PhysRevE.75.036218
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References
- J. A. Acebron, L. L. Bonilla, C. J. P. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137–185.
- J. T. Ariaratnam and S. H. Strogatz, Phase diagram for the Winfree model of coupled nonlinear oscillators, Phys. Rev. Lett. 86 (2001), 4278–4281.
- Dario Benedetto, Emanuele Caglioti, and Umberto Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci. 13 (2015), no. 7, 1775–1786. MR 3393174, DOI https://doi.org/10.4310/CMS.2015.v13.n7.a6
- J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature 211 (1966), 562.
- 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, DOI https://doi.org/10.1016/j.physd.2011.11.011
- Nikhil Chopra and Mark W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automat. Control 54 (2009), no. 2, 353–357. MR 2491964, DOI https://doi.org/10.1109/TAC.2008.2007884
- Pierre Degond and Sébastien Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys. 131 (2008), no. 6, 989–1021. MR 2407377, DOI https://doi.org/10.1007/s10955-008-9529-8
- Pierre Degond and Sébastien Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008), suppl., 1193–1215. MR 2438213, DOI https://doi.org/10.1142/S0218202508003005
- Florian Dörfler and Francesco Bullo, Synchronization in complex networks of phase oscillators: a survey, Automatica J. IFAC 50 (2014), no. 6, 1539–1564. MR 3214901, DOI https://doi.org/10.1016/j.automatica.2014.04.012
- Florian Dörfler and Francesco Bullo, On the critical coupling for Kuramoto oscillators, SIAM J. Appl. Dyn. Syst. 10 (2011), no. 3, 1070–1099. MR 2837521, DOI https://doi.org/10.1137/10081530X
- Jiu-Gang Dong and Xiaoping Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci. 11 (2013), no. 2, 465–480. MR 3002560, DOI https://doi.org/10.4310/CMS.2013.v11.n2.a7
- F. Giannuzzi, D. Marinazzo, G. Nardulli, M. Pellicoro, and S. Stramaglia, Phase diagram of a generalized Winfree model, Physical Review E 75 (2007), 051104.
- Seung-Yeal Ha, Hwa Kil Kim, and Sang Woo Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci. 14 (2016), no. 4, 1073–1091. MR 3491817
- Seung-Yeal Ha, Dongnam Ko, Jinyeong Park, and Sang Woo Ryoo, Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations 260 (2016), no. 5, 4203–4236. MR 3437585, DOI https://doi.org/10.1016/j.jde.2015.11.008
- Seung-Yeal Ha, Jinyeong Park, and Sang Woo Ryoo, Emergence of phase-locked states for the Winfree model in a large coupling regime, Discrete Contin. Dyn. Syst. 35 (2015), no. 8, 3417–3436. MR 3320132, DOI https://doi.org/10.3934/dcds.2015.35.3417
- Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432
- Yoshiki Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975) Lecture Notes in Phys., 39, Springer, Berlin, 1975, pp. 420–422. MR 0676492
- N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni, and R. E. Davis, Collective motion, sensor networks and ocean sampling, Proc. IEEE 95 (2007), 48–74.
- Stilianos Louca and Fatihcan M. Atay, Spatially structured networks of pulse-coupled phase oscillators on metric spaces, Discrete Contin. Dyn. Syst. 34 (2014), no. 9, 3703–3745. MR 3191000, DOI https://doi.org/10.3934/dcds.2014.34.3703
- G. Nardulli, D. Mrinazzo, M. Pellicoro, and S. Stramaglia, Phase shifts between synchronized oscillators in the Winfree and Kuramoto models, Available at http://arxiv.org/pdf/physics/0402100.pdf.
- W. Oukil, A. Kessi, and P. Thieullen, Periodic locked orbit in the Winfree model with $N$ oscillators, preprint.
- D. A. Paley, N. E. Leonard, R. Sepulchre, D. Grunbaum, and J. K. Parrish, Oscillator models and collective motion, IEEE Control Systems. 27 (2007), 89–105.
- L. Perea, G. Gómez, and P. Elosegui, Extension of the Cucker-Smale control law to space flight formation, J. Guidance, Control and Dynamics 32 (2009), 526–536.
- D. Dane Quinn, Richard H. Rand, and Steven H. Strogatz, Singular unlocking transition in the Winfree model of coupled oscillators, Phys. Rev. E (3) 75 (2007), no. 3, 036218, 10. MR 2358543, DOI https://doi.org/10.1103/PhysRevE.75.036218
- D. D. Quinn, R. H. Rand, and S. Strogatz, Synchronization in the Winfree model of coupled nonlinear interactions, A. ENOC 2005 Conference, Eindhoven, Netherlands, August 7–12, 2005 (CD-ROM).
- Arthur Winfree, 24 hard problems about the mathematics of 24 hour rhythms, Nonlinear oscillations in biology (Proc. Tenth Summer Sem. Appl. Math., Univ. Utah, Salt Lake City, Utah, 1978) Lectures in Appl. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1979, pp. 93–126. MR 564913
- A. Winfree, Biological rhythms and the behavior of populations of coupled oscillators. J. Theoret. Biol. 16 (1967), 15–42.
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Additional Information
Seung-Yeal Ha
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747, Republic of Korea
MR Author ID:
684438
Email:
syha@snu.ac.kr
Dongnam Ko
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Email:
pyeong@snu.ac.kr
Jinyeong Park
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
MR Author ID:
1099298
Email:
pjy40@snu.ac.kr
Sang Woo Ryoo
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
MR Author ID:
1099431
Email:
sangwoo.ryoo.1@gmail.com
Keywords:
The Kuramoto model,
partial locking states,
synchronization,
the Winfree model
Received by editor(s):
June 14, 2016
Published electronically:
August 12, 2016
Additional Notes:
The work of the first author was supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03. The work of the second author was supported by the fellowship of Cheong-Ahm Foundation. The work of the third author was supported by an NRF (National Research Foundation of Korea) grant funded by the Korean government (NRF-2014-Fostering Core Leaders of the Future Basic Science Program).
Article copyright:
© Copyright 2016
Brown University
