Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
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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.


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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
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
Email: pjy40@snu.ac.kr

Sang Woo Ryoo
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Email: sangwoo.ryoo.1@gmail.com

DOI: https://doi.org/10.1090/qam/1448
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

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