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On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators
1. | Department of Mathematics, Institute of Applied Mathematical Sciences and National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan |
2. | Department of Mathematical Sciences, School of Natural Science, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea |
We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the global convergence of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light of this, we show, under a suitable condition on the coupling strength and deviation of the intrinsic frequencies, that the diameter function of the phases is uniformly bounded. In a similar spirit, we also prove the global convergence of phase-locked synchronization for the Winfree model of $ N $ oscillators for $ N\ge2 $.
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137–185 Google Scholar |
[2] |
Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim,
Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[3] |
F. Dörfler and F. Bullo,
Synchronization in Complex Networks of Phase Oscillators: A Survey, Automatica, 50 (2014), 1539-1564.
doi: 10.1016/j.automatica.2014.04.012. |
[4] |
J.-G. Dong and X. Xue,
Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.
doi: 10.4310/CMS.2013.v11.n2.a7. |
[5] |
B. Ermentrout,
Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9.
doi: 10.1007/BF00276542. |
[6] |
B. Ermentrout,
An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.
doi: 10.1007/BF00164052. |
[7] |
S.-Y. Ha, H. K. Kim and S. W. Ryoo,
Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.
doi: 10.4310/CMS.2016.v14.n4.a10. |
[8] |
S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo,
Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations, 260 (2016), 4203-4236.
doi: 10.1016/j.jde.2015.11.008. |
[9] |
C.-H. Hsia, C.-Y. Jung and B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, preprint, arXiv: 1712.10111 Google Scholar |
[10] |
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69689-3. |
[11] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, New York, 1975,420–422. |
[12] |
J. Lunz, Complete synchronization of Kuramoto oscillators, J. Phys. A: Math. Theor., 44 (2011), 425102, 14 pp.
doi: 10.1088/1751-8113/44/42/425102. |
[13] |
S. H. Strogatz,
From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.
doi: 10.1016/S0167-2789(00)00094-4. |
[14] |
J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar |
[15] |
A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar |
show all references
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137–185 Google Scholar |
[2] |
Y.-P. Choi, S.-Y. Ha, S. Jung and Y. Kim,
Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[3] |
F. Dörfler and F. Bullo,
Synchronization in Complex Networks of Phase Oscillators: A Survey, Automatica, 50 (2014), 1539-1564.
doi: 10.1016/j.automatica.2014.04.012. |
[4] |
J.-G. Dong and X. Xue,
Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.
doi: 10.4310/CMS.2013.v11.n2.a7. |
[5] |
B. Ermentrout,
Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9.
doi: 10.1007/BF00276542. |
[6] |
B. Ermentrout,
An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.
doi: 10.1007/BF00164052. |
[7] |
S.-Y. Ha, H. K. Kim and S. W. Ryoo,
Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.
doi: 10.4310/CMS.2016.v14.n4.a10. |
[8] |
S.-Y. Ha, D. Ko, J. Park and S. W. Ryoo,
Emergent dynamics of Winfree oscillators on locally coupled networks, J. Differential Equations, 260 (2016), 4203-4236.
doi: 10.1016/j.jde.2015.11.008. |
[9] |
C.-H. Hsia, C.-Y. Jung and B. Kwon, On the synchronization theory of Kuramoto oscillators under the effect of inertia, preprint, arXiv: 1712.10111 Google Scholar |
[10] |
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69689-3. |
[11] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys., 39, Springer, New York, 1975,420–422. |
[12] |
J. Lunz, Complete synchronization of Kuramoto oscillators, J. Phys. A: Math. Theor., 44 (2011), 425102, 14 pp.
doi: 10.1088/1751-8113/44/42/425102. |
[13] |
S. H. Strogatz,
From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.
doi: 10.1016/S0167-2789(00)00094-4. |
[14] |
J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166. Google Scholar |
[15] |
A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42. Google Scholar |



Case | | | | | |
(Ⅰ) | | | | | |
(Ⅱ) | | | | | |
(Ⅲ) | | | | | |
(Ⅰ)' | | | | | |
(Ⅱ)' | | | | | |
(Ⅲ)' | | | | | |
Case | | | | | |
(Ⅰ) | | | | | |
(Ⅱ) | | | | | |
(Ⅲ) | | | | | |
(Ⅰ)' | | | | | |
(Ⅱ)' | | | | | |
(Ⅲ)' | | | | | |
(Ⅰ) | (Ⅱ) | (Ⅲ) | (Ⅰ)' | (Ⅱ)' | (Ⅲ)' | ||
3.80000 | 4.70000 | 4.60170 | 2.30000 | 4.70000 | 5.67130 | ||
0.23108 | 1.93260 | 0.67878 | 0.61164 | 1.88670 | 2.41560 | ||
0.34515 | 0.30511 | 0.27291 | 0.60397 | 0.30511 | 0.03716 | ||
6.05170 | 6.96100 | 0.48507 | 7.56320 | 8.10750 | 13.97020 | ||
0.15616 | 0.00375 | 0.41549 | 0.57795 | 1.22680 | 3.03850 | ||
0.98882 | 0.91498 | 0.99269 | 0.86806 | 0.75848 | 0.72047 | ||
6.18270 | 6.96220 | 0.21578 | 20.38000 | 28.04320 | 22.71800 | ||
0.00000 | 0.00000 | 0.00000 | 0.38512 | 1.01360 | 0.94404 | ||
0.99664 | 0.91483 | 0.99814 | 0.80857 | 0.61695 | 0.60180 | ||
6.18270 | 6.96220 | 0.21578 | 133.80190 | 228.90490 | 128.11660 | ||
0.00000 | 0.00000 | 0.00000 | 0.36309 | 0.83900 | 2.09790 | ||
0.99664 | 0.91483 | 0.99814 | 0.72070 | 0.60348 | 0.76208 | ||
6.18270 | 6.96220 | 0.21578 | 441.10500 | 782.61200 | 405.01650 | ||
0.00000 | 0.00000 | 0.00000 | 0.49383 | 1.24770 | 0.46759 | ||
0.99664 | 0.91483 | 0.99814 | 0.86642 | 0.40841 | 0.68921 |
(Ⅰ) | (Ⅱ) | (Ⅲ) | (Ⅰ)' | (Ⅱ)' | (Ⅲ)' | ||
3.80000 | 4.70000 | 4.60170 | 2.30000 | 4.70000 | 5.67130 | ||
0.23108 | 1.93260 | 0.67878 | 0.61164 | 1.88670 | 2.41560 | ||
0.34515 | 0.30511 | 0.27291 | 0.60397 | 0.30511 | 0.03716 | ||
6.05170 | 6.96100 | 0.48507 | 7.56320 | 8.10750 | 13.97020 | ||
0.15616 | 0.00375 | 0.41549 | 0.57795 | 1.22680 | 3.03850 | ||
0.98882 | 0.91498 | 0.99269 | 0.86806 | 0.75848 | 0.72047 | ||
6.18270 | 6.96220 | 0.21578 | 20.38000 | 28.04320 | 22.71800 | ||
0.00000 | 0.00000 | 0.00000 | 0.38512 | 1.01360 | 0.94404 | ||
0.99664 | 0.91483 | 0.99814 | 0.80857 | 0.61695 | 0.60180 | ||
6.18270 | 6.96220 | 0.21578 | 133.80190 | 228.90490 | 128.11660 | ||
0.00000 | 0.00000 | 0.00000 | 0.36309 | 0.83900 | 2.09790 | ||
0.99664 | 0.91483 | 0.99814 | 0.72070 | 0.60348 | 0.76208 | ||
6.18270 | 6.96220 | 0.21578 | 441.10500 | 782.61200 | 405.01650 | ||
0.00000 | 0.00000 | 0.00000 | 0.49383 | 1.24770 | 0.46759 | ||
0.99664 | 0.91483 | 0.99814 | 0.86642 | 0.40841 | 0.68921 |
Case | | | | | |
(Ⅰ) | | | | | |
(Ⅱ) | | | | | |
(Ⅲ) | | | | | |
(Ⅰ)' | | | | | |
(Ⅱ)' | | | | | |
(Ⅲ)' | | | | | |
Case | | | | | |
(Ⅰ) | | | | | |
(Ⅱ) | | | | | |
(Ⅲ) | | | | | |
(Ⅰ)' | | | | | |
(Ⅱ)' | | | | | |
(Ⅲ)' | | | | | |
(Ⅰ) | (Ⅱ) | (Ⅲ) | (Ⅰ)' | (Ⅱ)' | (Ⅲ)' | ||
5.70000 | 4.00000 | 5.74000 | 5.70000 | 4.00000 | 5.74710 | ||
4.27740 | 9.39060 | 54.67330 | 9.42280 | 15.11920 | 76.17700 | ||
0.45537 | 0.17338 | 0.03257 | 0.45537 | 0.17338 | 0.24487 | ||
13.18640 | 0.40287 | 14.95700 | 30.78020 | 81.84340 | 27.60650 | ||
0.00000 | 0.00000 | 0.00000 | 11.39260 | 19.27490 | 0.98300 | ||
0.94544 | 0.98766 | 0.77602 | 0.90994 | 0.42164 | 0.80234 | ||
13.18640 | 0.40287 | 14.95700 | 109.18220 | 316.97400 | 78.33540 | ||
0.00000 | 0.00000 | 0.00000 | 3.66710 | 9.38180 | 0.88459 | ||
0.94544 | 0.98766 | 0.77602 | 0.42567 | 0.64836 | 0.77224 | ||
13.18640 | 0.40287 | 14.95700 | 818.10600 | 2347.20000 | 549.54340 | ||
0.00000 | 0.00000 | 0.00000 | 10.20440 | 16.58980 | 0.72854 | ||
0.94544 | 0.98766 | 0.77602 | 0.78081 | 0.55872 | 0.77419 | ||
13.18640 | 0.40287 | 14.95700 | 2725.00000 | 7813.20000 | 1818.60000 | ||
0.00000 | 0.00000 | 0.00000 | 5.57280 | 11.35220 | 0.35697 | ||
0.94544 | 0.98766 | 0.77602 | 0.19058 | 0.57754 | 0.78170 |
(Ⅰ) | (Ⅱ) | (Ⅲ) | (Ⅰ)' | (Ⅱ)' | (Ⅲ)' | ||
5.70000 | 4.00000 | 5.74000 | 5.70000 | 4.00000 | 5.74710 | ||
4.27740 | 9.39060 | 54.67330 | 9.42280 | 15.11920 | 76.17700 | ||
0.45537 | 0.17338 | 0.03257 | 0.45537 | 0.17338 | 0.24487 | ||
13.18640 | 0.40287 | 14.95700 | 30.78020 | 81.84340 | 27.60650 | ||
0.00000 | 0.00000 | 0.00000 | 11.39260 | 19.27490 | 0.98300 | ||
0.94544 | 0.98766 | 0.77602 | 0.90994 | 0.42164 | 0.80234 | ||
13.18640 | 0.40287 | 14.95700 | 109.18220 | 316.97400 | 78.33540 | ||
0.00000 | 0.00000 | 0.00000 | 3.66710 | 9.38180 | 0.88459 | ||
0.94544 | 0.98766 | 0.77602 | 0.42567 | 0.64836 | 0.77224 | ||
13.18640 | 0.40287 | 14.95700 | 818.10600 | 2347.20000 | 549.54340 | ||
0.00000 | 0.00000 | 0.00000 | 10.20440 | 16.58980 | 0.72854 | ||
0.94544 | 0.98766 | 0.77602 | 0.78081 | 0.55872 | 0.77419 | ||
13.18640 | 0.40287 | 14.95700 | 2725.00000 | 7813.20000 | 1818.60000 | ||
0.00000 | 0.00000 | 0.00000 | 5.57280 | 11.35220 | 0.35697 | ||
0.94544 | 0.98766 | 0.77602 | 0.19058 | 0.57754 | 0.78170 |
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