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July  2019, 24(7): 3319-3334. doi: 10.3934/dcdsb.2018322

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

Received  February 2018 Revised  August 2018 Published  January 2019

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 $.

Citation: Chun-Hsiung Hsia, Chang-Yeol Jung, Bongsuk Kwon. On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3319-3334. doi: 10.3934/dcdsb.2018322
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. ChoiS.-Y. HaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.  Google Scholar

[7]

S.-Y. HaH. 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.  Google Scholar

[8]

S.-Y. HaD. KoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. ChoiS.-Y. HaS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae, J. Math. Biol., 29 (1991), 571-585.  doi: 10.1007/BF00164052.  Google Scholar

[7]

S.-Y. HaH. 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.  Google Scholar

[8]

S.-Y. HaD. KoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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

Figure 4.2.  The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 1.23691$
Figure 4.1.  The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 0.0201916$. The plots are in log scale in $t$
Figure 4.3.  The Winfree model (3.1) with $N = 5$, $\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}} = 1.15405$ where the matrix $K = K_1$ is given in (4.2). The plots are in log scale in $t$
Figure 4.4.  The Winfree model (3.1) with $N = 5$, $K = $, $\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}} = 13.8456$ where the matrix $K = K_2$ is given in (4.2)
Table 2.  Parameters for Kuramoto model experimented in Table 3. The notation $ U(a, b) $ is a uniform random distribution over $ [a, b] $
Case $N$ $K$ $\Omega$ $D(\Omega)/K$ $\Theta(0)$
(Ⅰ) $3$ $1$ $\{-0.1, 0.1, 0.0\}$ $0.2$ $\{1.5, -1.7, 2.1\}$
(Ⅱ) $5$ $2$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $1$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ) $20$ $1$ $U(-0.123, 0.123)$ $0.214915$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $1$ $\{-0.6, 0.9, 0.5\}$ $1.5$ $\{-3.0, -0.7, -2.0\}$
(Ⅱ)' $5$ $1$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $2$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ)' $20$ $1.5$ $U(-1.23, 1.23)$ $1.59328$ $U(-\pi, \pi)$
Case $N$ $K$ $\Omega$ $D(\Omega)/K$ $\Theta(0)$
(Ⅰ) $3$ $1$ $\{-0.1, 0.1, 0.0\}$ $0.2$ $\{1.5, -1.7, 2.1\}$
(Ⅱ) $5$ $2$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $1$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ) $20$ $1$ $U(-0.123, 0.123)$ $0.214915$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $1$ $\{-0.6, 0.9, 0.5\}$ $1.5$ $\{-3.0, -0.7, -2.0\}$
(Ⅱ)' $5$ $1$ $\{0.9, 0.1, -1.1, -0.9, 0.0\}$ $2$ $\{-1.4, 2.3, -1.8, 0.5, -2.4\}$
(Ⅲ)' $20$ $1.5$ $U(-1.23, 1.23)$ $1.59328$ $U(-\pi, \pi)$
Table 3.  The Kuramoto phases $ \Theta(t) $ and the modulus of the order parameter, $ |r| $, given in (4.1)
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$3.800004.700004.601702.300004.700005.67130
$D(\dot{\Theta}(t))$0.231081.932600.678780.611641.886702.41560
$|r|$0.345150.305110.272910.603970.305110.03716
$5$$D(\Theta(t))$6.051706.961000.485077.563208.1075013.97020
$D(\dot{\Theta}(t))$0.156160.003750.415490.577951.226803.03850
$|r|$0.988820.914980.992690.868060.758480.72047
$20$$D(\Theta(t))$6.182706.962200.2157820.3800028.0432022.71800
$D(\dot{\Theta}(t))$0.000000.000000.000000.385121.013600.94404
$|r|$0.996640.914830.998140.808570.616950.60180
$150$$D(\Theta(t))$6.182706.962200.21578133.80190228.90490128.11660
$D(\dot{\Theta}(t))$0.000000.000000.000000.363090.839002.09790
$|r|$0.996640.914830.998140.720700.603480.76208
$500$$D(\Theta(t))$6.182706.962200.21578441.10500782.61200405.01650
$D(\dot{\Theta}(t))$0.000000.000000.000000.493831.247700.46759
$|r|$0.996640.914830.998140.866420.408410.68921
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$3.800004.700004.601702.300004.700005.67130
$D(\dot{\Theta}(t))$0.231081.932600.678780.611641.886702.41560
$|r|$0.345150.305110.272910.603970.305110.03716
$5$$D(\Theta(t))$6.051706.961000.485077.563208.1075013.97020
$D(\dot{\Theta}(t))$0.156160.003750.415490.577951.226803.03850
$|r|$0.988820.914980.992690.868060.758480.72047
$20$$D(\Theta(t))$6.182706.962200.2157820.3800028.0432022.71800
$D(\dot{\Theta}(t))$0.000000.000000.000000.385121.013600.94404
$|r|$0.996640.914830.998140.808570.616950.60180
$150$$D(\Theta(t))$6.182706.962200.21578133.80190228.90490128.11660
$D(\dot{\Theta}(t))$0.000000.000000.000000.363090.839002.09790
$|r|$0.996640.914830.998140.720700.603480.76208
$500$$D(\Theta(t))$6.182706.962200.21578441.10500782.61200405.01650
$D(\dot{\Theta}(t))$0.000000.000000.000000.493831.247700.46759
$|r|$0.996640.914830.998140.866420.408410.68921
Table 1.  Parameters for Winfree model experimented in Table 4. The upper triangular entries of matrices $ K_i $, $ i = 3, \cdots, 8 $ are generated from a uniform random distribution over $ [0.5, 1.0] $ and the lower triangular entries by a symmetry. $ U(a, b) $ denotes a uniform random distribution over $ [a, b] $.
Case $N$ $K$ $\Omega$ $\displaystyle\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}}$ $\Theta(0)$
(Ⅰ) $3$ $K_3$ $\{1.7, 1.1, -1.7\}$ $2.56732$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ) $5$ $K_4$ $\{-1.1, -1.7, 0.9, 1.4, -0.4\}$ $2.52008$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ) $20$ $K_5$ $U(-28, 28)$ $37.4299$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $K_6$ $\{5.0, 2.1, -3.7\}$ $7.01639$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ)' $5$ $K_7$ $\{-2.1, -1.7, 0.9, 10.4, -8.4\}$ $12.6565$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ)' $20$ $K_8$ $U(-28, 28)$ $49.8966$ $U(-\pi, \pi)$
Case $N$ $K$ $\Omega$ $\displaystyle\max_{1\leq i \leq N}\frac{|\omega_i|}{k_{ii}}$ $\Theta(0)$
(Ⅰ) $3$ $K_3$ $\{1.7, 1.1, -1.7\}$ $2.56732$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ) $5$ $K_4$ $\{-1.1, -1.7, 0.9, 1.4, -0.4\}$ $2.52008$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ) $20$ $K_5$ $U(-28, 28)$ $37.4299$ $U(-\pi, \pi)$
(Ⅰ)' $3$ $K_6$ $\{5.0, 2.1, -3.7\}$ $7.01639$ $\{-0.9, 2.7, -3.0\}$
(Ⅱ)' $5$ $K_7$ $\{-2.1, -1.7, 0.9, 10.4, -8.4\}$ $12.6565$ $\{-0.8, 1.8, -0.2, 2.6, -1.4\}$
(Ⅲ)' $20$ $K_8$ $U(-28, 28)$ $49.8966$ $U(-\pi, \pi)$
Table 4.  The Winfree phases $ \Theta(t) $ and the modulus of the order parameter, $ |r| $, given in (4.1).
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$5.700004.000005.740005.700004.000005.74710
$D(\dot{\Theta}(t))$4.277409.3906054.673309.4228015.1192076.17700
$|r|$0.455370.173380.032570.455370.173380.24487
$5$$D(\Theta(t))$13.186400.4028714.9570030.7802081.8434027.60650
$D(\dot{\Theta}(t))$0.000000.000000.0000011.3926019.274900.98300
$|r|$0.945440.987660.776020.909940.421640.80234
$20$$D(\Theta(t))$13.186400.4028714.95700109.18220316.9740078.33540
$D(\dot{\Theta}(t))$0.000000.000000.000003.667109.381800.88459
$|r|$0.945440.987660.776020.425670.648360.77224
$150$$D(\Theta(t))$13.186400.4028714.95700818.106002347.20000549.54340
$D(\dot{\Theta}(t))$0.000000.000000.0000010.2044016.589800.72854
$|r|$0.945440.987660.776020.780810.558720.77419
$500$$D(\Theta(t))$13.186400.4028714.957002725.000007813.200001818.60000
$D(\dot{\Theta}(t))$0.000000.000000.000005.5728011.352200.35697
$|r|$0.945440.987660.776020.190580.577540.78170
$t$(Ⅰ)(Ⅱ)(Ⅲ)(Ⅰ)'(Ⅱ)'(Ⅲ)'
$0$$D(\Theta(t))$5.700004.000005.740005.700004.000005.74710
$D(\dot{\Theta}(t))$4.277409.3906054.673309.4228015.1192076.17700
$|r|$0.455370.173380.032570.455370.173380.24487
$5$$D(\Theta(t))$13.186400.4028714.9570030.7802081.8434027.60650
$D(\dot{\Theta}(t))$0.000000.000000.0000011.3926019.274900.98300
$|r|$0.945440.987660.776020.909940.421640.80234
$20$$D(\Theta(t))$13.186400.4028714.95700109.18220316.9740078.33540
$D(\dot{\Theta}(t))$0.000000.000000.000003.667109.381800.88459
$|r|$0.945440.987660.776020.425670.648360.77224
$150$$D(\Theta(t))$13.186400.4028714.95700818.106002347.20000549.54340
$D(\dot{\Theta}(t))$0.000000.000000.0000010.2044016.589800.72854
$|r|$0.945440.987660.776020.780810.558720.77419
$500$$D(\Theta(t))$13.186400.4028714.957002725.000007813.200001818.60000
$D(\dot{\Theta}(t))$0.000000.000000.000005.5728011.352200.35697
$|r|$0.945440.987660.776020.190580.577540.78170
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