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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  July 2019 Early access  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 and 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 [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 [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. [15] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.

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##### 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 [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 [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. [15] A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 1.23691$
The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 0.0201916$. The plots are in log scale in $t$
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 $t$
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)
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)' $20$ $1.5$ $U(-1.23, 1.23)$ $1.59328$ $U(-\pi, \pi)$
The Kuramoto phases $\Theta(t)$ and the modulus of the order parameter, $|r|$, given in (4.1)
 $t$ (Ⅰ) (Ⅱ) (Ⅲ) (Ⅰ)' (Ⅱ)' (Ⅲ)' $0$ $D(\Theta(t))$ 3.80000 4.70000 4.60170 2.30000 4.70000 5.67130 $D(\dot{\Theta}(t))$ 0.23108 1.93260 0.67878 0.61164 1.88670 2.41560 $|r|$ 0.34515 0.30511 0.27291 0.60397 0.30511 0.03716 $5$ $D(\Theta(t))$ 6.05170 6.96100 0.48507 7.56320 8.10750 13.97020 $D(\dot{\Theta}(t))$ 0.15616 0.00375 0.41549 0.57795 1.22680 3.03850 $|r|$ 0.98882 0.91498 0.99269 0.86806 0.75848 0.72047 $20$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 20.38000 28.04320 22.71800 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.38512 1.01360 0.94404 $|r|$ 0.99664 0.91483 0.99814 0.80857 0.61695 0.60180 $150$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 133.80190 228.90490 128.11660 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.36309 0.83900 2.09790 $|r|$ 0.99664 0.91483 0.99814 0.72070 0.60348 0.76208 $500$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 441.10500 782.61200 405.01650 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.49383 1.24770 0.46759 $|r|$ 0.99664 0.91483 0.99814 0.86642 0.40841 0.68921
 $t$ (Ⅰ) (Ⅱ) (Ⅲ) (Ⅰ)' (Ⅱ)' (Ⅲ)' $0$ $D(\Theta(t))$ 3.80000 4.70000 4.60170 2.30000 4.70000 5.67130 $D(\dot{\Theta}(t))$ 0.23108 1.93260 0.67878 0.61164 1.88670 2.41560 $|r|$ 0.34515 0.30511 0.27291 0.60397 0.30511 0.03716 $5$ $D(\Theta(t))$ 6.05170 6.96100 0.48507 7.56320 8.10750 13.97020 $D(\dot{\Theta}(t))$ 0.15616 0.00375 0.41549 0.57795 1.22680 3.03850 $|r|$ 0.98882 0.91498 0.99269 0.86806 0.75848 0.72047 $20$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 20.38000 28.04320 22.71800 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.38512 1.01360 0.94404 $|r|$ 0.99664 0.91483 0.99814 0.80857 0.61695 0.60180 $150$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 133.80190 228.90490 128.11660 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.36309 0.83900 2.09790 $|r|$ 0.99664 0.91483 0.99814 0.72070 0.60348 0.76208 $500$ $D(\Theta(t))$ 6.18270 6.96220 0.21578 441.10500 782.61200 405.01650 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 0.49383 1.24770 0.46759 $|r|$ 0.99664 0.91483 0.99814 0.86642 0.40841 0.68921
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)' $20$ $K_8$ $U(-28, 28)$ $49.8966$ $U(-\pi, \pi)$
The Winfree phases $\Theta(t)$ and the modulus of the order parameter, $|r|$, given in (4.1).
 $t$ (Ⅰ) (Ⅱ) (Ⅲ) (Ⅰ)' (Ⅱ)' (Ⅲ)' $0$ $D(\Theta(t))$ 5.70000 4.00000 5.74000 5.70000 4.00000 5.74710 $D(\dot{\Theta}(t))$ 4.27740 9.39060 54.67330 9.42280 15.11920 76.17700 $|r|$ 0.45537 0.17338 0.03257 0.45537 0.17338 0.24487 $5$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 30.78020 81.84340 27.60650 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 11.39260 19.27490 0.98300 $|r|$ 0.94544 0.98766 0.77602 0.90994 0.42164 0.80234 $20$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 109.18220 316.97400 78.33540 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 3.66710 9.38180 0.88459 $|r|$ 0.94544 0.98766 0.77602 0.42567 0.64836 0.77224 $150$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 818.10600 2347.20000 549.54340 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 10.20440 16.58980 0.72854 $|r|$ 0.94544 0.98766 0.77602 0.78081 0.55872 0.77419 $500$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 2725.00000 7813.20000 1818.60000 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 5.57280 11.35220 0.35697 $|r|$ 0.94544 0.98766 0.77602 0.19058 0.57754 0.78170
 $t$ (Ⅰ) (Ⅱ) (Ⅲ) (Ⅰ)' (Ⅱ)' (Ⅲ)' $0$ $D(\Theta(t))$ 5.70000 4.00000 5.74000 5.70000 4.00000 5.74710 $D(\dot{\Theta}(t))$ 4.27740 9.39060 54.67330 9.42280 15.11920 76.17700 $|r|$ 0.45537 0.17338 0.03257 0.45537 0.17338 0.24487 $5$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 30.78020 81.84340 27.60650 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 11.39260 19.27490 0.98300 $|r|$ 0.94544 0.98766 0.77602 0.90994 0.42164 0.80234 $20$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 109.18220 316.97400 78.33540 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 3.66710 9.38180 0.88459 $|r|$ 0.94544 0.98766 0.77602 0.42567 0.64836 0.77224 $150$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 818.10600 2347.20000 549.54340 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 10.20440 16.58980 0.72854 $|r|$ 0.94544 0.98766 0.77602 0.78081 0.55872 0.77419 $500$ $D(\Theta(t))$ 13.18640 0.40287 14.95700 2725.00000 7813.20000 1818.60000 $D(\dot{\Theta}(t))$ 0.00000 0.00000 0.00000 5.57280 11.35220 0.35697 $|r|$ 0.94544 0.98766 0.77602 0.19058 0.57754 0.78170
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