# American Institute of Mathematical Sciences

• Previous Article
Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients
• DCDS-B Home
• This Issue
• Next Article
Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation
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:

show all references

##### References:
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
 [1] Seung-Yeal Ha, Jinyeong Park, Sang Woo Ryoo. Emergence of phase-locked states for the Winfree model in a large coupling regime. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3417-3436. doi: 10.3934/dcds.2015.35.3417 [2] Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787 [3] Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001 [4] Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013 [5] Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks & Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018 [6] Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082 [7] Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943 [8] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006 [9] Zhuchun Li, Yi Liu, Xiaoping Xue. Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 345-367. doi: 10.3934/dcds.2019014 [10] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [11] Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084 [12] Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225 [13] Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161 [14] Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407 [15] Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351 [16] Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure & Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217 [17] Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253-261. doi: 10.3934/mbe.2011.8.253 [18] J. R. Fernández, R. Martínez, J. M. Viaño. Analysis of a bone remodeling model. Communications on Pure & Applied Analysis, 2009, 8 (1) : 255-274. doi: 10.3934/cpaa.2009.8.255 [19] Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491 [20] Fritz Gesztesy, Rudi Weikard, Maxim Zinchenko. On a class of model Hilbert spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5067-5088. doi: 10.3934/dcds.2013.33.5067

2018 Impact Factor: 1.008