On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators

Chun-Hsiung Hsia; Chang-Yeol Jung; Bongsuk Kwon

doi:10.3934/dcdsb.2018322

Article Contents

2019, Volume 24, Issue 7: 3319-3334. Doi: 10.3934/dcdsb.2018322

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

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: January 31, 2018

Revised: July 31, 2018

Early access: January 2019

Published: May 2019

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Abstract

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 $.

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Mathematics Subject Classification: Primary: 34D06; Secondary: 34D05.

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Figure 4.2. The Kuramoto model (1.1) with $N = 3$, $K = 1$, $D(\Omega)/K = 1.23691$

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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 $t$

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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 $t$

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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)

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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)'	$20$	$1.5$	$U(-1.23, 1.23)$	$1.59328$	$U(-\pi, \pi)$

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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)'
$0$	$D(\Theta(t))$	3.80000	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

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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)'	$20$	$K_8$	$U(-28, 28)$	$49.8966$	$U(-\pi, \pi)$

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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)'
$0$	$D(\Theta(t))$	5.70000	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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