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

• 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 $N\ge2$.

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

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

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

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

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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