×

On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators. (English) Zbl 1462.34080

Consider the general Kuramoto model \[ \displaystyle{\dot{\theta}_i = \omega_i + \frac{K}{N}\sum_{j=1}^N\sin(\theta_j-\theta_i)\ ;\ t>0\;,\;i=1,\overline{N}} \] and the Winfree model \[ \displaystyle{\dot{\theta}_i = \omega_i - \sum_{j=1}^Nk_{ij}(1+\cos\theta_j)\sin\theta_i\;,\;(k_{ij}=k_{ji}>0)\ ;\ t>0\;,\;i=\overline{1,N}} \] It is defined the global convergence of frequency synchronization as the property \[ \displaystyle{\lim_{t\rightarrow\infty}\dot{\theta}_i = \omega = \frac{1}{N}\sum_1^N\omega_i} \] for any initial conditions \(\theta_i(0)\), \(i=\overline{1,N}\). The paper gives sufficient conditions for the achievement of the aforementioned property in the case \(N=3\) for both Kuramoto and Winfree models.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI