Practical synchronization of Kuramoto system with an intrinsic dynamics

[Published in Networks and Heterogeneous Media 10 (2015), no. 4, 787 - 807.]

[Preprint version]

This is a joint work with Seung-Yeal Ha and Se Eun Noh.
(To see the Kuramoto model)

Let \zeta_i \in \mathbb R be a Kuramoto oscillators. Consider the Kuramoto system under external forcings:
\begin{equation}\label{prac-1}
\dot \zeta_i = \mathcal F_i(\zeta_i, t) + K\sum_{j=1}^N \psi_{ij} \sin(\zeta_j - \zeta_i),
\end{equation}
where the static matrix \Psi = (\psi_{ij}) satisfies symmetry and path connectedness. We study the practical synchronization of the Kuramoto dynamics of units distributed over networks. The practical synchronization is define by
\begin{equation*}\label{prac-2}
\lim_{K\to\infty} \limsup_{t\to\infty} D(\zeta(t))= 0.
\end{equation*}

The unit dynamics on the nodes of the network are governed by the interplay between their own intrinsic dynamics and Kuramoto coupling dynamics. Under some boundedness conditions, we show that the system \eqref{prac-1} yields a practical synchronization.