# Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators

[Analysis and Applications, published in online.]

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

We study the dynamic interplay between inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. When external fields and internal forces are exerted on a system of Kuramoto oscillators, each oscillator has its own distinct dynamics, so that there is no notion of collective dynamics in the ensemble, and complete synchronization is not observed in such systems. In this paper, we study a relaxed version of synchronization, namely the “practical synchronization”, of Kuramoto oscillators, emerging from the dynamic interplay between inertia and heterogeneous decoupled dynamics. We will show that for some class of initial configurations and parameters, the fluctuation of phases and frequencies around the average values will be proportional to the inverse of the coupling strength. We provide several numerical examples, and compare these with our analytical results.

# 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:
\label{prac-1}
\dot \zeta_i = \mathcal F_i(\zeta_i, t) + K\sum_{j=1}^N \psi_{ij} \sin(\zeta_j - \zeta_i),

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.