# 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.

# Remarks on the complete synchronization of the Kuramoto model

[Published in Nonlinearity, 28 (2015), no. 5, 1441-1462.]

[Preprint version]

This is a joint work with Seung-Yeal Ha and Hwa Kil Kim.
(To see the Kuramoto model)

So far, the previous results on the synchronization for the Kuramoto model had constraints on the initial configurations. The previous researches assumed that the initial positions of oscillators are confined in a half circle: $\displaystyle D(\Theta_0):=\max_{i,j}|\theta_{i0} - \theta_{j0}| \leq \pi$. However, it is known by numerical simulations that the synchronization occurs for any initial configurations with sufficiently large coupling strength $K$, except unstable equilbria. The previous analysis used the diameter $D(\Theta)$ as a Lyapunov functional to show the synchronization. We present an improved exponential synchronization estimate by extending the constraint on the initial configuration so that $D(\Theta_0) \geq \pi$. We use the dynamics of Kuramoto order paremeters defined by
\begin{equation*}\label{order-1}
re^{\mathrm i \phi} := \frac{1}{N} \sum_{j=1}^N e^{\mathrm i \theta_j}
\end{equation*}

The Kuramoto model can be expressed into the following form:
\begin{equation*}\label{Ku-order}
\dot \theta_i = \Omega_i - Kr\sin(\theta_i - \phi)
\end{equation*}

We show that the oscillators contract into a half circle region in finite time.

# Kuramoto model

Synchronization is collective behaviors of periodic motions, which is easily observed in many biologiclal, social and physical systems. Y. Kuramoto proposed a mathmatical model for synchronization in 1975. Kuramoto model describes the periodic motion as a rotation of phase on the unit circle. By using polar coordinate, the dynamics can be expressed in the following form:
\label{Ku-1}
\dot\theta_i = \Omega_i + \frac{K}{N} \sum_{j=1}^N \sin(\theta_j - \theta_i) \quad \mbox{for} \quad  i=1, \cdots, N

$\Omega_i$ and $K$ Kare called the natural frequency and the coupling strength, respectively.