# Emergent dynamics of Winfree oscillators on locally coupled networks

This is a joint work with Seung-Yeal Ha, Dongnam Ko, and Sang Woo Ryoo.

(To see the Winfree model)

We study the emergent dynamics of the Winfree model on a locally coupled static network. For the emergence of complete or partial oscillator deaths (COD, POD), we provided sufficient frameworks in terms of connection topology, coupling strength, and coupling functions. Moreover, our results for COD and POD covered generic initial data in the case of special sensitivity and influence functions $S(\theta)=-\sin\theta$ and $I(\theta) = 1 + \cos\theta$, in large coupling regimes. We also provided three qualitative estimates; in particular, we considered the existence of an attractor with positive measures, exponential $\ell_1-$contractivity, and the existence of an equilibrium inside the attractor of a large-coupling regime.

# Emergence of phase-locked states for the Winfree model in a large coupling regime

[Published in Discrete and Continuous Dynamical Systems - Series A, 35 (2015), no. 8, 3417-3436.]

[Preprint version]

This is a joint work with Seung-Yeal Ha and Sang Woo Ryoo.

(To see the Winfree model)

We study the large-time behavior of the globally coupled Winfree model in a large coupling regime. The Winfree model is the first mathematical model for the synchronization phenomenon in an ensemble of weakly coupled limit-cycle oscillators. For the dynamic formation of phase-locked states, we provide a sufficient framework in terms of geometric conditions on the coupling functions and coupling strength. We show that in the proposed framework, the emergent phase-locked state is the unique equilibrium state and it is asymptotically stable in an $\ell_1$-norm; further, we investigate its configurational structure. We also provide several numerical simulations, and compare them with our analytical results.

# Winfree model

In 1967, A. Winfree proposed a synchronization model, which is known for the first mathematical model describing synchronous phenomena.
\label{winf-1}
\dot \theta_i =\Omega_i + \frac{K}{N} S(\theta_i)\sum_{j=1}^N I(\theta_j), \quad i=1, \cdots, N,

where $S$ and $I$ are the sensitivity and influence functions, respectively. In the literatures, $S$ and $I$ are often assumed to be
\label{winf-2}
S(\theta) = -\sin\theta, \quad I(\theta) = 1+\cos\theta.

# 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.