# Emergence of partial locking states from the ensemble of Winfree oscillators

[Published in Quarterly of Applied Mathematics, 75 (2017), 39 – 68.]

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

(To see the Winfree model)

We study the emergence of partial locking states for a subsystem whose dynamics is governed by the Winfree model. The Winfree model is the first mathematical model for synchronization. Thanks to the lack of conservation laws except for the number of oscillators, it exhibits diverse asymptotic nonlinear patterns such as partial and complete phase locking, partial and complete oscillator death, and incoherent states. In this paper, we present two sufficient frameworks for a majority sub-ensemble to evolve to the phase-locked state asymptotically. Our sufficient frameworks are characterized in terms of the mass ratio of the subsystem compared to the total system, ratio of the coupling strength to the natural frequencies, and the phase diameter of the subsystem. We also provide several numerical simulations and compare their results to the analytical results.

# Collective synchronization of classical and quantum oscillators

[Published in EMS Surveys in Mathematical Sciences, 3 (2016), no. 2, 209 – 267.]

This is a joint work with Seung-Yeal Ha, Dongnam Ko, and Xiongtao Zhang.

Synchronization of weakly coupled oscillators is ubiquitous in biological and chemical complex systems. Recently, research on collective dynamics of many-body systems has been received much attention due to their possible applications in engineering. In this survey paper, we mainly focus on the large-time dynamics of several synchronization models and review state-of-art results on the collective behaviors for synchronization models. Following a chronological order, we begin our discussion with two classical phase models (Winfree and Kuramoto models), and two quantum synchronization models (Lohe and Schrödinger–Lohe models). For these models, we present several sufficient conditions for the emergence of synchronization using mathematical tools from dynamical systems theory, kinetic theory and partial differential equations in a unified framework.

# 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.