Communications in Mathematical Sciences

Volume 19 (2021)

Number 8

Emergent asymptotic patterns for the discrete and continuous Winfree models with inertia

Pages: 2217 – 2248

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n8.a7

Authors

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and Korea Institute for Advanced Study, Seoul, South Korea)

Myeongju Kang (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea; and Korea Institute for Advanced Study, Seoul, South Korea)

Woojoo Shim (Department of Mathematical Sciences, Seoul National University, Seoul, South Korea)

Abstract

We study emergent dynamics of the continuous Winfree model with inertia and its discrete analogue. The Winfree model is the first mathematical model for weakly-coupled oscillators modeling a collective synchronization of pulse-coupled oscillators. Unlike the Kuramoto-type models, the Winfree model does not conserve the total phase, so that its emergent dynamics becomes more interesting. In this paper, we provide sufficient conditions for the complete oscillator death to the Winfree model in the presence of inertia and the discrete-time analogue with or without inertia. Moreover, we also present a uniform-in-time convergence from the discrete model to the continuous model for zero inertia case, as the time-step tends to zero.

Keywords

discretization, inertia, synchronization, Winfree model

2010 Mathematics Subject Classification

34C15, 34D06, 82C22

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Received 27 July 2020

Accepted 11 May 2021

Published 7 October 2021