Collective behaviors of a Winfree ensemble on an infinite cylinder

Seung-Yeal Ha; Myeongju Kang; Bora Moon

doi:10.3934/dcdsb.2020204

Article Contents

2021, Volume 26, Issue 5: 2749-2779. Doi: 10.3934/dcdsb.2020204

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Collective behaviors of a Winfree ensemble on an infinite cylinder

1.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826 and, Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of)
2.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea (Republic of)
3.
Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea (Republic of)

^* Corresponding author: Bora Moon
^* Corresponding author: Bora Moon

Received: January 31, 2020

Revised: March 31, 2020

Early access: June 2020

Published: May 2021

The work of S.-Y. Ha is supported by the NRF grant (2017R1A2B2001864) and the work of B. Moon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

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Abstract

The Winfree model is the first phase model for synchronization and it exhibits diverse asymptotic patterns that cannot be observed in the Kuramoto model. In this paper, we propose a Winfree type model describing the aggregation of particles on the surface of an infinite cylinder. For a special case, our proposed model is in fact equivalent to the complex Winfree model. For the proposed model, we present a sufficient framework leading to the complete oscillator death and uniform $ \ell_p $-stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field limit. In addition, we also provide several numerical simulations for the particle and compare them with analytical results.

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Mathematics Subject Classification: Primary:70F99, 92B25;Secondary:35Q70.

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Figure 1. Time evolution of $ \frac{X(t)}{t} $ for two different initial data

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Figure 2. Initial and terminal configurations of $ (X_1, Y_1) $ and $ (X_2, Y_2) $

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Figure 3. Uniform $ \ell_1 $-stability

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