Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

Hansol Park

doi:10.3934/dcds.2021134

Article Contents

2022, Volume 42, Issue 2: 707-735. Doi: 10.3934/dcds.2021134

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Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

Hansol Park^,

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

^*Corresponding author: Hansol Park
^*Corresponding author: Hansol Park

Received: February 2021

Revised: June 2021

Early access: September 2021

Published: February 2022

The work of H. Park is supported by National Research Foundation of Korea (NRF-2020R1A2C3A01003881).

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Abstract

We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential $ \ell^1 $-stability and the existence of the equilibrium solution.

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Mathematics Subject Classification: Primary: 70G60, 34D06; Secondary: 70F10.

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Figure 1. Geometric visualization of $\mathcal{D}_{\frac{\pi}{2}-\alpha} $ when $d = 2 $

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Figure 2. Examples of $ \tilde{I} $

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Figure 3. Geometric visualization of $n_e(x_j)$ in Lemma 4.1

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Figure 4. Geometric visualization of the condition for $ \Omega_i$ when $d = 2 $

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