On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications

Jiu-Gang Dong; Seung-Yeal Ha; Doheon Kim

doi:10.3934/krm.2020022

Article Contents

2020, Volume 13, Issue 4: 653-676. Doi: 10.3934/krm.2020022

This issue Previous Article Next Article A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions

On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826
3.
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, South Korea

^* Corresponding author: Doheon Kim
^* Corresponding author: Doheon Kim

Received: May 2019

Revised: March 2020

Published: May 2020

The work of S.-Y. Ha was supported by the National Research Foundation of Korea(NRF-2020R1A2C3A01003881), and the work of D. Kim was supported by a KIAS Individual Grant (MG073901) at Korea Institute for Advanced Study

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Abstract

We study time-asymptotic interplay between time-delayed communication and Cucker-Smale (C-S) velocity alignment. For this, we present two sufficient frameworks for the asymptotic flocking to the continuous and discrete C-S models with $ q $-closest neighbors in the presence of time-delayed communications. Communication time-delays result from the finite-propagation speed of information and they are often ignored in the first place modeling of collective dynamics. In the absence of time-delays in communication, Cucker and Dong showed that the C-S model with $ q $-closest neighbors can exhibit a phase-transition like phenomenon for unconditional and conditional flockings depending on the size $ q $ relative to system size. In this paper, we investigate whether Cucker and Dong's result is robust with respect to the time-delayed communications or not. In fact, our flocking estimates show that the critical number of $ q $ for unconditional flocking is the same as in the case for zero time-delay, which shows the robustness of the Cucker and Dong's result with respect to small time-delay.

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Mathematics Subject Classification: Primary: 34D05; Secondary: 39A12, 70F99.

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Figure 1. The initial configuration $ ( {\boldsymbol{x}}_i, {\boldsymbol{v}}_i) $ $ (i = 1.\cdots,20) $

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Figure 2. The graphs of $ \mathcal D(V(t)) $ and $ \log \mathcal D(V(t)) $ for $ q = 9,10,11,12 $

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Figure 3. The graphs of $ \mathcal D(X(t)) $ and $ \log \mathcal D(X(t)) $ for $ q = 9,10,11,12 $

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Figure 4. Emergence of bi-cluster flocking for $ q = 9 $

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