# American Institute of Mathematical Sciences

August  2020, 13(4): 653-676. doi: 10.3934/krm.2020022

## 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

Received  May 2019 Revised  March 2020 Published  May 2020

Fund Project: 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

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.

Citation: Jiu-Gang Dong, Seung-Yeal Ha, Doheon Kim. On the Cucker-Smale ensemble with $q$-closest neighbors under time-delayed communications. Kinetic & Related Models, 2020, 13 (4) : 653-676. doi: 10.3934/krm.2020022
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##### References:
The initial configuration $( {\boldsymbol{x}}_i, {\boldsymbol{v}}_i)$ $(i = 1.\cdots,20)$
The graphs of $\mathcal D(V(t))$ and $\log \mathcal D(V(t))$ for $q = 9,10,11,12$
The graphs of $\mathcal D(X(t))$ and $\log \mathcal D(X(t))$ for $q = 9,10,11,12$
Emergence of bi-cluster flocking for $q = 9$
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