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

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 ocking 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 ockings 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 ocking estimates show that the critical number of q for unconditional ocking is the same as in the case for zero timedelay, which shows the robustness of the Cucker and Dong's result with respect to small time-delay. © American Institute of Mathematical Sciences.