# American Institute of Mathematical Sciences

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May  2022, 42(5): 2409-2432. doi: 10.3934/dcds.2021195

## Cucker-Smale model with time delay

 IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina

Received  May 2021 Revised  October 2021 Published  May 2022 Early access  December 2021

We study the flocking model for continuous time introduced by Cucker and Smale adding a positive time delay $\tau$. The goal of this article is to prove that the same unconditional flocking result for the non-delayed case is valid in the delayed case. A novelty is that we do not need to impose any restriction on the size of $\tau$. Furthermore, when the unconditional flocking occurs, velocities converge exponentially fast to a common one.

Citation: Mauro Rodriguez Cartabia. Cucker-Smale model with time delay. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2409-2432. doi: 10.3934/dcds.2021195
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We plot two simulations using system 33. The imagines at the top represent the case of $4$ particles with $K = 5$. The imagines at the bottom represent the case of $20$ particles with $K = 25$. The black solid lines show the center of positions $x_{\bf{c}}$ (left imagines) and the center of velocities $v_{\bf{c}}$ (right imagines). In both cases we set $\Delta t = 1/1000$, $\tau = 1$ and the initial conditions are constant in the velocities, satisfying $(d/dt) x^{(a)}_0 = v^{(a)}_0$ for each $a\in A$
Numerical simulation of Example 4.1. We use system 33 and set $\tau = 1$, $\Delta t = 1/1000$, $N = 2$ and $K = 1$
Numerical simulation of Example 4.2 with $\tau = 1$ and $\varepsilon = 0.2$. We use system 33 setting $\Delta t = 1/1000$, $N = 2$ and $K = 1$
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