Article Contents
Article Contents

# Cucker-Smale model with time delay

• 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.

Mathematics Subject Classification: 93D20, 34K25, 92D50.

 Citation:

• Figure 3.  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 $a\in A$

Figure 1.  Numerical simulation of Example 4.1. We use system 33 and set $\tau = 1$, $\Delta t = 1/1000$, $N = 2$ and $K = 1$

Figure 2.  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 $K = 1$

Figures(3)