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Cucker-Smale model with time delay

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  • 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.


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  • 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 $

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