Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles

Abstract

We study the communication time-delay effect on the emergent dynamics of thermodynamic Cucker–Smale (TCS) ensemble with temperature as an internal state. In the presence of time-delay, conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the TCS model, as it is. We show that temperature stays positive in the presence of small time-delay. As long as temperatures stay positive, the TCS model is well-posed by direct application of Cauchy–Lipschitz theory. When the size of time-delay and initial data are sufficiently small, we can derive differential inequalities for velocity and temperature diameters. Using these differential inequalities, we show that temperature and velocity diameters tend to zero exponentially fast, whereas the position diameter is uniformly bounded. We also present several numerical simulations and compare them with analytic results. © 2018 Elsevier Inc.