Unit speed flocking model with distance-dependent time delay

Abstract

We consider the emergence of flocking on the unit-speed model with a time-varying delay. We assume that the delay between two particles is a composition of a delay functional and the spatial distance between the particle's position and the other delayed particle's position. Therefore, the delay depends on the ensemble of all particles and is defined implicitly. For the local existence of the solution, we assume that the delay functional is a function and its norm is less than the speed of the particles. Using the unit-speed property and the local solution, we can obtain the global solution. To obtain the asymptotic flocking estimate, we assume that the delay functional is bounded by a linear function of the spatial distance and the coefficients of the linear function are sufficiently small. Under these assumptions on the delay functional, we present a sufficient condition of the initial configuration to obtain the exponential decay for the velocity diameter. We also validate our analytical results with several numerical experiments.