A collisionless singular discrete Cucker-Smale model with deterministic perturbations

Abstract

In this paper, we investigate two non-linearly perturbed extensions of the discrete Cucker-Smale model with singular coupling weights. The first perturbation is that all agents have non-identical free-will accelerations, and the second is that all agents have identical intrinsic dynamics with the Lipschitz property. For the first model, we apply the induction method and discrete energy method to show that agents avoid collisions for any time and flocking occurs under some initial conditions, if the diameter of agents’ free-will accelerations is summable. For the second model, we obtain collision-avoiding flocking occurrence under suitable initial data and the Lipschitz constant of the function for the intrinsic dynamics. We also provide several numerical examples to illustrate our main results.