Cucker-Smale model with normalized communication weights and time delay

TL;DR

This paper analyzes a delayed Cucker-Smale model with normalized communication, establishing conditions for flocking, deriving a mean-field limit, and exploring the system's long-term behavior through theoretical proofs and numerical simulations.

Contribution

It introduces a novel analysis of a Cucker-Smale model with time delay and normalized weights, including a Lyapunov functional and mean-field limit results.

Findings

Established sufficient conditions for asymptotic flocking.

Proved existence and stability of measure-valued solutions for the Vlasov-type equation.

Numerical simulations reveal flocking and oscillatory behaviors depending on delay size.

Abstract

We study a Cucker-Smale-type system with time delay in which agents interact with each other through normalized communication weights. We construct a Lyapunov functional for the system and provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. We also carry out a rigorous limit passage to the mean-field limit of the particle system as the number of particles tends to infinity.For the resulting Vlasov-type equation we prove the existence, stability and large-time behavior of measure-valued solutions. This is, to our best knowledge, the first such result for a Vlasov-type equation with time delay. We also present numerical simulations of the discrete system with few particles that provide further insights into the flocking and oscillatory behaviors of the particle velocities depending on the size of the time delay.