The Cucker-Smale model with time delay

TL;DR

This paper analyzes the Cucker-Smale flocking model incorporating a positive time delay, demonstrating that unconditional flocking persists for delay regardless of its size and establishing exponential decay of velocity differences.

Contribution

It extends the classical Cucker-Smale model by including time delay and proves that flocking behavior remains unconditional for all delays with exponential convergence.

Findings

Unconditional flocking occurs for all positive delays when β ≤ 1/2.

Velocity differences decay exponentially over time.

Flocking behavior is robust to the introduction of time delay.

Abstract

We study the classical Cucker-Smale model in continuous time with a positive time delay $\tau$. As in the non-delayed case, unconditional flocking occurs when $\beta \leq 1/2$ for every $\tau>0$. Furthermore, we prove the exponential decay for the diameter of the velocities.