Delay Induced Instabilities in Self-Propelling Swarms

TL;DR

This paper investigates how communication time delays affect the stability and collective behavior of self-propelling particle swarms, revealing a delay-induced transition characterized by a Hopf bifurcation.

Contribution

It introduces a model incorporating time delay into swarm dynamics and analytically links delay to a transition via Hopf bifurcation, expanding understanding of swarm stability.

Findings

Delay causes a transition to synchronized oscillations in swarms.

The transition is characterized by a Hopf bifurcation.

Analytical predictions match numerical simulations.

Abstract

We consider a general model of self-propelling particles interacting through a pairwise attractive force in the presence of noise and communication time delay. Previous work by Erdmann, et al. [Phys. Rev. E {\bf 71}, 051904 (2205)] has shown that a large enough noise intensity will cause a translating swarm of individuals to transition to a rotating swarm with a stationary center of mass. We show that with the addition of a time delay, the model possesses a transition that depends on the size of the coupling amplitude. This transition is independent of the initial swarm state (traveling or rotating) and is characterized by the alignment of all of the individuals along with a swarm oscillation. By considering the mean field equations without noise, we show that the time delay induced transition is associated with a Hopf bifurcation. The analytical result yields good agreement with numerical computations of the value of the coupling parameter at the Hopf point.