Delay-induced instabilities in self-propelling swarms

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 71, 051904 (2005)] 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 timedelay-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.The collective motion of multiagent systems has long been observed in biological populations including bacterial colonies [1][2][3], slime molds [4,5], locusts [6], and fish [7]. However, mathematical studies of swarming behavior have been performed for only a few decades. In addition to providing examples of biological pattern formation, the information gained from these mathematical investigations has led to an increased ability to intelligently design and control man-made vehicles [8][9][10][11][12].Many types of mathematical models have been used to describe coherent swarms. One popular approach is based on a continuum approximation in which scalar and vector fields are used to describe all of the relevant quantities [6,[13][14][15][16]. Another popular approach is based on treating every biological or mechanical individual as a discrete particle [7,14,15,17,18]. Depending on the problem, these individual-based models may be deterministic or stochastic.Regardless of the type of swarm model being used, one can see the emergence of ordered swarm states from an initial disordered state where individual particles have random velocity directions [13,14,17]. These ordered states may be translational or rotational in motion, and they may be spatially distributed or localized in clusters.In particular, it is known that a localized swarm state may transition to a new dynamical region as the system parameters or the noise intensity is changed. For example, it has been shown in [18] that a planar model of self-propelling particles interacting via a harmonic attractive potential in the presence of noise possesses a noise-induced transition whereby the translational motion of the swarm breaks down into rotational motion.Another aspect of swarm modeling that has not yet been considered is the effect of time delayed interactions arising from finite communication times between individuals. Much attention has been given to the effects of time delays in the context of physiology [19], optics [20], neurons [21], lasers [22], and many other types of systems. The aim of this ...

Hybrid dynamics in delay-coupled swarms with mothership networks

“…Moreover, it is known that time-delays in the agent interactions can produce a rotating state in which a swarm becomes highly aligned and localized [12,13]. The current research builds on the previous results for homogeneous networks, by generalizing the network topology.…”

“…Like the rotating state, many solutions are possible to Eqs. (13)(14) in general, depending on the parameter values, including multiple stable branches. This can lead to discontinuous jumps between hybrid states with different frequencies (as shown in Fig.7).…”

“…Eliminating τ in Eqs. (13)(14) gives a polynomial expression for the bifurcation frequency, ω (h) * :…”

“…Most studies of multiagent robotic systems with time delay have assumed global interactions or homogeneous topology [11,13,21]. In general, the effects of complex network structure on swarm behavior are much less explored, particularly with time-delayed interactions, even though topology is known to strongly influence many processes and produce interesting new dynamics [1,[22][23][24].…”

“…Several recent works in swarm pattern formation have focused on time-delay effects, which can produce new patterns and bistability [11][12][13]. Delays model the finite time required for agents to send and process information in real systems.…”