Capturing pattern bi-stability dynamics in delay-coupled swarms

Abstract

-Swarms of large numbers of agents appear in many biological and engineering fields. Dynamic bi-stability of co-existing spatio-temporal patterns has been observed in many models of large population swarms. However, many reduced models for analysis, such as mean-field (MF), do not capture the bifurcation structure of bi-stable behavior. Here, we develop a new model for the dynamics of a large population swarm with delayed coupling. The additional physics predicts how individual particle dynamics affects the motion of the entire swarm. Specifically, (1) we correct the center of mass propulsion physics accounting for the particles' velocity distribution; (2) we show that the model we develop is able to capture the pattern bi-stability displayed by the full swarm model.

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.…”

“…This typically leads to coherent rotation, known as the rotating state, which is shown in Fig.1(d). Together the three states comprise the known dynamical modes for swarming networks with delay [11,12]. Phase diagrams are shown in Fig.2 for bimodal networks.…”

“…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.…”

“…where r i is the position of the ith agent in 2-dimensions, dots denote time derivatives, A ij is the connection matrix, J is the coupling strength between neighbors in the network, and τ is the characteristic time delay between agent interactions [12,21]. For simplicity and analytic tractability, we assume that the mutual forces are springlike: ∇ r i U (r i (t) − r j (t − τ )) = r i (t) − r j (t − τ ), though sufficiently small repulsive terms do not alter the dynamics [11].…”

“…where r i is the position of the ith agent in 2-dimensions, dots denote time derivatives, A ij is the connection matrix, J is the coupling strength between neighbors in the network, and τ is the characteristic time delay between agent interactions [11,20]. For simplicity, we assume that the mutual forces are spring-like:…”