  It is known from both theory and experiments that introducing time delays
into the communication network of mobile-agent swarms produces coherent
rotational patterns. Often such spatio-temporal rotations can be bistable with
other swarming patterns, such as milling and flocking. Yet, most known
bifurcation results related to delay-coupled swarms rely on inaccurate
mean-field techniques. As a consequence, the utility of applying macroscopic
theory as a guide for predicting and controlling swarms of mobile robots has
been limited. To overcome this limitation, we perform an exact stability
analysis of two primary swarming patterns in a general model with time-delayed
interactions. By correctly identifying the relevant spatio-temporal modes that
determine stability in the presence of time delay, we are able to accurately
predict bistability and unstable oscillations in large swarm simulations--
laying the groundwork for comparisons to robotics experiments.
