  Swarming behavior, where coherent motion emerges from the interactions of
many mobile agents, is ubiquitous in physics and biology. Moreover, there are
many efforts to replicate swarming dynamics in mobile robotic systems which
take inspiration from natural swarms. In particular, understanding how swarms
come apart, change their behavior, and interact with other swarms is a research
direction of special interest to the robotics and defense communities. Here we
develop a theoretical approach that can be used to predict the parameters under
which colliding swarms form a stable milling state. Our analytical methods rely
on the assumption that, upon collision, two swarms oscillate near a
limit-cycle, where each swarm rotates around the other while maintaining an
approximately constant density. Using our methods, we are able to predict the
critical swarm-swarm interaction coupling (below which two colliding swarms
merely scatter) for nearly aligned collisions as a function of physical swarm
parameters. We show that the critical coupling corresponds to a saddle-node
bifurcation of a limit-cycle in the constant-density approximation. Finally, we
show preliminary results from experiments in which two swarms of micro UAVs
collide and form a milling state, which is in general agreement with our
theory.
