Symmetry and reduction in collectives: low-dimensional cyclic pursuit.

We investigate low-dimensional examples of cyclic pursuit in a collective, wherein each agent employs a constant bearing (CB) steering law relative to exactly one other agent. For the case of three agents in the plane, we characterize relative equilibria and pure shape equilibria of associated closed-loop dynamics. Re-scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis. Study of bifurcation of a family of collinear equilibria dependent on a single CB control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space. For collectives in three dimensions, with an appropriate notion of CB pursuit strategy and corresponding steering law, the two-agent case proves to be explicitly integrable. These results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue, and raise interesting possibilities for behaviour in biological contexts.