Collective motion of ring-coupled planar particles

We study stabilization of collective motion of N constant-speed, planar particles with less than all-to-all coupling. Our interest is in circular motions of the particles around the fixed center of mass of the group, as has been studied previously with all-to-all coupling. We focus on coupling defined by a ring, i.e., each particle communicates with exactly two other particles. The Kuramoto model of coupled oscillators, restricted to "ring" coupling, serves as our model for controlling the relative headings of the particles. Each phase oscillator represents the heading of a particle. We prove convergence to a set of solutions that correspond to symmetric patterns of the phases about the unit circle. The exponentially stable patterns are generalized regular polygons, determined by the sign of the coupling gain K.