  Synchronization is a universal phenomenon, seen in systems as diverse as
superconducting Josephson junctions and discharging pacemaker cells. Here the
elements have rhythmic state variables whose mutual influence promotes temporal
order. A parallel form of order is seen in swarming systems, such as schools of
fish or flocks of birds. Now the degrees of freedom are the individuals'
positions, which get redistributed through interactions to form spatial
structures. Systems capable of both swarming and synchronizing, dubbed
swarmalators, have recently been proposed [O'Keeffe, Kevin P., and Steven H.
Strogatz. "Swarmalators: Oscillators that sync and swarm." arXiv preprint
arXiv:1701.05670 (2017)] and analyzed in the continuum limit. Here we extend
this work by studying finite populations of swarmalators, whose phase
similarity affects both their spatial attraction and repulsion. We find ring
states, and compute criteria for their existence and stability. Larger
populations can form annular distributions, whose density and inner and outer
radii we calculate explicitly. These states may be observable in groups of
Japanese tree frogs, magnetic colloids, and other systems with an interplay
between swarming and synchronization.
