  In some physical and biological swarms, agents effectively move and interact
along curved surfaces. The associated constraints and symmetries can affect
collective-motion patterns, but little is known about pattern stability in the
presence of surface curvature. To make progress, we construct a general model
for self-propelled swarms moving on surfaces using Lagrangian mechanics. We
find that the combination of self-propulsion, friction, mutual attraction, and
surface curvature produce milling patterns where each agent in a swarm
oscillates on a limit cycle, with different agents splayed along the cycle such
that the swarm's center-of-mass remains stationary. In general, such patterns
loose stability when mutual attraction is insufficient to overcome the
constraint of curvature, and we uncover two broad classes of stationary
milling-state bifurcations. In the first, a spatially periodic mode undergoes a
Hopf bifurcation as curvature is increased which results in unstable
spatiotemporal oscillations. This generic bifurcation is analyzed for the
sphere and demonstrated numerically for several surfaces. In the second, a
saddle-node-of-periodic-orbits occurs in which stable and unstable milling
states collide and annihilate. The latter is analyzed for milling states on
cylindrical surfaces. Our results contribute to the general understanding of
swarm pattern-formation and stability in the presence of surface curvature, and
may aid in designing robotic swarms that can be controlled to move over complex
surfaces.
