Stability of milling patterns in self-propelled swarms on surfaces

TL;DR

This paper develops a Lagrangian mechanics model to analyze the stability of milling patterns in self-propelled swarms on curved surfaces, revealing how surface curvature influences pattern bifurcations and stability.

Contribution

It introduces a general framework for understanding swarm pattern stability on curved surfaces, including bifurcation analysis for different geometries.

Findings

Milling patterns involve agents oscillating on limit cycles with stationary center-of-mass.

Surface curvature can destabilize milling patterns when mutual attraction is insufficient.

Identifies two classes of bifurcations: Hopf bifurcation on spheres and saddle-node bifurcation on cylinders.

Abstract

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.