Symmetry and reduction in collectives: cyclic pursuit strategies

We specify and analyse models that capture the geometry of purposeful motion of a collective of mobile agents, with a focus on planar motion, dyadic strategies and attention graphs which are static, directed and cyclic. Strategies are formulated as constraints on joint shape space and are implemented through feedback laws for the actions of individual agents, here modelled as self-steering particles. By reduction to a labelled shape space (using a redundant parametrization to account for cycle closure constraints) and a further reduction through time rescaling, we characterize various special solutions (relative equilibria and pure shape equilibria) for cyclic pursuit with a constant bearing (CB) strategy. This is accomplished by first proving convergence of the (nonlinear) dynamics to an invariant manifold (the CB pursuit manifold), and then analysing the closedloop dynamics restricted to the invariant manifold. For illustration, we sketch some low-dimensional examples. This formulation—involving strategies, attention graphs and sensor-driven steering laws— and the resulting templates of collective motion, are part of a broader programme to interpret the mechanisms underlying biological collective motion.