Emergent behaviors of a first-order particle swarm model on the hyperboloid

Abstract

We study the emergent dynamics of a first-order particle swarm model (PSM) on the hyperboloid with a constant negative curvature that corresponds to the special case (p, q) = (1, d) of the PSM on the indefinite special orthogonal group SO(p, q) in the work of Ritchie et al. [Chaos 28, 053116 (2018)]. For the proposed PSM on the hyperboloid, we first establish the global existence of a solution via the extension of a local solution to a global one by the continuity argument and then show that the solutions are uniformly bounded by a quantity only depending on the initial data and the coupling strength although the underlying manifold is not compact. In this paper, we consider both attractive and repulsive couplings. For an attractive regime, we show that the complete synchronization occurs for all initial data, whereas for a repulsive regime, we show that distances between particles diverge to infinity, as time goes on. Finally, we present several numerical results consistent with our theoretical results.