Nonlinear stability of flock solutions in second-order swarming models

Abstract In this paper we consider interacting particle systems which are frequently used to model collective behaviour in animal swarms and other applications. We study the stability of orientationally aligned formations called flock solutions, one of the typical patterns emerging from such dynamics. We provide an analysis showing that the nonlinear stability of flocks in second-order models entirely depends on the linear stability of the first-order aggregation equation. Flocks are shown to be nonlinearly stable as a family of states under reasonable assumptions on the interaction potential. Furthermore, it is tested numerically that commonly used potentials satisfy these hypotheses and the nonlinear stability of flocks is investigated by an extensive case study of uniform perturbations.

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