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 behavior 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, we numerically verify that commonly used potentials satisfy these hypotheses and investigate the nonlinear stability of flocks by an extensive case-study of uniform perturbations.

22 pages, 1 figure, 1 table

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• The Dynamics of Interacting Swarms
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• Single to Double Mill Small Noise Transition via Semi-Lagrangian Finite Volume Methods
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• Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials
• Explicit Equilibrium Solutions For the Aggregation Equation with Power-Law Potentials
• Reduced fluid models for self-propelled particles interacting through alignment