Emergent behaviors in group ring flocks

TL;DR

This paper introduces a first-order aggregation model on a group ring, demonstrating that under positive coupling, the system's flow converges to an equilibrium manifold influenced by the underlying group's structure.

Contribution

The paper develops a novel aggregation model on group rings and analyzes its asymptotic behavior, revealing the dependence of equilibrium structures on group properties.

Findings

Flow tends to an equilibrium manifold asymptotically

Lyapunov functional decreases along the flow

Equilibrium structure depends on the underlying group

Abstract

We present a first-order aggregation model on a group ring, and study its asymptotic dynamics. In a positive coupling strength regime, we show that the flow generated by the proposed model tends to an equilibrium manifold asymptotically. For this, we introduce a Lyapunov functional which is non-increasing along the flow, and using the temporal decay of the nonlinear functional and the LaSalle invariance principle, we show that the flow converges toward an equilibrium manifold asymptotically. We also show that the structure of an equilibrium manifold is strongly dependent on the structure of an underlying group.