Grassmannian reduction of Cucker-Smale systems and dynamical opinion games

TL;DR

This paper introduces a novel reduction method for analyzing collective behavior in alignment models with self-propulsion, demonstrating exponential alignment and stable opinion consensus through Grassmannian-based analysis.

Contribution

It develops a new Grassmannian reduction technique to simplify the analysis of complex multi-agent systems with self-propulsion and friction, revealing exponential alignment and stable opinion equilibria.

Findings

Exponential alignment for initial velocities within a sector of opening less than π.

Existence of a unique stable Nash equilibrium representing consensus.

Global attraction of the system's dynamics to the consensus opinion.

Abstract

In this note we study a new class of alignment models with self-propulsion and Rayleigh-type friction forces, which describes the collective behavior of agents with individual characteristic parameters. We describe the long time dynamics via a new method which allows to reduce analysis from the multidimensional system to a simpler family of two-dimensional systems parametrized by a proper Grassmannian. With this method we demonstrate exponential alignment for a large (and sharp) class of initial velocity configurations confined to a sector of opening less than $\pi$. In the case when characteristic parameters remain frozen, the system governs dynamics of opinions for a set of players with constant convictions. Viewed as a dynamical non-cooperative game, the system is shown to possess a unique stable Nash equilibrium, which represents a settlement of opinions most agreeable to all agents. Such an agreement is furthermore shown to be a global attractor for any set of initial opinions.