Grassmannian reduction of cucker-smale systems and dynamical opinion games

Daniel Lear; David N. Reynolds; Roman Shvydkoy

doi:10.3934/dcds.2021095

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2021, Volume 41, Issue 12: 5765-5787. Doi: 10.3934/dcds.2021095 open access

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Grassmannian reduction of cucker-smale systems and dynamical opinion games

Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA

^* Corresponding author: lear@uic.edu
^* Corresponding author: lear@uic.edu

Received: October 31, 2020

Revised: April 30, 2021

Early access: July 2021

Published: December 2021

This work was supported in part by NSF grant DMS-1813351

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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 us 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.

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Mathematics Subject Classification: Primary: 92D25; Secondary: 35Q35.

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Figure 1. Isosceles triangles in the proof of Lemma 4.3

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