# American Institute of Mathematical Sciences

December  2021, 41(12): 5765-5787. doi: 10.3934/dcds.2021095

## 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

Received  November 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

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

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.

Citation: Daniel Lear, David N. Reynolds, Roman Shvydkoy. Grassmannian reduction of cucker-smale systems and dynamical opinion games. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5765-5787. doi: 10.3934/dcds.2021095
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