Open AccessGrassmannian reduction of cucker-smale systems and dynamical opinion games
Author(s) -
Daniel Lear,
David N. Reynolds,
Roman Shvydkoy
Publication year - 2021
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2021095
Subject(s) - attractor , grassmannian , nonholonomic system , class (philosophy) , mathematics , reduction (mathematics) , pure mathematics , nash equilibrium , exponential function , set (abstract data type) , mathematical economics , computer science , mathematical analysis , geometry , artificial intelligence , robot , programming language , mobile robot
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 \begin{document}$ \pi $\end{document} . 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.