Daniel Lear Claveras : Grassmannian reduction of Cucker-Smale systems and dynamical opinion games

Posted by Ian Tobasco , part of the Analysis and Applied Mathematics Seminar.

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Nov. 30, 2020, 4 p.m.

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Abstract

In this work, 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. Joint work with David N. Reynolds & Roman Shvydkoy