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Grassmannian reduction of cucker-smale systems and dynamical opinion games

  • * Corresponding author: lear@uic.edu

    * Corresponding author: lear@uic.edu 

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

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  • 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.

    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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