Abstract

We will discuss three closely related models that govern the collective dynamics of agents. All three models, have an external Rayleigh-friction type forcing that drives each individual agent along with a coupling mechanism which pushes them towards an agreement whether that be velocity alignment, consensus, or synchronization. The first context is a flocking and alignment model, where the famous Cucker-Smale model (2007) has been appended with the friction and self-propulsion force, driving each agent to a common velocity. The result is achieved via a grassmannian reduction method that reduces the problem to an essentially two dimensional system, where we can take advantage of the dissipative structure. Reducing to a first-order system the model can be seen as a nonlinear consensus model for opinion dynamics, where the friction force functions as a stubbornness term. This model is related to the Taylor model (1968) of opinion dynamics, however, the nonlinear force in our model captures more rich asymptotic behavior, including stable disagreement fixed points. Finally, allowing for complex-valued stubbornness parameters results in the coupled Stuart-Landau model of synchronization, which is closely related to the Kuramoto model (1975). We will investigate the various asymptotic states of this model focusing on the effects of allowing heterogeneous amplitude parameters.