Winter School: An introduction to kinetic models in the emergence of complex behavior in social and economic systems

Models of collective behavior and synchronization

Vladislav Panferov

California State University, Northridge

Lecture 1: The Kuramoto model and related kinetic equations I will describe some classical and more modern results on the dynamics of the Kuramoto model of coupled oscillators and its analogs used to describe collective motion of animals and flocking phenomena. The classical problem studied in relation to this model is the time-dependent version of phase transition between order and discorder in a system of oscillators which happens as the coupling strength parameter passes through a critical value. The techniques to approach the problem include bifurcation analysis applied to a kinetic mean-field version of the model, and I will describe some of the results and methods. Lecture 2: I will focus on relations between the Kuramoto model and models used to describe collective behavior of animals, such as the Vicsek and Cucker-Smale equations. I will talk about kinetic equation description for the flocking problem, some rigorous results and open problems in this area, including macroscopic limits of the flocking-type models, and the derivation of the macroscopic Vicsek model as an asymptotic limit based on kinetic equation.