Abstract:
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. |