Dynamical systems defined on networks have applications in many fields in science and engineering. In particular, it is important to understand when networks exhibit synchronous or other types of coherent collective behaviors. Other questions include whether such coherent behavior is stable with respect to random perturbation, or how to described the detailed structure of this behavior during its evolution. We will examine the mathematical challenges of understanding these questions for networked dynamical systems with a particular focus on the dynamics of the Kuramoto oscillator model.
This will be the last in his sequence of an introductory lecture on Hypocoercivity for Langevin dynamics. For those who have not attended the previous lectures and are familiar with Langevin dynamics, the talk should be accessible. We will continue our discussion on convergence to equilibrium for second-order Langevin dynamics using the Poincare approach. We'll recap convergence in H^1(\mu) and then we'll talk about the direct L^2(\mu) method of Dolbeault, Mouhot, and Schmeiser, also called the DMS approach.