Bifurcations in Vlasov and Kuramoto models

Jeudi 18 mars 14:00-15:00 - David Métivier - École Polytechnique

Résumé : A wide variety of physical systems are governed over certain time scales by mean-field forces rather than “collisions” between their constituents. The appropriate kinetic description is then a Vlasov, or Vlasov-like equation. In this category, we find the Vlasov-Poisson equation at the heart of plasma physics, Collisionless Boltzmann Equation (or Vlasov-Newton equation) describing self-gravitating systems, but also coupled oscillator systems such as the Kuramoto model. Vlasov-like equations possess both regular features (such as an infinite number of conserved quantities) and chaotic ones (such as the development of infinitely fine structures in phase space), which make both the understanding of their qualitative behavior and their numerical simulation famously difficult problems.
In this talk, I will address the question : What happens close to weakly unstable stationary states, how can we describe the dynamics with simple low-dimensional equations, i.e., do a bifurcation analysis ? I will present a short review of these questions for the Vlasov and Kuramoto equations. I will finish by showing a recent result on an asymptotic exact finite-dimensional reduction of the Vlasov equation close to some stationary states.

Bifurcations in Vlasov and Kuramoto models  Version PDF