Minisymposium: Calculs rigoureux en systèmes dynamiques
A bifurcation diagram of steady states for a system of PDEs via rigorous numerics, Part I: Theoretical description of the method
Vanicat, Matthieu
ENS Cachan, France

In this talk, I will present a method using rigorous numerics to compute several smooth branches of steady states for a system of PDEs, in the case of three reaction-diffusion equations depending on a diffusion parameter. The problem of showing the existence of steady states is similar to the one of finding a Newton-like operator fixed point. The corresponding operator is constructed explicitly in a neighbourhood of a numerical approximated solution. The proof of existence and uniqueness lies on the uniform contraction principle. An effective way to check the hypotheses of this theorem is to construct the radii polynomials which control the truncation error of the numerical approximation, and to show that these radii polynomials are simultaneously negative. This check is done rigorously using interval arithmetic. The implementation issues and results will be discussed in details in a talk by Maxime Breden.

Lundi, 17 juin, 15h30
Salle Des Plaines A