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SUMMARY:Computer-assisted proofs for dynamical systems - Maxime Breden (EN
 S Paris-Saclay &amp\; Université Laval)
DTSTART:20170615T140000Z
DTEND:20170615T150000Z
UID:TALK72804@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:To understand the global behavior of a nonlinear system\, the 
 first step is\nto study its invariant set. Indeed\, specific solutions lik
 e steady states\, periodic orbits and connections between them are buildin
 g blocks that organize the global dynamics. While there are many deep\, ge
 neral and theoretical mathematical results about the existence of such sol
 utions\, it is often difficult to apply them to a specific example. Beside
 s\, when dealing with a precise application\, it is not only the existence
  of these solutions\, but also their qualitative properties that are of in
 terest. In that case\, a powerful and widely used tool is numerical simula
 tions\, which is well adapted to the study of an explicit system and can p
 rovide insights for problems where the nonlinearities hinder the use of pu
 rely analytical techniques.\nHowever\, one can do even better. Using numer
 ical results as a starting\npoint\, and combining them with a posteriori e
 stimates\, one can then get rigorous results and prove the existence of a 
 genuine solution close to the numerical one. In this talk\, I will explain
  how such computer-assisted theorem can be obtained. I will then focus on 
 some examples where these techniques can be useful\, namely to study non h
 omogeneous steady states of cross-diffusion systems\, and to prove the exi
 stence of periodic solutions of the Navier-Stokes equations in a Taylor-Gr
 een flow.
LOCATION:MR 14\, CMS
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