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CATEGORIES:Applied and Computational Analysis
SUMMARY:Computer-assisted proofs for dynamical systems - M
axime Breden (ENS Paris-Saclay &\; Université L
aval)
DTSTART;TZID=Europe/London:20170615T150000
DTEND;TZID=Europe/London:20170615T160000
UID:TALK72804AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/72804
DESCRIPTION:To understand the global behavior of a nonlinear s
ystem\, the first step is\nto study its invariant
set. Indeed\, specific solutions like steady state
s\, periodic orbits and connections between them a
re building blocks that organize the global dynami
cs. While there are many deep\, general and theore
tical mathematical results about the existence of
such solutions\, it is often difficult to apply th
em to a specific example. Besides\, when dealing w
ith a precise application\, it is not only the exi
stence of these solutions\, but also their qualita
tive properties that are of interest. In that case
\, a powerful and widely used tool is numerical si
mulations\, which is well adapted to the study of
an explicit system and can provide insights for pr
oblems where the nonlinearities hinder the use of
purely analytical techniques.\nHowever\, one can d
o even better. Using numerical results as a starti
ng\npoint\, and combining them with a posteriori e
stimates\, one can then get rigorous results and p
rove the existence of a genuine solution close to
the numerical one. In this talk\, I will explain h
ow such computer-assisted theorem can be obtained.
I will then focus on some examples where these te
chniques can be useful\, namely to study non homog
eneous steady states of cross-diffusion systems\,
and to prove the existence of periodic solutions o
f the Navier-Stokes equations in a Taylor-Green fl
ow.
LOCATION:MR 14\, CMS
CONTACT:Carola-Bibiane Schoenlieb
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