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SUMMARY:Gérard Iooss (LJAD)\, Heteroclinics and orthogonal domain walls
DTSTART:20240326T100000Z
DTEND:20240326T110000Z
DTSTAMP:20240529T115800Z
UID:indico-event-11778@indico.math.cnrs.fr
DESCRIPTION:We consider the pattern defect in Bénard-Rayleigh convection
which consists in orthogonal domain walls connecting a set of convective r
olls with another set of rolls orthogonal to the first set. This is unders
tood as an heteroclinic orbit of a reversible system where the $x$-coordin
ate plays the role of time. This appears as a perturbation of the heterocl
inic orbit of a 6-dimensional reversible normal form system (see Buffoni e
t al [1]). We prove analytically on this 6-dimensional system (see [2]) th
e existence\, local uniqueness\, and analyticity in parameters\, of the he
teroclinic connection between two equilibria\, each corresponding to a sys
tem of convective rolls. The 3-dimensional unstable manifold of one equili
brium\, intersects transversally the 3-dimensional stable manifold of the
other equilibrium\, both manifolds lying on a 5-dimensional invariant mani
fold. This gives an analytic proof of the result obtained by a variational
method in [1]. We study the linearized operator along this heteroclinic\,
and study the complete perturbed system given by the now 8-dimensional no
rmal form (a phase is added for the limiting periodic equilibrium). We are
then able (see [3]) to prove\, via an adapted Lyapunov-Schmidt method\, t
he persistence of the heteroclinic corresponding to the orthogonal wall\,
showing near the bifurcation\, the existence of a one parameter family of
walls\, with a clear link between wave numbers at $\\pm\\infty$.Keywords.
reversible dynamical systems\, invariant manifolds\, bifurcations\, hetero
clinic connection\, domain walls in convectionReferences[1] B.Buffoni\, M.
Haragus\, G.Iooss. Heteroclinic orbits for a system ofamplitude equations
for orthogonal domain walls. J. Diff. Equ. (2023). (link) [2] G.Iooss. He
teroclinic for a 6-dimensional reversible system related with or-thogonal
domain walls in convection. Preprint 2023 (submitted to JDDE). [3] G.Ioos
s. Existence of orthogonal domain walls in Bénard-Rayleigh con-vection. P
reprint 2023 (submitted to JMFM).\n\nhttps://indico.math.cnrs.fr/event/117
78/
LOCATION:Salle de conférence (laboratoire Dieudonné)
URL:https://indico.math.cnrs.fr/event/11778/
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