Travelling water-waves, as a paradigm for bifurcations in reversible infinite dimensional ``dynamical'' systems
type: Item
instance of: scholarly article
title: Travelling water-waves, as a paradigm for bifurcations in reversible infinite dimensional ``dynamical'' systems
zbMATH Open document ID: 0932.37049
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/228717
full work available at URL: Iooss.MAN.html
review text: The author is interested in bifurcations of solutions lying in a neighborhood \(O\). To this end at first the author restricts himself to reversible finite dimensional systems for which he studies periodic, quasiperiodic, homoclinic to \(O\), and homoclinic to periodic solutions. Using center manifold reduction and normal form theory the author manages to prove the persistence of large classes of reversible solutions under higher order terms. The obtained results are applied to water-wave problems, where 2D travelling waves in a potential flow are considered. Indeed, in case of finite depth layers, the author shows that the problem of finding small bounded solutions is reducible to a finite dimensional center manifold, on which the system reduces to reversible ODE. Some open problems are discussed as well.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 37J15
zbMATH DE Number: 1184398
zbMATH Keywords: periodics quasiperiodic
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: reversible systems
zbMATH Keywords: center manifold
zbMATH Keywords: water-wave problem
author: Gérard Iooss
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184398