On the discretization of a partial differential equation in the neighborhood of a periodic orbit
type: Item
instance of: scholarly article
title: On the discretization of a partial differential equation in the neighborhood of a periodic orbit
zbMATH Open document ID: 0791.65071
DOI: 10.1007/BF01385746
author: François Alouges
author: A. Debussche
published in: Numerische Mathematik
publication date: 1994-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/133729
review text: This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equation and its approximation are closed in a certain sense, provided that different initial data are allowed. This answers partly an open problem posed by \textit{W.-J. Beyn} [ibid. 51, 103- 122 (1987; Zbl 0617.65082)]. Examples of application to dissipative partial differential equations are provided.
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 569152
zbMATH Keywords: evolution equation
zbMATH Keywords: time discretization
zbMATH Keywords: hyperbolic periodic orbit
zbMATH Keywords: invariant closed curve
zbMATH Keywords: semigroups
Wikidata QID: Q115393649
MaRDI profile type: Publication
cites work: On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point
cites work: Regular attractors of semigroups and evolution equations
cites work: On the Numerical Approximation of Phase Portraits Near Stationary Points
cites work: On invariant closed curves for one-step methods
cites work: Q3793649
cites work: Invariant curves for numerical methods
cites work: Invariant curves of one-step methods
cites work: Two concepts for numerical periodic solutions of ODE's
cites work: Dimension of the attractors associated to the Ginzburg-Landau partial differential equation
cites work: Q4039920
cites work: Upper Semicontinuity of Attractors for Approximations of Semigroups and Partial Differential Equations
cites work: Lower semicontinuity of attractors of gradient systems and applications
cites work: Geometric theory of semilinear parabolic equations
cites work: Invariant manifolds
cites work: Stable Attracting Sets in Dynamical Systems and in Their One-Step Discretizations
cites work: A note on multistep methods and attracting sets of dynamical systems
cites work: Floquet theory for partial differential equations
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Q5201632
OpenAlex ID: W2035783537
description: scientific article; zbMATH DE number 569152