Approximate and real trajectories for generic dynamical systems
type: Item
instance of: scholarly article
title: Approximate and real trajectories for generic dynamical systems
zbMATH Open document ID: 0821.58036
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-09-21T00:00:00Z
review text: The authors show that generic dynamical systems have a weak shadowing property. In particular, they show that, for a generic dynamical system \(\varphi\) in the space of discrete continuous dynamical systems on a compact manifold with the usual \(C^ 0\) topology, given \(\varepsilon > 0\) there exists \(\delta > 0\) such that \(\delta\)-trajectories of \(\varphi\) are \(\varepsilon\)-close to real trajectories. A \(\delta\)-trajectory is an idealization of the notion of a ``locally accurate'' numerical approximation to a real trajectory. The shadowing result in this paper involves a weaker property than the pseudo-orbit tracing property (POTP). The authors show only that \(\delta\)-trajectories of generic dynamical systems are ``weakly \(\varepsilon\)-traced'' by true trajectories. However, there are no dimensional restrictions on this result. The authors also present some results on the inverse problem of tracing real trajectories by \(\delta\)-trajectories.
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 65J99
zbMATH DE Number: 743909
zbMATH Keywords: weak shadowing
zbMATH Keywords: numerical solutions of differential equations
zbMATH Keywords: dynamical systems
zbMATH Keywords: shadowing property
zbMATH Keywords: real trajectories
author: Robert M. Corless
author: Sergei Yu. Pilyugin
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1027
OpenAlex ID: W2051061875
DOI: 10.1006/JMAA.1995.1027
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description: scientific article; zbMATH DE number 743909