On bifurcations arising from unstable equilibria and invariant sets
type: Item
instance of: scholarly article
title: On bifurcations arising from unstable equilibria and invariant sets
zbMATH Open document ID: 0931.54035
author: Peter Seibert
author: José S. Florio
published in: Journal of Differential Equations
publication date: 2000-02-27T00:00:00Z
review text: A locally asymptotically compact family of semidynamical systems, indexed by parameters \(\lambda\) on a metric space is considered. The main result of the paper states that if an isolated compact set \(M\) invariant for every value of the parameter \(\lambda\) is unstable for a certain value \(\lambda^0\) of the parameter \(\lambda\) and stable for values close to \(\lambda^0\) (that is: unstability is changing into stability when a certain critical parameter value \(\lambda^0\) is passed), then a bifurcation takes place which may be one of the kinds: extracritical, i.e. \(M\) splits into more than one invariant sets when the parameter value \(\lambda^0\) is passed, or critical, i.e. for \(\lambda^0\) the invariant sets accumulate at \(M\) (including the case when these sets adhere to \(M\) in the form of homoclinic orbits; this case is called a weak-critical bifurcation). The authors illustrate the above by giving two elementary but instructive examples.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 1247706
reviewed by: Andrzej Pelczar
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3495
OpenAlex ID: W2041260572
cites work: Q5614853
cites work: Q4536860
cites work: Q4039920
cites work: Liapunov direct method in approaching bifurcation problems
cites work: Semidynamical systems in infinite dimensional spaces
cites work: On the foundations of bifurcation theory
cites work: On the reduction to a subspace of stability properties of systems in metric spaces
DOI: 10.1006/JDEQ.1998.3495
description: scientific article; zbMATH DE number 1247706