On bifurcations arising from unstable equilibria and invariant sets
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.
scientific article; zbMATH DE number 1247706