×

Stability of parametrized families of gradient vector fields. (English) Zbl 0533.58018

This paper studies the structural stability of a \(C^{\infty}\) one parameter family of gradient vector fields defined on a closed \(C^{\infty}\) manifold. Let \(X^ g\!_ 1(M)\) be the set of such vector fields endowed with the \(C^{\infty}\) Whitney topology. The main result of the paper is the following. Theorem. There exists an open and dense G C \(X^ g\!_ 1(M)\) such that if \(\{X_{\mu}\}\) is in G then \(\{\chi_{\mu}\}\) is structurally stable.
Reviewer: M.Teixeira

MSC:

37C75 Stability theory for smooth dynamical systems
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
57R25 Vector fields, frame fields in differential topology
PDFBibTeX XMLCite
Full Text: DOI