Palis, J.; Takens, F. Stability of parametrized families of gradient vector fields. (English) Zbl 0533.58018 Ann. Math. (2) 118, 383-421 (1983). 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 Cited in 2 ReviewsCited in 16 Documents 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 Keywords:Morse-Smale gradient fields; genericity; structural stability PDFBibTeX XMLCite \textit{J. Palis} and \textit{F. Takens}, Ann. Math. (2) 118, 383--421 (1983; Zbl 0533.58018) Full Text: DOI