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


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
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