Stability of parametrized families of gradient vector fields (Q789797)

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.