I am asking for a reference that contains a proof of Theorem 4, which is on page 315 of the following text:

Hirsch, Morris W., and Stephen Smale. Differential equations, dynamical systems, and linear algebra. Vol. 60. Academic press, 1974.

Let $W$ be an open set in a vector space and $\mathcal{V}(W)$ be the set of all $C^1$ vector fields on $W$. Let $D^n = \{ x \in \mathbb{R}^n : \lvert x \rvert \leq 1 \}$. Consider in $\mathcal{V}(D^n)$ the set $\mathrm{grad}(D^n)$ of gradient vector fields that point inward on $D^n$.

Theorem 4 The set of structurally stable systems contained in $\mathrm{grad}(D^n)$ is open and dense in $\mathrm{grad}(D^n)$.


1 Answer 1


This came out of J. Palis' 1967 Thesis:

J. Palis "On Morse-Smale dynamical systems" Topology 8, 1969, 385--405.

But that dealt with dimension $\leq 3$. The result you mention seems to first appear as a corollary in

J. Palis and S. Smale "Structural stability theorems" in Global analysis proceedings Symp. Pure Math., 14 AMS, 1970, 223--231.

  • $\begingroup$ Using your information to start my search, I found an even earlier reference in: Smale, Stephen. "On gradient dynamical systems." The Annals of Mathematics 74.1 (1961): 199-206. in the form of Theorem A $\endgroup$ Feb 8, 2013 at 2:19
  • $\begingroup$ I thought I didn't do that bad from home (without library access to the papers). Thanks for completing the reference hunt :) $\endgroup$ Feb 8, 2013 at 2:52

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