Source : Morse Theory and Floer Homology Authors : Michèle Audin, Mihai Damian

In this note we define what are pseudo-gradient fields, that are related to the critical points of a Morse function .

We just work on $\mathcal{M}=\mathbb{R}^n$, but the reasoning can be extended to a generic manifold. Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$, then its gradient is a map $\nabla f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ that in general is defined by

$$\nabla f(x)^Tv = df_x(v),\quad v\in T_x\mathbb{R}^n,,x\in\mathbb{R}^n.$$

Then main properties of the vector field $X(x)=-\nabla f(x)$ are the following

1. $X(x)=0$ if and only if $x$ is a critical point of $f$,

2. $\mathcal{L}_Xf \leq 0$, and $=0$ only on the critical points of $f$

Let now $V\subset \mathbb{R}^n$, and $f:V\rightarrow\mathbb{R}$ a Morse function. Then $X\in\mathfrak{X}(V)$ is a pseudo-gradient field or pseudo-gradient adapted to $f$ if

1. $\mathcal{L}_X f\leq 0$ and $=0$ if and only if $x$ is a critical point of $f$,

2. In a Morse chart in the neighbourhood of a critical point, $X$ coincides with $-\nabla f$.

A quite generic family of vector fields satisfying at least the first property is

$$X(x) = -P(x)\nabla f(x),\quad P(x)^T=P(x)>0$$

(see Howse, James W., et al. “An application of gradient-like dynamics to neural networks.” Conference Record Southcon. IEEE, 1994.)

Here expand on how pseudo-gradients allow to characterise the stable and unstable manifolds of the equilibria of $-\nabla f(x)$. These facts allow to prove that these vector fields are structurally stable provided the stable and unstable manifolds of the critical points intersect transversely (see Morse–Smale system - Wikipedia ).

Add some important information from Morse-Smale systems - Scholarpedia where it is studied when these systems are structurally stable.