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We know index of function $F:\mathbb{R}^n\to\mathbb{R}$ at critical point $x_0\in\mathbb{R}^n$ is the number of negative eigen values of Hessian matrix $DF^2(x_0)$. By intuition, it's also the number of decreasing direction. Let's call $F$'s index $F$-index. When $F$ is viewed as height function, then sink's $F$-index is a $0$, and source($2$),saddle($1$).

For dynamical system $X'=DF(X)$, negative eigen value of $DF^2(x_0)$ implies moving back to critical point $x_0$. In this case, sink's $F$-index is $2$, source($0$), saddle($1$).

The concept of source/sink/saddle is different according to the same index. Is it the reason that gradient flow is defined as $X'=-DF(X)$ by convention?

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The idea of gradient flow going against the gradient comes from natural sciences, not from the desire to match two mathematical definitions. In physics, a system tends to minimize its potential energy, and therefore evolves in the direction opposite the gradient. The force with potential $U$ is $-\nabla U$. The direction of net flow of some substance in a mixture is opposite to the gradient of its concentration.

This point of view influences the definitions of index of Morse-Smale dynamical systems. As seen here or here, they are commonly expressed in the form $\dot X=-\nabla F(X)$. The index of a stationary point of $F$ was already defined as the number of negative Hessian eigenvalues. From the dynamical system point of view, the index is taken as the dimension of the unstable manifold, so that it coincides with the previous notion of index.

My point is, it's the $-\nabla$ that influences the definitions of index, not the other way around.

When someone decides to fight the influence of physics and define gradient flows as $\dot X=\nabla F(X)$, they end up with slightly inconsistent definitions; no big deal.

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  • $\begingroup$ Both indices are index of $F$ at critical point $x_0$, i.e. number of negative eigen values of $D^2F(x_0)$. I don't mention the degree/index of a vector field stationary point. $\endgroup$ Commented Jul 6, 2014 at 17:22
  • $\begingroup$ @MartinWang In the question you have two versions of index: the first one gives 0 for a sink, the other gives 2. What is the definitions of the second one? $\endgroup$ Commented Jul 6, 2014 at 17:25
  • $\begingroup$ It's still the number of negative eigen values of $D^2F(x_0)$. A sink in the sense of vector field is corresponding to the index $2$, which is not literally consistent with the sink concept by intuition. $\endgroup$ Commented Jul 7, 2014 at 1:01
  • $\begingroup$ Actually, it's called Morse Index. In my understanding, when considering a function $F(X)$, the morse index at critical point is number of negative eigen values of hessian matrix. however, when considering a gradient flow $X'=F(X)$, the morse index is defined as number of positive eigen values. $\endgroup$ Commented Jul 7, 2014 at 1:15
  • $\begingroup$ @MartinWang Okay, I clarified the answer. $\endgroup$ Commented Jul 7, 2014 at 3:16

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