Difference between equilibrium points and critical points

Question

I am trying to figure out a qualitative difference between a critical point and an equilibrium point in the context of autonomous ODE

Let us consider the following Cauchy problem:

$ y'(t) = f(y(t))$, with $y(0)=y_0$.

Formally, I know the difference between the two. However in the above context we have:

$y_1\in \mathbb{R}^m$ is a critical point $ \Leftrightarrow y_1'(t) = 0 \Leftrightarrow f(y_1(t)) = 0 \Leftrightarrow y_1$ is an equilibrium point.

Is there an error in this reasonning?

Thank you for your time.

Answer 1

There are critical points where it isn't that the derivative is zero, but that it doesn't exist. So your very first "iff" doesn't quite work there.