Suppose I have a dynamical system of the form
$$
\frac{dx}{dt} = f(x)
$$
Most of the frameworks I am familiar with for analyzing such systems revolve around finding the fixed points $x^*$ where $f(x^*) = 0$. However, suppose this system does not have any fixed points. So I'm wondering whether you can look at its derivative, i.e
$$
g(x) = \frac{d^2x}{dt} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t} = \frac{\partial f}{\partial x}f(x)
$$
Can we get any interesting information by finding the fixed points of $g(x)$? I am thinking that in some cases a system may not have steady states in $x$, but it may achieve steady states in the velocities of $x$. An example might be an engine, where the engine is net producing stuff, but it might stabilize to a constant rate of production.
Also, I believe that $\frac{\partial f}{\partial x}$ is the Jacobian, so if $f(x)$ is never 0, we just need to find the kernel of the Jacobian.
Ideas and references would be much appreciated, thanks!