I am looking at a system in the form of: $$ \dot{\bf{x}}=\bf{f}(x) $$ and now I wish to find some stability properties of the equilibrium points, i.e. where $\bf f(x)=0$. Particularly, the uniqueness of the stable fixed points is of my interest. One of the ways I come up with is, it is possible to construct a Lyapunov function $$ V=\bf f(x)^Tf(x) $$ and prove that $\frac{dV}{dt}<0$. Obviously, this is overkill, since the local minima in $V$ do not imply fixed points, which only happens at $V=0$. $\frac{dV}{dt}<0$ simply implies that the magnitude of time derivatives decrease until we reach the fixed point.
Hence, I am wondering whether it would be possible to find a Lyapunov function to be applied here, whose time derivative smaller than zero would be necessary and sufficient for the system to be stable. Alternatively, are there any other good theorems for this type of differential equation (which is written in the form of its first order time derivative)?
Many thanks in advance for your attention and answers :)