I want to try to understand the following theorem from Nonlinear Systems (Hassan Khalil, 2000):
Let $x=0$ be an equilibrium point for $$\dot{x} = f(x),$$ in which $f$ is locally Lipschitz. Let $V: D \to R$ be a continouly differentiable function such that $V(0)=0$ and $V(x_0)>0$ for some $x_0$ with arbitrarily small $||x_0||$. Define a set $U$ as a ball of radius $r$ centered at the origin and suppose that $\dot{V}(x)>0$ in $U$. Then, $x=0$ is unstable.
To me, it would imply that swapping $V(x_0)>0 \rightarrow V(x_0)<0$ and $\dot{V}(x)>0 \rightarrow \dot{V}(x)<0$ in the theorem can also be used for the assessment of instability. If I am not mistaken then we could also formulate the theorem as follows.
Let $x=0$ be an equilibrium point for $$\dot{x} = f(x),$$ in which $f$ is locally Lipschitz. Let $V: D \to R$ be a continouly differentiable function such that $V(x)$ is indefinite/positive definite/negative definite (includes $V(0)=0$). Define a set $U$ as a ball of radius $r$ centered at the origin and suppose that $\dot{V}(x)>0$ or $\dot{V}(x)<0$ / positive definite / negative definite in $U$. Then, $x=0$ is unstable.
Is this interpretation of the theorem correct?