So I have learned about Lyapunov theory to study the stability of equilibrium points are now we want to apply it to the study of gradient systems. So suppose we have $x'=-\nabla V(x)$ and we have that $a$ is an equilibrium point for this equation that is $\nabla V(a)=0$. Now if we have that $a$ is an isolated local minimum we can use the Lyapunov function $H(x):=V(x)-V(a)$ to see that this is an asymptotically stable point. If $a$ is an isolated local maximum we can use $-H(x)$ to see that it is unstable, but what happens if $a$ is an isolated saddle point ? How can we study the stability in this case? One way I thought about it would be to use the Hartman-Grobman theorem and we know that the linearization of this dynamical system will be unstable and so since they have homeomorphic flows I guess this would also be unstable, but I am not completely sure this works, or if there is another way to see this.
I guess my biggest doubt is that if we can use the Hartman-Grobman theorem to study the stability of the sistem from the linearized equation.
Any help is appreciated, thanks in advance.
Best Answer
You could use Hartman–Grobman (if $a$ is hyperbolic), but it's perhaps overkill. There's the simpler Lyapunov instability theorem which says that if there is a differentiable function $H$ which is defined in a neighbourhood of $a$ and does not have a local minimum at $a$, and $\dot H<0$ on a punctured neighbourhood of $a$, then $a$ is unstable.