# Stability of equilibrium points in Gradient Systems , Lyapunov functions and Hartman-Grobman Theorem

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.

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.
• Hm thx. The Lyapunov instability theorem that I was aware of was that if there exists a function $V\geq 0$ in a neighborhood of $a$ such that $\frac{d}{dt}V(x(t))>0$ for $x\in U-\{a\}$, then we have that the point is unstable, the only problem of using that here would be that I could not be sure that $V\geq 0$ since the function would be $V(a)-V(x)$ and $a$ is a saddle point. Dec 21, 2020 at 6:56
• You don't need $V \ge 0$ everywhere in the neighborhood, that's an unnecessarily strong assumption. See for example Hahn, Stability of Motion, Theorem 25.4, or Arrowsmith & Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, Theorem 3.5.3. Dec 21, 2020 at 10:10