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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.

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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.

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  • $\begingroup$ 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. $\endgroup$
    – Someone
    Dec 21, 2020 at 6:56
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    $\begingroup$ 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. $\endgroup$ Dec 21, 2020 at 10:10
  • $\begingroup$ Hi, just one question reguarding the Hartman-Grobman theorem , it makes sense to me that we can extract the asymptotic behavior from the linearized system but I would like to see a proof and I am not quite being able to do one myself. Do you know any reference where I could look this up ? @Hans Lundmark $\endgroup$
    – Someone
    Dec 23, 2020 at 14:20
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    $\begingroup$ @Lost: It's no wonder if you can't prove that on your own, since it's a rather difficult theorem! Here are a few books which claim to give proofs (I haven't read them too carefully, so I can't tell which one would be easiest to understand): C. Chicone, Ordinary Differential Equations with Applications; C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos; P. Hartman, Ordinary Differental Equations. $\endgroup$ Dec 23, 2020 at 15:14

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