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What is the proper definition for the Asymptotic stability of a scalar autonomous equation. For eg. $x'=f(x),$ where $x \in \mathbb{R},$ (say) the equilibrium is at a point $x_0.$ So $f(x_0)=0.$ And how do we go and determine whether it is asymptotically stable or not ?

If it's a system you linearize the system, evaluate the Jacobian at the critical points and find the eigen values of the matrix. All eigen values having a negative real part is the criterion for asymptotic stability. Do we have to come up with a proper Lyapunov function to achieve this ? Any help in understanding this is much appreciated.

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    In dimension $1$, the Jacobian is a $1\times1$ matrix, with unique entry and unique eigenvalue $f'(x)$, hence the asymptotic stability at a point such that $f(x)=0$ means that $f'(x)<0$. – Did Sep 07 '16 at 13:07
  • @Did thank you. The question that I have is in two parts: What it means for $x_0$ be asymptotically stable and how would you go about deciding whether $x_0$ is asymptotically stable. I was wondering for the latter an analogous Lyapunov argument exists. –  Sep 07 '16 at 13:10
  • My comment answers both, no? – Did Sep 07 '16 at 13:15
  • @Did Right. I was looking for two separate solutions. This was from a previous preliminary exam. –  Sep 07 '16 at 13:16

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