# Critical point stability by linearization

Given a autonomous vector-valued DE: $\dot{y} = f(y)$. Let $Df$ be the Jacobian of $f$.

I want to characterize the stability of critical points. So if $t_0$ is a critical point (i.e. $f(t_0) = 0$), then we have $f(y) = Df(y_0) \cdot (y-y_0) + O(|y-y_0|^2)$.

Then I'd like to use the stability of $\dot{x} = Ax$, where $A = Df(y_0)$, to say something about the stability of $\dot{y} = f(y)$ at critical point $t_0$. I can prove this result for asymptotic (and uniform asymptotic stability), but don't see a proof for liapunov (i.e. plain old regular) stability.

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