# Why can you determine the stability of a system by taking the eigenvalues of the Jacobian?

Why can you determine the stability of a system by taking the eigenvalues of the Jacobian? I know it's an elementary question but it's been a while.

Thank you!

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On which king of system are you working? – Davide Giraudo Feb 4 '12 at 17:38
The system is $\dot{x} = f(x, u)$, $y = h(x, u)$ – badatmath Feb 4 '12 at 18:03

Let's consider a system $\dot x = f(x)$, $x \in \Omega \subset \mathbb{R}^n$, $f(0) = 0$. If a function $f(x)$ is sufficiently smooth then we can rewrite this system as $\dot x = Ax + g(x)$, where $g(x) = o(|x|)$ and $A$ is actually the Jacobi matrix of $f(x)$ in $x = 0$. The Lyapunov-Poincare theorem states then that if for any eigenvalue $\lambda$ of matrix $A$ we have $\Re (\lambda) < 0$ then $x(t) \equiv 0$ is asymptotically stable solution, else if there exists $\lambda$ such that $\Re(\lambda) > 0$ then our system is unstable.
There exists a generalisation of this result to the case of nonstationary systems $\dot x = f(t,x)$.