I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium point. Somebody can help me?
I am searching the following Theorem:
Let us consider the non-linear system $\dot{x}=f(x)$ with an arbitrary equilibrium point $\bar{x}$. This equilibrium point $\bar{x}$ is:
- asymptotically stable if all eigenvalues of $Df(\bar{x})$ have negative real part;
- unstable if at least one eigenvalue of $Df(\bar{x})$ has positive real part;
where $Df(\bar{x})$ is the Jacobin matrix of $f$ applied in the equilibrium point $\bar{x}$.
Thank you very much!
Ana