Consider an autonomous ODE $$\dot x = f(x)$$ where $x\in\mathbb{R}^n$ and $x_0$ is an equilibrium point. I was wondering if the following statement is true:
If all eigenvalues of the Jacobian matrix $\frac{\partial f}{\partial x}(x_0)$ have negative real part, then $x_0$ is exponentially stable.
Note that the above statement is obvious if we replace "exponentially" with "asymptotically ".
Yes, $x_{0}$ is exponentially stable. As far as I know, that is called linearization. :)