I have this nonlinear differential equation $d\textbf{x}/dt=f(\textbf{x})$, where $\textbf{x}\in \mathbb{R}^n$. There are results which guarantee the convergence of the dynamical system to $\textbf{x}=\textbf{0}$, and the simplest of them deal with linearising the equation about $\textbf{x}=\textbf{0}$.

But I would love to say more about the convergence. For example if $f(\textbf{x})=A\textbf{x}$, then we have results bounding the exponential convergence, with rate of convergence related to the greatest eigenvalue of $A$. Are there linearising or approximating results pertaining to the exponential stability of dynamical systems when $f(\textbf{x})$ is nonlinear? I would love to be directed to textbooks or journal papers.

EDIT: I would like to add one more relevant question. It can be proved that if the linearised system is exponentially stable, then the nonlinear system also is exponentially stable. Does the converse hold? Is there a necessary and sufficient condition using Lyapunov methods?