I have an autonomous dynamical system $\dot{\mathbf x}(t)=\mathbf f(\mathbf x(t))$ on $\mathbb R^2$, and I found that solutions are future asymptotic to an equilibrium point $\mathbf a$, i.e.
$\lim_{t\to\infty}\mathbf x(t)=\mathbf a$
for all initial conditions in the domain of interest.
I am now interested in how fast solutions approach this equilibrium point, i.e. I would like to obtain an asymptotic expansion (EDIT: valid for large $t$) of the form
$\mathbf x(t)\simeq\mathbf a+(\text{term decaying with $t$})+\dots$.
How can I determine the decaying term, and is the procedure different for hyperbolic and degenerate equilibrium points?
I do have the book by Perko, Differential Equations and Dynamical Systems, 3rd ED. So if it is actually in there, and I just cannot find me, it would be great if someone could point me to the respective part of the book as well!
Thanks for help!