I'd like to know how fast (or slow) a dynamical system will return to equilibrium, especially if it is near an unstable equilibrium.
The question is vague, but that is intentional. I'm looking for any reading material that could help me in this respect.
Yes you can perform something called linear stability analysis that if the forcing function is linear. This same method can be applied to non linear systems.
The first few chapters of this book deal with what I think is what you want. http://bookstore.siam.org/cl21/