Question: Suppose we have a smooth dynamical system on $R^n$ (defined by a system of ODEs). Assuming that the system has a finite set of fixed points, I am interested in knowing (or obtaining references about) what is the behaviour of its fixed point structure under perturbations of the ODEs. More specifically, i would like to know under which conditions the total number of fixed points remains the same. By perturbations I mean generic changes in the system of equations... the more general the better.

I am sorry if the question is too basic, my interest comes from the study of the so-called renormalization flow in field theories. In particular, it would be important for me to generate an intuition about the conditions under which the approximations performed over a dynamical system alters its fixed point structure.

Answer: 
A good reference for this sort of thing is Guckenheimer and Holmes, *Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*, Springer-Verlag, 1983, if you're not already familiar with it (and even if you are, for that matter). Chapter 3 in particular is relevant to your question.
If you are just interested in the number of the fixed points of the flow, you can just put your question in terms of zeros of the vector field; in this case the right tool is the topological degree, which is stable under small perturbations of the field in the uniform norm; and in absolute value is (generically) a lower bound on the number of the zeros. Of course, if the vector field is variational (it's the gradient if a functional) much stronger invariants are available (all the Morse complex machinery).