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.
System of ODEs is a vector field. If there's finite number of fixed points and periodic orbits and the system is Morse-Smale then it is structurally stabel under $C^1$ small perturbations. In particular, all fixed points survive.

More details are here
<http://www.scholarpedia.org/article/Morse-Smale_systems>

The manifold needs to be compact. So one extends $R^n$ to $S^n$ by adding an extra fixed point.