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: 
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.
In order to develop your intuition, you might want to start
with a flow having no fixed points -- so a constant nonzero vector field, say
in the plane. Perturb it by a nice vector field, eg. polynomial 
(or polynomial times a positive function decaying to zero at infinity) until you get zeros for your vector field. Now, what is it you want to show, or preserve?
Do you want to estimate how big the perturbation need be (in terms of
the size of the initial nonzero vector field) in order for zeros 
to be created? 

I think you need to be a bit more specific about where
you are headed, or what restrictions there are on your initial vector field
and its allowed perturbations to get anywhere here.