My question consider the reason for calculating normal form.
given vector field: $$\dot{x}=F(x),\ \ x\in \mathbb{R^n}$$ we try to bring the system to normal form by a sequence of transformation but what is the reason for it ? what information the normal form give to us?
If one has a reference with examples for the information we get from normal form I will be glad.
Thank you.
Well, the reason for this to be able to say something meaningful about the original system. As an example, consider an equation $$ \dot x=f(x,\alpha) $$ that depends on a scalar parameter $\alpha$. Assume that the equilibrium $\hat x$ is non-hyperbolic at $\alpha=0$ (i.e., $f'(\hat x,0)=0$). Then under quite general assumption on $f$ it is possible to prove that there exists an invertible change of variable and parameter such that in new coordinate we have the following normal form: $$ \dot y=\mu\pm y^2. $$ This last equation is very simple to analyze (it exhibits what is called fold bifurcation) and hence the original system also experiences the same bifurcation.
A more extensive treatment along with additional references can be found at http://www.scholarpedia.org/article/Normal_forms