The question is mainly about $\it flows$, not maps (i.e., continuous time, not discrete time).

- Is it known if the study of polynomial dynamical systems on $\mathbb R^n$ can be reduced to the study of polynomial dynamical systems on $\mathbb R^n_+$?

In other words, given a polynomial vector field $f_1$ on $\mathbb R^n$, is it known if always (or under what conditions on $f_1$) there must exist a polynomial vector field $f_2$ on $\mathbb R^n_+$ such that the dynamical system $\frac{dx}{dt} = f_1(x)$ on $\mathbb R^n$ is topologically equivalent to the dynamical system $\frac{dx}{dt} = f_2(x)$ on $\mathbb R^n_+$?

Also, consider a domain $D\subset\mathbb R^n$. Under what conditions on $D$ it it true that any dynamical system given by a polynomial vector field $f_1$ on $D$ is topologically equivalent to a dynamical system given by some polynomial vector field $f_2$ on $\mathbb R^n_+$?

Are there any special results for $n=2$?

Same question for $\it rational \ dynamical \ systems$ instead of polynomial dynamical systems.

In particular, is it known for what domains $D\subset\mathbb R^n$ does there exist a rational function homeomorphism $h:D\to\mathbb R^n_+$ such that $its \ inverse \ is \ also \ a \ rational \ function$? (Note that this part of the question is just about the existence of a single rational map, and not about dynamics).