Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$.

Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) such that $$x_1^y = f(y).$$

The motivation comes from the study of particular flows on the bi-dimensional torus and their first return map. The specific question is quite long to explain in detail, but if you are interested I can try to state it better

*Note added by B. Kloeckner:* this is the improved version of the question, pasted from math.SE. I voted to reopen this version, as I think the question, while deserving this improvement by its authors, was unduly migrated, possibly because some of the voters did not realize that the ODE to be found is autonomous.