Let $f\colon\mathbb{R}^n\to\mathbb{R}^n$. When is it possible to find some $g\in C^1([0,1]\times\mathbb{R}^n, \mathbb{R}^n)$, uniformly Lipschitz continuous w.r.t the second argument, such that if $u_{x_0}$ is the solution of the IVP $$ \dot{x}(t) = g(t,x(t)),\quad x(0) = x_0,$$ we have $f = x_0\mapsto u_{x_0}(1)$? I.e., what do we need from $f$ for it to be describable by the (time-dependent) flow given by an ODE? How can find that ODE?
I figured these might be questions answered in the analysis of dynamical systems (or not because of the time-dependency?); can somebody give a brief answer or direction on where to go looking?
Many thanks in advance.
Edit: The question is on mathoverflow now.
First, of course if you are looking for an autonomous differential equation this is impossible (think of $f(x)=-\frac 12 x$, clearly there is no one dimensional system $\dot x=g(x)$ that would correspond to it). If, however, you allow an ODE on a manifold then the inverse problem always can be solved. It is called a suspension system. You can find some general remarks about such systems in Section 1.5.1 of Kuznetsov's book Applied bifurcation theory. In Chapter 9 of the same book the author constructs explicit systems of ODEs approximating certain maps.