Short version (as suggested by Mariano, and as in title):
For every diffeomorphism $f: \Omega \rightarrow \Omega$, (where $\Omega$ is a compact subset of $\mathbb{R}^n$) that is the time-1 map of a "non-autonomous flow", is it also the time-1 map of an autonomous flow?
Consider the following situation:
For $x \in \Omega \subseteq \mathbb{R}^n$ where $\Omega$ is compact, Let $\phi(x, t)$ be defined by the following boundary conditions: $$ \phi(x, 0) = x, \quad \phi(x, 1) = f(x), \quad f \in C^\infty $$
and the differential equation: $$ \frac{d \phi(x, t)}{dt} = v(\phi(x, t), t) $$
Of course, the set of functions $v(\phi(x, t))$ is a proper subset of $v(\phi(x, t), t)$. I'm trying to find a $v$ that satisfies the ODE and boundary conditions.
My question is, is there some $f(x)$ for which the differential equation $$ \frac{d \phi(x, t)}{dt} = v(\phi(x, t), t) $$ has a solution but
$$ \frac{d \phi(x, t)}{dt} = v(\phi(x, t)) $$ does not?