On a conjecture of De Giorgi related to homogenization

For a periodic vector field $\bf F$, let ${\bf X}^\epsilon$ solve the dynamical system \begin{equation*} \frac{d{\bf X}^\epsilon}{dt} = {\bf F}\left(\frac {{\bf X}^\epsilon}\epsilon\right) . \end{equation*} In \cite{DeGiorgi} Ennio De Giorgi enquiers whether from the existence of the limit ${\bf X}^0(t):=\lim\limits_{\epsilon\to 0}{\bf X}^\epsilon(t)$ one can conclude that $\frac{d{\bf X}^0}{dt}= constant$. Our main result settles this conjecture under fairly general assumptions on $\bf F$, which may also depend on $t$-variable. Once the above problem is solved, one can apply the result to the transport equation, in a standard way. This is also touched upon in the text to follow.