Fine asymptotic expansion of the ODE's flow

In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X (t, x) = b X(t, x) with X(0, x) = x $\in$ R d , where b is a regular Z dperiodic vector field in R d. More precisely, we provide various conditions on b to obtain a "fine" asymptotic expansion of X of the type: |X(t, x) -- x -- t $\zeta$(x)| $\le$ M < $\infty$, which is uniform with respect to t $\ge$ 0 and x $\in$ R d (or at least in a subset of R d), and where $\zeta$(x) for x $\in$ R d , are the rotation vectors induced by the flow X. On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) -- x -- t $\zeta$(x) reads as $\Phi$ X(t, x) -- $\Phi$(x), which yields immediately the desired expansion when the vector-valued function $\Phi$ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Y d and of vector-valued functions $\Phi$. On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not. It turns out that for any two-dimensional flow X associated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector $\xi$, the fine asymptotic expansion of X holds in R 2 if, and only if, $\xi$ 1 /$\xi$ 2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors $\zeta$(x) is investigated in a similar way. Finally, several examples and counterexamples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.