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 ∈ R^d , where b is a regular Z^d periodic 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 ζ(x)| ≤ M < ∞, which is uniform with respect to t ≥ 0 and x ∈ R^d (or at least in a subset of R^d), and where ζ(x) for x ∈ 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 ζ(x) reads as Φ(X(t, x)) − Φ(x), which yields immediately the desired expansion when the vector-valued function Φ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on the torus Y^d and of vector-valued functions Φ. 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 ξ, the fine asymptotic expansion of X holds in R^2 if, and only if, ξ_1/ξ_2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors ζ(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. In particular, the case of some Euler flows is investigated.