Fine asymptotic expansion of the ODE's flow

TL;DR

This paper investigates the precise asymptotic behavior of solutions to certain periodic nonlinear ODEs, establishing conditions for detailed expansions based on the properties of the vector field and rotation vectors, with applications in Diophantine conditions.

Contribution

It provides new necessary and sufficient conditions for fine asymptotic expansions of ODE flows, linking them to the structure of the vector field and Diophantine properties of rotation vectors.

Findings

Asymptotic expansion holds if rotation vector ratios are Diophantine.

Characterization of vector fields for which the expansion is bounded.

Examples illustrating when the expansion fails or holds under various conditions.

Abstract

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.