Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications

<p style='text-indent:20px;'>This paper deals with the long time asymptotics of flow <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> solution to vector-valued ODE: id="M2">\begin{document}$ X'(t, x) = b(X(t, x)) for id="M3">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, id="M4">\begin{document}$ X(0, x a point torus id="M5">\begin{document}$ Y_d $\end{document}</tex-math></inline-formula>. We assume that vector field id="M6">\begin{document}$ b reads as id="M7">\begin{document}$ \rho\, \Phi where id="M8">\begin{document}$ \rho is non negative regular function and id="M9">\begin{document}$ vanishing in id="M10">\begin{document}$ In this work, singleton condition means Herman rotation set id="M11">\begin{document}$ {\mathsf{C}}_b composed average values id="M12">\begin{document}$ respect invariant probability measures id="M13">\begin{document}$ id="M14">\begin{document}$ \{\zeta\} This first allows us obtain id="M15">\begin{document}$ when id="M16">\begin{document}$ nonlinear current field. Then, we prove general perturbation result assuming id="M17">\begin{document}$ uniform limit id="M18">\begin{document}$ positive sequence id="M19">\begin{document}$ (\rho_n)_{n\in \mathbb{N}} satisfying id="M20">\begin{document}$ \rho\leq\rho_n id="M21">\begin{document}$ {\mathsf{C}}_{\rho_n\Phi} id="M22">\begin{document}$ \{\zeta_n\} It turns out id="M23">\begin{document}$ either remains singleton, or enlarges closed line segment id="M24">\begin{document}$ [0_{ \mathbb{R}^d}, \lim_n\zeta_n] id="M25">\begin{document}$ \mathbb{R}^d provide various corollaries according positivity not some weighted harmonic id="M26">\begin{document}$ These results are illustrated by different examples which highlight alternative satisfied id="M27">\begin{document}$ Finally, homogenize linear transport equation induced oscillating velocity id="M28">\begin{document}$ b(x/{\varepsilon}) $\end{document}</tex-math></inline-formula>.</p>