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

Abstract

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