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

This paper deals with the long time asymptotics of the flow $ X $ solution to the vector-valued ODE: $ X'(t, x) = b(X(t, x)) $ for $ t\in \mathbb{R} $, with $ X(0, x) = x $ a point of the torus $ Y_d $. We assume that the vector field $ b $ reads as $ \rho\, \Phi $, where $ \rho $ is a non negative regular function and $ \Phi $ is a non vanishing regular vector field in $ Y_d $. In this work, the singleton condition means that the Herman rotation set $ {\mathsf{C}}_b $ composed of the average values of $ b $ with respect to the invariant probability measures for the flow $ X $ is a singleton $ \{\zeta\} $. This first allows us to obtain the asymptotics of the flow $ X $ when $ b $ is a nonlinear current field. Then, we prove a general perturbation result assuming that $ \rho $ is the uniform limit in $ Y_d $ of a positive sequence $ (\rho_n)_{n\in \mathbb{N}} $ satisfying $ \rho\leq\rho_n $ and $ {\mathsf{C}}_{\rho_n\Phi} $ is a singleton $ \{\zeta_n\} $. It turns out that the limit set $ {\mathsf{C}}_b $ either remains a singleton, or enlarges to the closed line segment $ [0_{ \mathbb{R}^d}, \lim_n\zeta_n] $ of $ \mathbb{R}^d $. We provide various corollaries of this result according to the positivity or not of some weighted harmonic means of $ \rho $. These results are illustrated by different examples which highlight the alternative satisfied by $ {\mathsf{C}}_b $. Finally, the singleton condition allows us to homogenize the linear transport equation induced by the oscillating velocity $ b(x/{\varepsilon}) $.