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

Abstract

This paper deals with the long time asymptotics X(t, x)/t of the flow X solution to the autonomous vector-valued ODE: X (t, x) = b(X(t, x)) for t $\in$ R, with X(0, x) = x a point of the torus Y d := R d /Z d. We assume that the vector field b reads as the product $\rho$ $\Phi$, where $\rho$ : Y d $\rightarrow$ [0, $\infty$) is a non negative regular function and $\Phi$ : Y d $\rightarrow$ R d is a non vanishing regular vector field. In this work, the singleton condition means that the rotation set C b composed of the average values of b with respect to the invariant probability measures for the flow X is a singleton {$\zeta$}, or equivalently, that lim t$\rightarrow$$\infty$ X(t, x)/t = $\zeta$ for any x $\in$ Y d. This combined with Liouville's theorem regarded as a divergence-curl lemma, first allows us to obtain the asymptotics of the flow X when b is a 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$\in$N satisfying for any n $\in$ N, $\rho$ $\le$ $\rho$ n and C $\rho$n$\Phi$ is a singleton {$\zeta$ n }. It turns out that the limit set C b either remains a singleton, or enlarges to the closed line set [0, lim n $\zeta$ n ] of R d. We provide various corollaries of this perturbation result involving or not the classical ergodic condition, according to the positivity or not of some harmonic means of $\rho$. These results are illustrated by different examples which show that the perturbation result is limited to the scalar perturbation of $\rho$, and which highlight the alternative satisfied by the rotation set C b. Finally, we prove that the singleton condition allows us to homogenize in any dimension the linear transport equation induced by the oscillating velocity b(x/$\epsilon$) beyond any ergodic condition satisfied by the flow X.