Revisiting the asymptotics of the flow for some dynamical systems on the torus

2019/12/19 by Briane, Marc, Hervé, Loïc
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.09213

Abstract

In this paper we study the large time asymptotics of the flow of a dynamical system X'=b(X) posed in the d-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if b vanishes at different points, we only assume that the set of the averages of b with respect to the invariant probability measures for the flow is reduced to a singleton. We also rewrite the Liouville theorem which holds for any invariant probability measure μ, namely μ b is divergence free, as a divergence-curl formula satisfied by any regular periodic function. The combination of these two tools turns out to be a new approach to get the asymptotics for some flows. This allows us to obtain the desired asymptotics in any dimension when b = a ξ with a a possibly vanishing periodic nonnegative function and ξ a nonzero vector in R^d, or when b = A∇ v with A a periodic nonnegative symmetric matrix-valued function and v a periodic function.

Related

• Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications
• Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations
• Limit theorems for toral translations
• Specific properties of the ODE's flow in dimension two versus dimension three
• Parabolic dynamics and Anisotropic Banach spaces