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

TL;DR

This paper investigates the long-term behavior of flows on the torus for certain dynamical systems, introducing new methods that relax classical ergodicity assumptions and apply to broader classes of vector fields.

Contribution

It presents a novel approach to analyze asymptotics of flows on the torus without requiring unique ergodicity, using divergence-curl formulas and invariant measure properties.

Findings

Established asymptotic behavior for flows with vanishing or gradient vector fields.

Extended results to higher dimensions and more general vector fields.

Provided new tools for analyzing long-term dynamics on the torus.

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 $\mu$, namely $\mu\,b$ 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 $b = a\,\xi$ with $a$ a possibly vanishing periodic nonnegative function and $\xi$ a nonzero vector in $R^d$, or when $b = A\nabla v$ with $A$ a periodic nonnegative symmetric matrix-valued function and $v$ a periodic function.