# Ordinary differential equations defined by a trigonometric polynomial field: Behavior of the solutions

## TL;DR

This paper investigates solutions to ODEs driven by trigonometric polynomial fields, showing they possess a rotation vector and are weakly almost periodic, revealing structured long-term behavior.

## Contribution

It proves the existence of a rotation vector for solutions of such ODEs and characterizes their boundedness and almost periodicity.

## Key findings

- Solutions admit a rotation vector 
- The function x(t) - 
ho t is bounded
- Solutions are weakly almost periodic functions

## Abstract

We consider the ordinary differential equations defined by a trigonometric polynomial field, we prove that any solution $x$ admits a "rotation vector" $\rho\in \mathbb{R}^n$. More precisely, the function $t\mapsto x(t)-\rho t$ is bounded on time and it is a "weak almost periodic" function of "slope" $\rho$.