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

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" $\rho$.