When Periodicity Fails to Guarantee the Existence of Rotation

In this manuscript, we present a simple counterexample of a smooth $C^{\infty}$ periodic system modulo $\mathbb{Z}^3$, where the rotation vector exists only in the weak sense and not in the strong sense (also referred to as frequencies in physics and biology). This example highlights the subtle yet important distinction between these two concepts of rotation vectors in the context of periodic flows. It also raises open questions regarding the role of arithmetic properties, such as Liouville numbers, in the dynamics of such systems, pointing to areas for future research in the theory of oscillating systems.