In this manuscript, we construct an explicit counterexample of a smooth \(C^{ \infty}\), periodic dynamical system on the torus \(\mathbb{ T}^3\) for which the rotation vector exists in a weak sense, but fails to exist in the strong sense of bounded deviation (also referred to as { \it { frequencies}} in parts of the physics and biology literature). The construction exploits Liouville-type arithmetic properties and demonstrates that smoothness and periodicity alone do not ensure bounded deviation, even within the class of integrable systems.
Categories: math.DS