When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on the 3-torus with a Nilpotent Linearization

Abstract

In this manuscript, we construct an explicit counterexample of smooth infinitely differentiable, periodic dynamical system on the 3-torus for which the rotation vector exists in the weak sense but fails to exist in the strong sense of bounded deviation. The construction uses Liouville-type arithmetic resonances and shows that periodicity and infinite differentiability alone do not guarantee a bounded drift deviation, even for integrable flows. To the best of our knowledge, this is the first example of a smooth integrable torus flow with unbounded rotational deviation whose Jacobian is everywhere nilpotent.: the Jacobian matrix is strictly nilpotent, all its eigenvalues are identically zero, and consequently no local exponential stretching or contraction occurs anywhere. The unbounded deviation from the linear drift is generated purely by the accumulation of infinitely many incommensurable frequencies, without any amplification mechanism. This purely neutral local dynamics makes the example particularly relevant for coupled phase oscillator models, where the linearized dynamics around a synchronized state is typically nilpotent or neutral. Moreover, this work extends the theory of trigonometric polynomial fields. It was previously shown that when the frequency spectrum $\Lambda_f$ is finite, a strong rotation vector always exists. The present counterexample demonstrates that as soon as the spectrum becomes infinite (while retaining full regularity) the strong rotation vector can disappear. Thus, the finiteness of the spectrum cannot be replaced by smoothness alone.