When Periodicity Fails to Guarantee the Existence of Rotation

TL;DR

Problem addressed: the paper investigates rotation vectors for a smooth $C^{\infty}$ periodic system modulo $\mathbb{Z}^3$ and constructs a counterexample where the rotation vector exists only in the weak sense. Approach: define an integrable ODE with $h_1(z)=r$, $h_2(z)=1$, $h_3(z)=\sum_{m\in\mathbb{N}} \frac{1}{p_m^{m-1}} \cos(2\pi(z_1 p_m - z_2 q_m))$, where $r$ is a Liouville number and $(p_m,q_m)$ satisfy $0<|rp_m - q_m| \le p_m^{-(m-1)}$, yielding a solution $x(t)=(rt,t,\phi_r(t))$ with $\phi_r(t)=\int_0^t \sum_{m\in\mathbb{N}} \frac{1}{p_m^{m-1}} \cos(2\pi(r p_m - q_m)s) ds$. Result: exchanging sum and integral gives $\phi_r(t)=\sum_{m\in\mathbb{N}} \frac{1}{p_m^{m-1}(rp_m - q_m)} \sin(2\pi(r p_m - q_m)t)$, and one shows $\lim_{t\to\infty} t^{-1} x_3(t)=0$ while $x_3(t)$ is unbounded. Conclusion: this provides a counterexample where the rotation vector exists in the weak sense but not in the strong sense, highlighting a link between Liouville arithmetic and rotation-vector behavior and motivating open questions about higher-dimensional arithmetic and oscillator interactions.

Abstract

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.

Theorems & Definitions (2)

• Definition 1: Liouville Number
• Remark 2