Overview
This research explores the long-term behavior of periodic dynamical systems, particularly on the 3-torus, through the concept of rotation sets. A rotation set describes all possible average motions (rotation vectors) that the system can exhibit over time.
We focus on how these rotation sets are influenced by the system’s arithmetic structure and distinguish between:
Strong rotation vectors, which are uniformly defined for all orbits,
Weak rotation vectors, which arise only as statistical or ergodic averages.
Mathematical Context and Background
This research program lies at the intersection of analysis and algebra, particularly in the fields of number theory and Diophantine approximation. It builds upon previous results presented in the article published in the journal Dynamical Systems, titled Ordinary Differential Equations Defined by a Trigonometric Polynomial Field: Behavior of the Solutions:
https://www.tandfonline.com/doi/abs/10.1080/14689367.2023.2170212
Key Result and Open Question
One of the main results of our upcoming study is presented in the following working paper.
When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on the 3-torus with a Nilpotent Linearization, https://arxiv.org/abs/2504.21006.
A technical note on the arithmetic cone of smooth periodic vector fields https://arxiv.org/abs/2607.13102
Liouville Numbers: The construction relies on Liouville numbers (which are extremely well approximated by rationals) that generate an accumulation of arithmetic resonances in the third component.
Nilpotent Linearization: The Jacobian of the vector field is strictly nilpotent everywhere (all of its eigenvalues are zero). Consequently, the unbounded deviation does not stem from any local exponential stretching or amplification mechanism, but purely from the sum of incommensurable oscillations.
Arithmetic Cone and Uniform Boundedness: The arithmetic cone represents all direction vectors for which the spectral sum remains uniformly bounded along rational approximations. For finite trigonometric polynomials, this cone covers the entire space, whereas for pathological smooth fields, it forms a strict subset because of small-divisor resonances.
Spectrally Admissible Fields: This class consists of smooth periodic vector fields whose augmented arithmetic cone fills the entire space. It encompasses all finite trigonometric polynomials while excluding counterexamples driven by Liouville-type arithmetic resonances.
Goals of the Project
To understand how the arithmetic structure of a system shapes its rotation set.
To develop a classification framework for periodic ODEs based on the strength and nature of their rotation vectors.
To explore the implications of this classification in areas like celestial mechanics, quasi-periodic systems, and statistical physics.