Interval maps mimicking circle rotations

TL;DR

This paper studies a family of real line maps that mimic circle rotations, revealing conditions for periodic orbits and attracting behaviors, with numerical insights into complex phenomena.

Contribution

It introduces a specific two-parameter family of maps modeling circle rotation dynamics and proves the existence of attracting periodic orbits under certain conditions.

Findings

Existence of attracting periodic orbits for rational parameters

Behavior closely resembles circle rotations with large parameters

Numerical evidence of complex dynamical phenomena

Abstract

We investigate the dynamics of maps of the real line whose behavior on an invariant interval is close to a rational rotation on the circle. We concentrate on a specific two-parameter family, describing the dynamics arising from models in game theory, mathematical biology and machine learning. If one parameter is a rational number, $k/n$, with $k,n$ coprime, and the second one is large enough, we prove that there is a periodic orbit of period $n$. It behaves like an orbit of the circle rotation by an angle $2\pi k/n$ and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena. While we do not give rigorous proofs for them, we provide convincing explanations.