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 2πk/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.
•There are continuous interval maps with behavior mimicking circle rotations.•Families of interval maps with different attracting periodic orbits can look similar.•The same maps appear across various disciplines, including mathematical biology and machine learning.