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 focus 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 π k/n and attracts trajectories of Lebesgue almost all starting points. We also discover numerically other interesting phenomena.