The pacemaking system of the heart is complex; a healthy heart constantly integrates and responds to extracardiac signals, resulting in highly complex heart rate patterns with a great deal of variability. In the laboratory and in some pathological or age related states, however, dynamics can show reduced complexity that is more readily described and modeled. Reduced heart rate complexity has both clinical and dynamical significance – it may provide warning of impending illness or clues about the dynamics of the heart’s pacemaking system. Here we describe simple and interesting heart rate dynamics that we have observed in premature human infants – reversible transitions to large- amplitude periodic oscillations – and we show that they give early warning of bacterial infections in premature infants, and we show that the appearance and disappearance of these periodic oscillations can be described by a simple mathematical model, a Hopf bifurcation.