Dynamics of a hybrid thermostat model with discrete sampling time control

The dynamics of a simple thermostat model is described. In the model the control system samples the temperature at regular but discrete time intervals rather than by continuous monitoring. The model exhibits quasi-periodic oscillations and banding, where the response falls into two or more bands of phase space representing either better or poorer control. A return circle map is derived which explains the observed dynamics. Some extensions of these results to the case where the flow is non-linear are also given.