We consider the heat equation in a multidimensional domain with nonlocal hysteresis feedback control in a boundary condition. Thermostat is our prototype model.

By reducing the problem to a discontinuous infinite dynamical system, we construct all periodic solutions with exactly two switchings on the period and study their stability. In the problem under consideration, the hysteresis gap (the difference between the switching temperatures) is of especial importance.

If the hysteresis gap is large enough, then the constructed periodic solution is in fact unique and globally stable. For small values of hysteresis gap coexistence of several periodic solutions with different stability properties is proved to be possible.

This is a joint work with Pavel Gurevich.