The Thermostat Problem

Abstract

A paradigm model for an air-conditioning system is studied: heat flux to and from one end of a bar is a (nonlinear) function of the temperature at the other end. The behaviour of this model is studied through semi-discretisation in the spatial variable and local linearisation. This procedure produces an autonomous system of ordinary differential equations whose local stability may be studied through its spectrum of eigenvalues and related back to the original problem through the Hopf bifurcation theorem. It will be shown that the proportionality constant ° is a bifurcation parameter which gives rise to three qualitatively different solutions: one stable, where the temperature tends exponentially to zero; one stable that is bounded by an envelope which tends exponentially to zero; and an unstable solution that oscillates with ever increasing amplitude. An almost local problem is also studied with similar results: the three qualitative solutions arise as before with the bifurcation parameter decreasing as the problem becomes closer to the local problem. Integral equation characterisations of the nonlinear problem are developed and existence and uniqueness are demonstrated. For the linear problem the general analytic solution is provided and its numerical evaluation is discussed.