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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 Hopfdoi:10.5540/tema.2002.03.01.0015 fatcat:2e7fqaskgzaxpaectfgyprl2zi