The auxiliary heated Tokamak is considered as an open thermodynamic system in which the auxiliary power is applied externally to a resistively relaxed state of maximum entropy. It results in the magnetic configuration of the plasma maintaining, under the external action, the same form of the profiles as in the purely ohmic state with maximum entropy. The sensitivity of the plasma, with respect to slight resistive perturbations at the edge, is studied, exploiting its tendency to adhere to the structure of an ideally conductive equilibrium as far as the local dependence of the current is linear in time. Assuming the existence of a q-dependent region at the edge with very high thermal diffusivity, a q=const boundary surface exists with constant temperature. This situation results in a link between the temperature and the q=const surfaces of the slightly resistive equilibrium (or between the resistivity and the current) and in the existence of bifurcating states related to a resistive instability whose nonlinear evolution and possible saturation is sensitive on whether the initial disturbance is heating or cooling the edge.