An aerosol provision system, such as an electronic cigarette, comprises a heating element for generating an aerosol from a source liquid and control circuitry for controlling a supply of electrical power from a power supply, such as battery / cell, to the heating element. The control circuitry is configured to measure an indication of a derivative of an electrical characteristic of the heating element with respect to time, for example a first time derivative or a second time derivative of a resistance of the heating element (or a related parameter, such as conductance, current draw, power draw or voltage drop). Based on the measured time derivative, the control circuitry is configured to determine whether or not a fault condition, e.g. localised heating of the heating element, has arisen for the electronic aerosol provision system. The overall change in the electrical characteristic for the heating element caused by the localised heating may be small and so difficult to reliably identify, but the rate at which the change occurs can be expected to be relatively high, which can mean the time derivative of the local characteristic is more reliable indicator of the occurrence of the fault condition.Un sistema de disposicion de aerosol, tal como un cigarrillo electronico, que comprende un elemento de calentamiento para la generacion de un aerosol a partir de un liquido de fuente y circuiteria de control para controlar un suministro de energia electrica a partir de un suministro de energia, tal como una bateria/celda, al elemento de calentamiento. La circuiteria de control esta configurada para medir una indicacion de un derivado de una caracteristica electrica del elemento de calentamiento respecto al tiempo, por ejemplo un primer deriva del tiempo o un segundo derivado del tiempo de una resistencia del elemento de calentamiento (o un parametro relacionado, tal como conductancia, consumo de corriente, consumo de energia o caida de voltaje). Basado en el derivado del tiem
