The progressive breakdown of ultrathin (≈2nm) SiO2 gate oxides subjected to constant electrical stress is investigated using a simple equivalent circuit model. It is shown how the interplay among series, parallel, and filamentary conductances that represent the breakdown path and its surroundings leads under certain hypothesis to a sigmoidal current-time characteristic compatible with the experimental observations. The dynamical properties of the breakdown trajectories are analyzed in terms of the logistic potential function, the Lyapunov exponent, and the system's attractor. It is also shown that the current evolution is compatible with Prigogine's minimum entropy production principle. © 2011 American Institute of Physics.