This work elucidates some aspects of the nonlinear dynamics of a thermally-activated locally-active memristor based on a micro-structure consisting of a bi-layer of Nb₂O₅ and Nb₂O_x materials. Through application of techniques from the theory of nonlinear dynamics to an accurate and simple mathematical model for the device, we gained a deep insight into the mechanisms at the origin of the emergence of local activity in the memristor. This theoretical study sets a general constraint on the biasing arrangement for the stabilization of the negative differential resistance effect in locally active memristors and provides a theoretical justification for an unexplained phenomenon observed at HP labs. As proof-of-principle, the constraint was used to enable a memristor to induce sustained oscillations in a one port cell. The capability of the oscillatory cell to amplify infinitesimal fluctuations of energy was theoretically and experimentally proved.