This brief elucidates a recent observation, based on the numerical integration of a predictive model of a ReRAM cell, manufactured at Forschungszentrum Jülich, under the as-sumption that a particular symmetric triangular periodic voltage waveform is let fall continuously between the device terminals. Averaging the model state equation in time, a rigorous methodol-ogy is developed to identify all the admissible levels, around which the memory state of the non-volatile memristor may oscillate asymptotically. This work provides proof of evidence for the benefits that a theoretic approach may provide to circuit design with inherently-nonlinear memristive nanostructures.