Tremendous efforts are made towards the development and realization of memristors for memory technology. Furthermore, memristor-based neuron and synapse models are considered in several investigations on neuromorphic systems. Some work is devoted to take advantage of peculiar nonlinear dynamics emerging in memristors to extend or improve the functionalities of state-of-the-art circuits and systems, both in the digital domain, where their use has been proposed in logic gate design, and in the analog domain, where volatile memristors with negative differential resistance effect, capable to amplify infinitesimal fluctuations of energy, have been adopted to design interesting transistor-less circuits. While in an increasingly number of cases, memristor models based on charge transport phenomena can be verified through measurements to a high degree of accuracy, the systematic design of circuits exploiting the rich dynamical behavior of memristors is still restricted to certain cases. For some memristors, even circuits with a few number of elements may exhibit different nonlinear phenomena, which can be only described by results obtained in numerical simulations. In this manuscript a nonlinear system theory-based approach will be introduced and discussed in detail. It is based on the determination of nonlinear characteristic functions allowing the characterization of circuit properties to a high degree of accuracy. These real-valued functions represent circuit elements or sub-circuits, and may be used in an automated circuit design approach. Results showing the performance of the proposed method will be given and discussed in this contribution.