Electric oscillations constitute the essential physical phenomenon utilized for the development of oscillatory neural networks. At the core of these developments there are the memristic systems. One of the essential ingredients for these systems to work as oscillators, is a negative differential resistance (NDR). This can be achieved by two ways according to the I-V characteristic curve that can resemble an N or an S letter (N-NDR or S-NDR type systems respectively). The archetypal S-type systems are formed by sub-stoichiometric transition metal oxides as VO2, NbO2 and TaOx sandwiched between two metal electrodes. In this work, we have studied from a mathematical point of view this type of systems, specifically the bifurcation points as well as the frequencies at the oscillatory regimes. At the most simple level, a memristor behaving as an S-NDR type oscillator can be modelled as a nonlinear element which depends on an internal variable in a particular way, a parallel capacitor and a series resistance. This system can be driven by imputing a constant current (galvanostatically) or a constant voltage (potentiostatically). We have associated to it two coupled differential equations that include two variables. In this context oscillations may arise depending on the value of some controlling parameters such as input intensity and the capacitance. In general, while low values of the capacitance lead the system to a stable stationary point, higher values at the Hopf bifurcation, produce well defined sinusoidal oscillations. Form this point on, relaxation oscillations are found when increasing the capacitance value.