One of signatures of a memristor

In this paper, a model of memristor oscillator with a piecewise function is taken into account. The classification and stability of equilibria of the model are discussed in detail. One of main findings is that there exist at least three types of singular continuum (pinched hysteresis loop) whose loci always pass through the origin in the voltage-current plane, and whose loop areas will shrink to zero, which is one of signatures of a memristor which distinguishes it from non-memristive devices. By using qualitative analysis and bifurcation theory, we have obtained two Hopf bifurcation surfaces and a unique unstable periodic orbit, whose existence and uniqueness are proved in a strict mathematical way. By using numerical simulations, we have given different phase portraits of equilibria, singular continuum and periodic orbits, which verify all analytical results. Our results demonstrate that the three types of singular continuum and a unique unstable periodic orbit may become some of important signatures of a memristor distinguishing from non-memristive devices, and the associated memristor circuits with piecewise functions may reflect more real aspects of functional roles of a memristor in electronic circuits.