Unravelling the dynamical richness of 3D canonical memristor oscillators

In the study of the dynamical richness of canonical memristor oscillators, two main drawbacks are to be overcome. First, the dimensional issue, since at least a 3D dynamical system must be analyzed; secondly, one must deal with discontinuous models, when a piecewise linear characteristics for the memristor is adopted. Such discontinuity arises after taking derivatives of the piecewise linear characteristics of the memristor, in order to work with state variables within the realm of currents and voltages. Here it is shown that reduced (2D), continuous models suffice for developing a sound mathematical analysis of the oscillator dynamics. Starting from the 3D discontinuous vector field that rules the dynamics of currents and voltages, topologically equivalent 2D continuous models are derived. The key point is the adequate exploitation of an existing conserved quantity that arises as a consequence of the conservation laws for fluxes and charges. Our analysis confirms the numerical results of existing works devoted to these oscillators, and shows rigorously the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions, what justifies the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.