Feature preserving discretization of memristive circuits using an input-output approach

It is known that continuous-time (CT) dynamic analog circuits containing ideal memristors and, more generally, memelements, admit for structural reasons first integrals and, consequently, enjoy the foliation feature of the state space. This important property makes it possible the coexistence of a huge number of different attractors, even when the memelement has a constitutive relation of a general shape. This paper considers the problem of how discrete-time (DT) memristive circuits preserving this foliation feature can be derived by the CT ones. Specifically, a discretization procedure of CT circuits is proposed in input-output setting, instead of the state space one, which applies to memristors as well as memcapacitors and meminductors, and it is not limited to the Forward Euler discretization operator. It is shown that the procedure provides DT maps which preserve the foliation feature of the original CT circuits for any discretization time step T, thus enabling extreme multistability. Moreover, it is highlighted that by increasing the time step T the obtained DT maps can exhibit chaotic attractors even if the CT memristive circuits display convergent behaviors, a phenomenon which is known as computational chaos. As an example, it is shown that, by applying the proposed procedure to a simple CT RC-memristive circuit with the memristor charge depending quadratically on the flux, the obtained DT circuits embed the logistic and the Henon maps depending on the chosen discretization operator.