First integrals can explain coexistence of attractors, multistability, and loss of ideality in circuits with memristors

In this paper a systematic procedure to compute the first integrals of the dynamics of a circuit with an ideal memristor is presented. In this perspective, the state space results in a layered structure of manifolds generated by first integrals, which are associated, via the choice of the initial conditions, to different exhibited behaviors. This feature turns out to be a powerful investigation tool, and it can be used to disclose the coexistence of attractors and the so called “extreme multistability,” which are typical of the circuits with ideal memristors. The first integrals can also be exploited to study the energetic behavior of both the circuit and of the memristor itself. How to extend these results to the other ideal memelements and to more complex circuit configurations is shortly mentioned. Moreover, a class of ideal memristive devices capable of inducing the same first integrals layered in the state space is introduced. Finally, a mechanism for the loss of the ideality is conceived in terms of spoiling the first integrals structure, which makes it possible to develop a non-ideal memristive model. Notably, this latter can be interpreted as an ideal memristive device subject to a dynamic nonlinear feedback, thus highlighting that the non-ideal model is still affected by the first integrals influence, and justifying the importance of studying the ideal devices in order to understand the non-ideal ones.