Extreme Multistability of Memristor Circuits Operating in the Complex Domain

Recent years have witnessed a remarkable surge of interest for discretized versions of Continuous-Time (CT) memristor circuits. A large part of the results aim at showing that, when the discretization step size is not small, Discrete-Time (DT) memristor circuits can display a much more complex dynamics with respect to the CT counterparts. A strong motivation is that the maps thus obtained are of potential interest themselves for engineering applications and they can be easily implemented via a dedicated digital hardware. Recently, a DT version of the ideal memristor introduced by Leon Chua, capable of preserving the first integrals and invariant manifolds of CT memristor circuits, has been proposed. It has been shown that its dynamics can embed classic real DT maps, such as the logistic and Hénon ones. The goal of this paper is to extend the original CT and DT models to the complex domain in order to derive DT memristor circuits whose dynamics can embed classic complex maps. Under the assumption that the CT memristor has an analytic nonlinearity, the paper first introduces a complex DT counterpart via the principle of analytic continuation, also providing a discretization scheme that preserves the invariant manifolds. Then, the paper focuses on a DT circuit with a complex memristor and a capacitor, which generates a two-dimensional complex map in the Voltage–Current Domain (VCD) and a one-dimensional complex map on each invariant manifold, i.e. in the Flux–Charge Domain (FCD). It is shown that, by a suitable choice of the circuit parameters, discretization steps and memristor nonlinearity, it is possible to embed in the FCD the classic quadratic and cubic complex maps. On this basis, this paper explores the property of extreme multistability for the map in the VCD, showing that there is a continuum of both different chaotic dynamics on Julia sets embedded in the invariant manifolds and different periodic dynamics of any integer period. Notably, all these dynamics coexist for the same set of circuit parameters and memristor nonlinearity, thus demonstrating the extreme richness of dynamics that can be generated via the proposed complex DT memristor circuits.