Mauro Di Marco

Memristor circuits are gaining increasing importance in modern electronics due to their ability to mimic neurons, synapses, and, more generally, the computation capabilities of biological neural systems. The ideal memristor model was introduced by L. Chua in a seminal paper back in 1971. Later on, several more complex memristor models, such as the popular Voltage ThrEshold Adaptive Memristor model, have been developed to better describe the state‐dependent current‐voltage behavior of physical memristor devices, in particular Resistive Random Access Memories, a nanotechnology of great appeal nowadays. While the study of circuits with ideal memristors is reaching a mature state, very few results are available on the analysis of the dynamics of circuits with real memristors. As demonstrated in this paper, peculiar dynamical properties of cells with ideal memristors are also displayed by some analogue electrical circuits with real memristors. In particular, simple circuits, including, as dynamical elements, one capacitor and a memristive one‐port, composed of a pair of back‐to‐back Resistive Random Access Memory devices described through the Voltage ThrEshold Adaptive Memristor model, admit physical quantities, referred to as Invariants of Motion, which are conserved over time. As a consequence, their state space can be foliated into Invariant Manifolds, which induce the coexistence of infinitely many different monostable and bistable quiescent steady states. A second‐order cell from the proposed class may be toggled between regimes of quiescent monostability and bistability by controlling the initial condition‐dependent index of the manifold along which its dynamical variables are constrained to lie at all times. Moreover, at any equilibrium point, the voltage across its capacitor vanishes, which allows the memristive one‐port to hold its state without energy consumption. These attractive properties make the proposed cells ideal candidates to serve as sensing‐and‐memcomputing units for innovative Cellular Neural Networks to be deployed for solving time‐critical edge computing problems in resource‐constrained environments.

Convergence of nonlinear circuits towards equilibrium points (EPs) is one of the most basic properties both from a theoretic and a practical viewpoint. It is especially relevant for nonlinear circuits modeling neural networks, since a convergent network with multiple stable EPs is tailor made to implement content addressable memories (CAMs) storing multiple patterns as stable EPs or to solve combinatorial optimization problems in real time. Convergence has been widely investigated in the last few decades. By far, the available convergence results can be applied to circuits without memristors, while the study of convergence in presence of memristors is only in its infancy. In this paper, a class of nonlinear circuits containing memristors, capacitors, passive or active resistors and independent sources, is considered. Active resistors are crucial, since they permit to obtain circuits with multiple stable EPs. A number of basic results on convergence are obtained via the flux-charge analysis method (FCAM) and a fundamental reciprocity principle for the considered class of circuits. The conditions for convergence are robust, i.e., they hold also for perturbations of the circuit parameters and memristor nonlinearities involved. The results are illustrated via selected examples where use is made of the celebrated HP memristor model.

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.

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.

Convergence of dynamic feedback neural networks (NNs), as the Cohen-Grossberg, Hopfield and cellular NNs, has been for a long time a workhorse of NN theory. Indeed, convergence in the presence of multiple stable equilibrium points (EPs) is crucial to implement content addressable memories and solve several other signal processing tasks in real time. There are two typical ways to use a convergent NN, i.e.: a) let the activations evolve while maintaining fixed weights and inputs (activation dynamics) or b) adapt the weights while maintaining fixed activations (weight dynamics). As remarked in a seminal paper by Hirsch, there is another interesting possibility, i.e., let the neuron interconnection weights evolve while simultaneously running the activation dynamics (weight-activation dynamics). The weight-activation dynamics is of importance also because it is more plausible than the other two types for modeling neural systems. The paper breaks new ground by analyzing for the first time in a systematic way the convergence properties of the weight-activation dynamics for a class of memristor feedback dynamic NNs. The main result is that, under suitable assumptions on the structure of the memristor interconnections, the solutions (weights and activations) converge to an EP, except at most for a set of initial conditions with zero measure. The result includes the most important case where the NN has multiple stable EPs.

In recent years, many contributions have been reported in the literature showing that memristor circuits can exhibit very rich oscillatory and chaotic behaviors. In this paper, we look for conditions ensuring that memristor circuits are nonoscillatory, i.e. they do not display oscillations and more complex attractors and enable convergence toward some of the infinite nonisolated equilibrium points. Specifically, we consider the class of memristor circuits composed by the interconnection of a linear time-invariant two-terminal (one port) element and an ideal memristor, which can be either flux-controlled or charge-controlled and whose characteristic satisfies a slope-bounded condition. First, exploiting the well-known fact that any circuit with an ideal memristor admits first integrals, and hence its state space is decomposed in a continuum of invariant manifolds, a state space representation of the circuit dynamics on each invariant manifold is derived in an explicit way. Then, conditions to ensure the absence of oscillatory behaviors on each manifold are obtained by employing the 2-additive compound matrix of the Jacobian of these representations. It is shown that the memristor circuit is nonoscillatory if there exists a common Lyapunov function for two suitable 2-additive compound matrices, which is obtained by solving two linear matrix inequalities. Two memristor circuits are analyzed in some detail to illustrate the application of the results and their degree of conservatism.

The paper considers a wide class of nonlinear circuits with an ideal memristor, capacitors, inductors and current or voltage sources. A fundamental dynamical property is that each memristor circuit in this class admits a first integral (invariant of motion or preserved quantity), i.e., a function which is constant along the solutions. The first main result is that we can put the state equations of each circuit in a universal form, known as linear-gradient form, given by a state-dependent skew-symmetric matrix times the gradient of the first integral. This is a simple and general form which is both of theoretic and practical interest. First of all, it makes manifest the existence of a first integral. Moreover, it admits an elegant discrete-time (DT) analogue. Indeed, the linear-gradient form, combined with geometric discretization methods and the concept of discrete gradients, yields a DT version of each memristor circuit that exactly preserves the first integral for any discretization time step. This is relevant, since the existence of a first integral is a fragile property that is in general destroyed by typical discretization schemes used in the literature no matter how small the step size is. On one hand, the proposed discretization scheme can be used for constructing numerical algorithms that better approximate the solutions of memristor circuits for small step sizes. On the other hand, for larger step sizes the obtained DT memristor circuits can be of interest by themselves since they are able to easily generate complex dynamics potentially useful for engineering applications (computational chaos). Furthermore, the paper shows that, thanks to the decomposition of the state space in invariant manifolds, the derived DT circuits exhibit extreme multistability, i.e., the coexistence of infinitely many different attractors for fixed set of circuit parameters, memristor nonlinearity and step size.

In the last few years the literature has witnessed a remarkable surge of interest for maps implemented by discrete-time (DT) memristor circuits. One main reason is that from numerical simulations it appears that even for simple memristor circuits the maps can easily display complex dynamics, including chaos and hyperchaos, which are of relevant interest for engineering applications. Goal of this manuscript is to investigate on the reasons underlying this type of complex behavior. To this end, the manuscript considers the map implemented by the simplest memristor circuit given by a capacitor and an ideal flux-controlled memristor or an inductor and an ideal charge-controlled memristor. In particular, the manuscript uses the DT flux-charge analysis method (FCAM) introduced in a recent paper to ensure that the first integrals and foliation in invariant manifolds of continuous-time (CT) memristor circuits are preserved exactly in the discretization for any step size. DT-FCAM yields a two-dimensional map in the voltage-current domain (VCD) and a manifold-dependent one-dimensional map in the flux-charge domain (FCD), i.e., a one-dimensional map on each invariant manifold. One main result is that, for suitable choices of the circuit parameters and memristor nonlinearities, both DT circuits can exactly embed two classic chaotic maps, i.e., the logistic map and the tent map. Moreover, due to the property of extreme multistability, the DT circuits can simultaneously embed in the manifolds all the dynamics displayed by varying one parameter in the logistic and tent map. The paper then considers a DT memristor Murali-Lakshmanan-Chua circuit and its dual. Via DT-FCAM these circuits implement a three-dimensional map in the VCD and a two-dimensional map on each invariant manifold in the FCD. It is shown that both circuits can simultaneously embed in the manifolds all the dynamics displayed by two other classic chaotic maps, i.e., the Hénon map and the Lozi map, when varying one parameter in such maps. In essence, these results provide an explanation of why it is not surprising to observe complex dynamics even in simple DT memristor circuits.

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves.