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.

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.

Nonvolatile memristive devices display nonlinear characteristics suitable for implementing circuits exhibiting oscillations or more complex dynamic behaviors, including chaos. However, the results presented in related works are mostly limited to simulations and employing ideal memristor models whose resistance is governed by a charge-flux relation that is not connected to real devices, thus hindering the realization of such nonlinear oscillators. In this work, we present the framework for the physical implementation of a tunable memristor Chua’s circuit, which is based on a nonvolatile memristive device that provides the nonlinear conductance required by the circuit and the possibility to tune it for the purpose of selecting among different oscillation patterns. We first establish design guidelines to guarantee complex oscillations in the tunable memristor Chua’s circuit. Further, we physically implement the circuit after characterizing and modeling the tunable current-voltage characteristic of a real device. Our circuit successfully generates different oscillation patterns just by programming the nonvolatile memristive device to different states. The devised design guidelines and device modeling were used to extend the experimental work and draw further requirements for device properties for a successful circuit implementation.

The article considers a large class of delayed neural networks (NNs) with extended memristors obeying the Stanford model. This is a widely used and popular model that accurately describes the switching dynamics of real nonvolatile memristor devices implemented in nanotechnology. The article studies via the Lyapunov method complete stability (CS), i.e., convergence of trajectories in the presence of multiple equilibrium points (EPs), for delayed NNs with Stanford memristors. The obtained conditions for CS are robust with respect to variations of the interconnections and they hold for any value of the concentrated delay. Moreover, they can be checked either numerically, via a linear matrix inequality (LMI), or analytically, via the concept of Lyapunov diagonally stable (LDS) matrices. The conditions ensure that at the end of the transient capacitor voltages and NN power vanish. In turn, this leads to advantages in terms of power consumption. This notwithstanding, the nonvolatile memristors can retain the result of computation in accordance with the in-memory computing principle. The results are verified and illustrated via numerical simulations. From a methodological viewpoint, the article faces new challenges to prove CS since due to the presence of nonvolatile memristors the NNs possess a continuum of nonisolated EPs. Also, for physical reasons, the memristor state variables are constrained to lie in some given intervals so that the dynamics of the NNs need to be modeled via a class of differential inclusions named differential variational inequalities.

Controlling multistability, i.e., designing control laws for switching among different attractors, is an emerging issue in the area of memristor circuits. The paper considers the Chua’s memristor circuit which is known to display infinitely many attractors, each one contained in an invariant manifold of the circuit state space. The problem of switching among these attractors via pulse-programmed feedforward control laws, which are implementable via a unique current/voltage source, is investigated. In particular, it is shown that if the shape of the voltage source in series to the inductor is suitably designed, then it is possible to switch in finite time from one attractor to another.

Random mechanisms including mutations are an internal part of evolutionary algorithms, which are based on the fundamental ideas of Darwin’s theory of evolution as well as Mendel’s theory of genetic heritage. In this paper, we debate whether pseudo-random processes are needed for evolutionary algorithms or whether deterministic chaos, which is not a random process, can be suitably used instead. Specifically, we compare the performance of 10 evolutionary algorithms driven by chaotic dynamics and pseudo-random number generators using chaotic processes as a comparative study. In this study, the logistic equation is employed for generating periodical sequences of different lengths, which are used in evolutionary algorithms instead of randomness. We suggest that, instead of pseudo-random number generators, a specific class of deterministic processes (based on deterministic chaos) can be used to improve the performance of evolutionary algorithms. Finally, based on our findings, we propose new research questions.

This article introduces a new class of memristor neural networks (NNs) for solving, in real-time, quadratic programming (QP) and linear programming (LP) problems. The networks, which are called memristor programming NNs (MPNNs), use a set of filamentary-type memristors with sharp memristance transitions for constraint satisfaction and an additional set of memristors with smooth memristance transitions for memorizing the result of a computation. The nonlinear dynamics and global optimization capabilities of MPNNs for QP and LP problems are thoroughly investigated via a recently introduced technique called the flux-charge analysis method. One main feature of MPNNs is that the processing is performed in the flux-charge domain rather than in the conventional voltage-current domain. This enables exploiting the unconventional features of memristors to obtain advantages over the traditional NNs for QP and LP problems operating in the voltage-current domain. One advantage is that operating in the flux-charge domain allows for reduced power consumption, since in an MPNN, voltages, currents, and, hence, power vanish when the quick analog transient is over. Moreover, an MPNN works in accordance with the fundamental principle of in-memory computing, that is, the nonlinearity of the memristor is used in the dynamic computation, but the same memristor is also used to memorize in a nonvolatile way the result of a computation.

Memristors are more and more seen as the basic elements for the development of new unconventional computational schemes. One of the appealing property of memristor circuits is multistability, i.e., the coexistence in the state space of a rich variety of different attractors. This paper considers the problem of controlling multistability for a circuit with a charge-controlled memristor which displays infinite stable equilibrium points and limit cycles. Specifically, it is shown how voltage and current sources can be pulse programmed in order to steer the circuit dynamics from one stable equilibrium point to a different stable equilibrium point within a given finite time interval.

Circuits containing memelements (memory elements) are suitable for the design of new unconventional computational schemes. The coexistence of a rich variety of different attractors is one of the appealing property of these circuits, which has stimulated the so-called “multistability control” problem. This paper considers the multistability control problem for a circuit with a charge-controlled memristor. It is shown how pulse control inputs can be generated via an external current generator in order to drive the system dynamics from an attractor to another one in a given finite time interval.

In this paper, a nonlinear approach to modeling DC-DC PWM converters in continuous conduction mode (CCM) is presented. Specifically, a converter nonlinear model is implemented in MATLAB and SIMULINK for both the case without parasitic components (ideal case) and with parasitic components (non-ideal case). The implementation, which is based on the analytical computation of the periodic solutions in the high frequency steady-state operating condition, makes it possible to perform a systematic analysis of the dynamical behavior induced by the various perturbation sources acting on the converters. A comparison of the dynamics generated in the ideal and nonideal cases by step changes in duty cycle, input voltage and load resistance is reported for both the boost and buck converters.

DC-DC pulse-width modulated (PWM) converters are nonlinear systems that require control circuits to obtain the desired output voltage. These control circuits are often designed using linearized models of the converters, so modeling is a relevant topic for the design of performing closed-loop converters. In this paper, the nonlinear nature of DC-DC PWM converters operated under continuous conduction mode (CCM) is considered. It is first shown how the exact periodic solutions can be analytically computed for the nonlinear converter model in the high-frequency steady-state operating condition. Also, a nonlinear variation model, which characterizes the closed-loop dynamics, induced by perturbations of the steady state, is provided. The nonlinear models and the analytical periodic solutions are implemented in MATLAB and SIMULINK and applied to a boost converter subject to parasitic components. In particular, the behavior to step changes in duty cycle, input voltage, and load resistance are compared with the results obtained using small-signal circuit averaging techniques.

The paper considers the problem of designing controllers to stabilize periodic orbits in a class of sinusoidally forced nonlinear systems. This problem is formulated as an absolute stability problem of a linear periodic feedback system, in order to employ the well-known circle criterion. In this setting, we provide an LMI-based synthesis of the optimal stabilizing controller, i.e., the one ensuring the largest obtainable stability bounds.

The issue of the loss of complete stability for a class of cellular neural networks (CNNs) is analyzed. It is shown that there are CNNs in this class for which a Hopf bifurcation is present, even if the interconnection matrix is arbitrarily close to some symmetric matrix. This shows that, in the general case, complete stability is not robust with respect to perturbations of nominal symmetric interconnection matrices.

The authors propose a theoretical framework for backpropagation (BP) in order to identify some of its limitations as a general learning procedure and the reasons for its success in several experiments on pattern recognition. The first important conclusion is that examples can be found in which BP gets stuck in local minima. A simple example in which BP can get stuck during gradient descent without having learned the entire training set is presented. This example guarantees the existence of a solution with null cost. Some conditions on the network architecture and the learning environment that ensure the convergence of the BP algorithm are proposed. It is proven in particular that the convergence holds if the classes are linearly separable. In this case, the experience gained in several experiments shows that multilayered neural networks (MLNs) exceed perceptrons in generalization to new examples.>