Ideal Memristor Research Articles

Flux–Charge Analysis of Two-Memristor-Based Chua's Circuit: Dimensionality Decreasing Model for Detecting Extreme Multistability

In this paper, from a new perspective of flux and charge, we present in-depth analyses of two ideal memristor emulators and the fifth-order memristive Chua's circuit constructed based on them. The constitutive flux–charge relations of the two adopted memristor emulators are first formulated, and their initial-dependent characteristics are numerically revealed and experimentally verified. Thereafter, with these two constitutive relations, a third-order dimensionality decreasing flux–charge model for the fifth-order memristive Chua's circuit is constructed, in which five extra constant system parameters are introduced to indicate the initial states of the five dynamic elements. Numerical simulations confirm that this newly constructed model possesses several determined equilibria and maintains the initial-dependent dynamics of the original voltage–current model. Thus, the complex and sensitive initial state-related extreme multistability phenomenon can be deeply explored through theoretical analyses and hardware measurements. It is demonstrated that the sensitive extreme multistability phenomenon becomes detectable in the flux–charge domain, which is efficient for exploring the inner mechanisms and further seeking possible applications of this special phenomenon.

Computation of Pinched Hysteresis Loop Area From Memristance-vs-State Map

The voltage-current characteristic of memristor driven by sinusoidal signal has the shape of hysteresis loop pinched at the origin. The lobe area of the hysteresis loop has been computed so far either from the voltage-current plane for all types of memristors or from the constitutive relation in the flux-charge plane of ideal memristor. In this brief, we provide an alternative approach in computing the lobe area from the memristance-versus-state map of ideal and ideal generic memristors driven by sinusoidal or periodic continuous piecewise linear signal with zero dc component. The applicability of the proposed approach is demonstrated through a number of examples.

Hidden extreme multistability and dimensionality reduction analysis for an improved non-autonomous memristive FitzHugh–Nagumo circuit

Due to the introduction of ideal memristors, extreme multistability has been found in many autonomous memristive circuits. However, such extreme multistability has not yet been reported in a non-autonomous memristive circuit. To this end, this paper presents an improved non-autonomous memristive FitzHugh–Nagumo circuit that possesses a smooth hyperbolic tangent memductance nonlinearity, from which coexisting infinitely many attractors are obtained. By utilizing voltage–current circuit model, a three-dimensional non-autonomous dynamical model is established, based on which the initial-dependent dynamics is explored by numerical plots and extreme multistability is thereby exhibited. To confirm that the improved non-autonomous memristive circuit operates in hidden oscillating patterns, an accurate two-dimensional non-autonomous dimensionality reduction model with initial-related parameters is further built by using incremental integral transformation, upon which stability analysis and bifurcation behaviors are elaborated. Because the equilibrium state of the dimensionality reduction model is always a stable node-focus, hidden extreme multistability with coexisting infinitely many attractors is truly confirmed. Finally, PSIM circuit simulations validate the initial-related hidden dynamical behaviors.

Designed Memristor Circuit for Self‐Limited Analog Switching and its Application to a Memristive Neural Network

AbstractAnalog memristors enable compact neuromorphic computing with low power consumption. One of the issues with the technology is slow precise analog data programming. In this study, a novel analog data programming method utilizing a self‐limited set switching is proposed. The method can transfer any resistance values from reference resistors to the target memristor accurately inside a crossbar array by performing an appropriate voltage clocking. An ideal memristor model based on the method is proposed and a Ti‐doped NbOx charge trap memristor is evaluated as a promising candidate for applications. The characteristic error of the Ti‐doped NbOx memristor device is about 5% on average, compared to the ideal memristor, and configuring optimum parallel resistors in the circuit further improves this to 2.95%. The method is then applied to program a memristive neural network and this error is confirmed negligible; thus the proposed method is viable.

Φ memristor: Real memristor found

In this work, we invented the Φ memristor to exhibit the direct flux-charge interaction, in which a wire carrying a controlled amount of current is strung through a magnetic core, and, simultaneously, sensing the possibly induced voltage by the switched flux. This work confirms the existence of the ideal memristor postulated almost 50 years ago. In order to study this and its positive/negative integer-/fraction-order counterparties, the flux-charge relationship (and its approximations), the complete differential conformal transformation, and the complete triangular periodic table of elementary circuit elements are developed. The ideal Φ memristor, fractional memristor, mem-inductor/mem-capacitor, and negative memristor are predicted within the context of this new 3-in-1 memristor physics, and their new synaptic functionalities for a brain-like computer are studied experimentally. The Editor and Publisher of Journal of Applied Physics are retracting this paper, effective 8 January 2021, due to concerns regarding the validity of the conclusions.

Dimensionality Reduction Analysis for Detecting Initial Effects on Synchronization of Memristor-Coupled System

This paper presents a 9-D memristor-coupled system with three ideal memristors and investigates its initial effects on synchronization using dimensionality reduction analysis method. The 9-D memristor-coupled system is yielded from two identical 4-D ideal memristor-based hyper-jerk systems via coupling an ideal memristor, from which the initials-dependent synchronization with parallel offset for larger coupling strength is numerically exhibited. To explore the initial effects on synchronization, an equivalent 6-D dimensionality reduction model is built using state variable mapping (SVM) method, from which the implicit initials of the 9-D memristor-coupled system are transformed into the explicitly initials-related system parameters of such a 6-D dimensionality reduction model. Thus, the inherent initial mismatches between the two coupled identical 4-D subsystems are explicitly expressed as the initials-related parameter mismatches between the two coupled non-identical 3-D dimensionality reduction subsystems. The initials-related boundedness of the error system between the two non-identical 3-D dimensionality reduction subsystems is derived by Lyapunov analysis method, upon which the initial effects on synchronization with parallel offset are expounded quantitatively. Furthermore, the initials-dependent synchronization is well confirmed by the numerical simulations, which demonstrates that the initials do have great influence on synchronization dynamics of the coupled memristive system.

Dynamical attractors of memristors and their networks

It is shown that the time-averaged dynamics of memristors and their networks periodically driven by alternating-polarity pulses may converge to fixed-point attractors. Starting with a general memristive system model, we derive basic equations describing the fixed-point attractors and investigate attractors in the dynamics of ideal, threshold-type and second-order memristors, and memristive networks. A memristor potential function is introduced, and it is shown that in some cases the attractor identification problem can be mapped to the problem of potential function minimization. Importantly, the fixed-point attractors may only exist if the function describing the internal state dynamics depends on an internal state variable. Our findings may be used to tune the states of analog memristors, and also be relevant to memristive synapses subjected to forward- and back-propagating spikes.

Dimensionality Reduction Reconstitution for Extreme Multistability in Memristor‐Based Colpitts System

In this paper, a four‐dimensional (4‐D) memristor‐based Colpitts system is reaped by employing an ideal memristor to substitute the exponential nonlinear term of original three‐dimensional (3‐D) Colpitts oscillator model, from which the initials‐dependent extreme multistability is exhibited by phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3‐D dimensionality reduction model is built using the state variable mapping (SVM) method, which allows the implicit initials of the 4‐D memristor‐based Colpitts system to be changed into the corresponding explicitly initials‐related system parameters of the 3‐D dimensionality reduction model. The initials‐related equilibria of the 3‐D dimensionality reduction model are derived and their initials‐related stabilities are discussed, upon which the dynamical mechanism is quantitatively explored. Furthermore, the initials‐dependent extreme multistability is depicted by two‐parameter plots and the coexistence of infinitely many attractors is demonstrated by phase portraits, which is confirmed by PSIM circuit simulations based on a physical circuit.

A simple test for ideal memristors

An ideal memristor is defined as a resistor with memory that, when subjected to a time-dependent current, , its resistance RM(q) depends only on the charge q that has flowed through it, so that its voltage response is . It has been argued that a clear fingerprint of these ideal memristors is a pinched hysteresis loop in their I– curves. However, a pinched I– hysteresis loop is not a definitive test of whether a resistor with memory is truly an ideal memristor because such a property is shared also by other resistors whose memory depends on additional internal state variables, other than the charge. Here, we introduce a very simple and unambiguous test that can be utilized to check experimentally if a resistor with memory is indeed an ideal memristor. Our test is based on the duality property of a capacitor-memristor circuit whereby, for any initial resistance states of the memristor and any form of the applied voltage, the final state of an ideal memristor must be identical to its initial state, if the capacitor charge finally returns to its initial value. In actual experiments, a sufficiently wide range of voltage amplitudes and initial states are enough to perform the test. The proposed test can help resolve some long-standing controversies still existing in the literature about whether an ideal memristor does actually exist or it is a purely mathematical concept.

Chain of magnetic tunnel junctions as a spintronic memristor

In the context of neuromorphic computation, spintronic memristors are investigated for their use as synaptic weights. In this paper, we propose and experimentally demonstrate a resistive synaptic device based on ten magnetic tunnel junctions (MTJs) connected in a serial configuration. Our device exhibits multiple resistance levels that support its use as a synaptic element. It allows for two operating knobs: external magnetic field and voltage pulses (Spin-Transfer Torque). Moreover, it can be operated in different ways. When varying continuously the amplitude of the voltage pulse and/or the magnetic field, eleven resistance states can be reached. In contrast, if the initial state of the chain is reset between every step, a very large number of levels are reached. Ideally, a total of 2N resistance levels could be accessible. This coincides well with the desired analog-like behavior in ideal memristors. Since this device consists of a scalable number of N MTJs, and MTJ technology is continuously optimized and improved, the proposed memristor shows promise as a scalable synapse solution for neuromorphic hardware implementations.

The case for rejecting the memristor as a fundamental circuit element

The memory resistor with the moniker memristor was a harmless postulate in 1971. Since 2008 a device that claims to be the memristor is on the prowl, seeking recognition as a fundamental circuit element, sometimes wanting electronics textbooks to be rewritten, always promising remarkable digital, analog and neuromorphic computing possibilities. A systematic discussion about the fundamental nature of the device is almost absent within the memristor community. Advocates use incomplete constitutive relationships, ignore concepts of activity/passivity and aver that nonlinearity is central to their case. Few researchers have examined these claims. Our report investigates the assertion that the memristor is a fundamental passive circuit element, from the fresh perspective that electrical engineering is the science of charge management. We demonstrate with a periodic table of fundamental elements that the 2008 memristor is not the 1971 postulate and neither of them is fundamental. The ideal memristor is an unphysical active device and any physically realizable memristor is a nonlinear composition of resistors with active hysteresis. We also show that there exists only three fundamental passive circuit elements.

Flux-Charge Description of Circuits With Non-Volatile Switching Memristor Devices

This brief aims at defining a class ${\mathcal{ D}}$ of extended memristors by using a combination of fundamental algebraic circuit elements corresponding to ideal memristors and nonlinear resistors. By massaging the characteristics of the constitutive elements, such class ${\mathcal{ D}}$ of extended memristors is able to approximate rectifying effects and asymmetric pinched hysteresis loops of real non-volatile switching memristor devices. Finally, it is shown that if the nonlinear resistors are described in terms of piecewise linear characteristics, then the flux-charge analysis method permits to analyze the dynamics of nonlinear circuits with extended memristors in ${\mathcal{ D}}$ .

Complex Dynamics in Arrays of Memristor Oscillators via the Flux–Charge Method

The key intent of the work is to analyze complex dynamics and synchronization phenomena in a 1-D array of $N$ diffusively coupled memristor-based oscillatory/chaotic circuits, i.e., each uncoupled oscillator is a 3rd–order memristor-based Chua’s circuit obtained by replacing the nonlinear resistor with an ideal flux-controlled memristor. It is shown that the state space $ \mathbb {R}^{4N}$ in the voltage–current domain of the array can be decomposed in $\infty ^{N}~3N$ -dimensional manifolds which are positively invariant for the nonlinear dynamics. Moreover, on each manifold the array obeys a different reduced-order dynamics in the flux-charge domain. These basic properties imply that two main types of bifurcations can occur, i.e., standard bifurcations on a fixed invariant manifold induced by changing the circuit parameters and bifurcations due to the variation of initial conditions and invariant manifold, but for fixed circuit parameters. The latter bifurcation phenomena are referred to as bifurcations without parameters. The reduced dynamics on invariant manifolds, and their analytic expressions, are the key tools for a comprehensive analysis of synchronization phenomena in the array of memristor-based Chua’s circuits. The main results are proved via a recently introduced technique for studying memristor-based circuits in the flux-charge domain.

About Fingerprints of Chua's Memristors

In this paper, we present an analysis of some selected fingerprints of Chua's memristors. First, the memristor is presented not as a concrete nanodevice but, more generally, as a manifestation of permanent natural principle. Next, an overview of the current classification of memristors is given in the light of various mechanisms of the memristance adjustment by internal state and by terminal variables. It is shown that only the ideal and ideal generic memristors have their stamps in Chua's table of fundamental elements and in the equivalent storeyed structure, both of them being based on the idea of predictive modeling. A brief overview on the memristor fingerprints is presented. The homothety fingerprint is discussed in detail, and all existing elements complying with this fingerprint are identified. Finally, it is argued that the most natural fingerprint of the original memristors introduced by L. Chua in 1971 should be sought in the charge-flux coordinates of their constitutive relations.

Memristor-Based Canonical Chua’s Circuit: Extreme Multistability in Voltage-Current Domain and Its Controllability in Flux-Charge Domain

This paper investigates extreme multistability and its controllability for an ideal voltage-controlled memristor emulator-based canonical Chua’s circuit. With the voltage-current model, the initial condition-dependent extreme multistability is explored through analyzing the stability distribution of line equilibrium point and then the coexisting infinitely many attractors are numerically uncovered in such a memristive circuit by the attraction basin and phase portraits. Furthermore, based on the accurate constitutive relation of the memristor emulator, a set of incremental flux-charge describing equations for the memristor-based canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established. As a result, the initial condition-dependent dynamics in the voltage-current domain is converted into the system parameter-associated dynamics in the flux-charge domain, which is confirmed by numerical simulations and circuit simulations. Therefore, a controllable strategy for extreme multistability can be expediently implemented, which is greatly significant for seeking chaos-based engineering applications of multistable memristive circuits.

Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria

Memristor-based nonlinear dynamical system easily presents the initial condition-dependent dynamical phenomenon of extreme multistability, i.e., coexisting infinitely many attractors, which has been received much attention in recent years. By introducing an ideal and active flux-controlled memristor into an existing hypogenetic chaotic jerk system, an interesting memristor-based chaotic system with hypogenetic jerk equation and circuit forms is proposed. The most striking feature is that this system has four line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. Stability of these line equilibria are analyzed, and coexisting infinitely many attractors’ behaviors with the variations of the initial conditions are investigated by bifurcation diagrams, Lyapunov exponent spectra, attraction basins, and phased portraits, upon which the forming mechanism of extreme multistablity in the memristor-based hypogenetic jerk system is explored. Specially, unusual transition behavior of long term transient period with steady chaos, completely different from the phenomenon of transient chaos, can be also found for some initial conditions. Moreover, a hardware circuit is design and fabricated and its experimental results effectively verify the truth of extreme multistablity.

Sunpatiens compact hot coral: memristors in flowers.

Leon Chua postulated the theory of a memristor - a resistor with memory - in 1971, and the first solid-state memristor was built in 2008. Memristors exist in vivo as components of plasma membranes in plants, fruits, roots and seeds. A memristor is a nonlinear element; its current-voltage characteristic is similar to that of a Lissajous pattern. Here, we found memristors in flowers. Electrostimulation by bipolar periodic sinusoidal or triangular waves of an androecium, a spur, petals and a pedicel in Sunpatiens flowers induces hysteresis loops with a pinched point at low frequencies between 0.1mHz and 1mHz. At high frequencies, the pinched hysteresis loop transforms to a non-pinched hysteresis loop instead of a single line I=U/R for ideal memristors because the amplitude of electrical current depends on capacitance of a flower's tissue and electrodes, frequency and direction of scanning. The discovery of memristors in Sunpatiens (Impatiens spp.) creates a new direction in the modelling and understanding of electrophysiological phenomena in flowers.

Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system.

In this paper, two kinds of novel ideal active flux-controlled smooth multi-piecewise quadratic nonlinearity memristors with multi-piecewise continuous memductance function are presented. The pinched hysteresis loop characteristics of the two memristor models are verified by building a memristor emulator circuit. Using the two memristor models establish a new memristive multi-scroll Chua's circuit, which can generate 2N-scroll and 2N+1-scroll chaotic attractors without any other ordinary nonlinear function. Furthermore, coexisting multi-scroll chaotic attractors are found in the proposed memristive multi-scroll Chua's circuit. Phase portraits, Lyapunov exponents, bifurcation diagrams, and equilibrium point analysis have been used to research the basic dynamics of the memristive multi-scroll Chua's circuit. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design.

Electrical Nonlinearity Emulation Technique for Current-Controlled Memristive Devices

Emerging memristor technology is recently drawing widespread attention due to its potential for diverse applications. Due to the lack of real solid-state memristive devices, there have been many initiatives to develop memristor emulators to study their behavior and applications. One of the most widely used ideal memristor models developed by the HP Lab does not fit the anticipated nonlinear behaviors of a real memristor. In this paper, we propose the concept and the design of a practical memristor emulator, which can be used to mimic the behavior of the well-known current controlled memristor models like- Simmons tunneling barrier model and the ThrEshold adaptive memristor model . Our proposed emulator model mimics the behavior of the electrical nonlinearity of the fabricated memristor. Prior emulators can only emulate the linear electrical behavior. In addition to the mathematical modeling and analysis of the proposed emulator, we provide SPICE simulation and experimental results. Furthermore, the proposed emulator has been used to verify some applications like Wien Oscillators. Finally, a brief comparison with the previously published emulators is presented to highlight the advantages of the proposed design.

Hysteresis versus PSM of ideal memristors, memcapacitors, and meminductors

The classical fingerprint of the ideal memristor with differentiable constitutive relation has recently been extended with a piece of knowledge that the area of the pinched hysteresis loop of a memristor driven by sinusoidal signal with fixed amplitude and increasing frequency disappears with an integer power of the frequency. This fingerprint is extended to the memcapacitor and meminductor, and it is also shown that the way the hysteresis disappears can be determined directly from the behaviour of the parameter versus state map (PSM) of the element in the neighbourhood of the initial operating point.