Pinched Hysteresis Loop with Nonlinear Electronic Components: From Memristor to Hysteristor Concepts

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Introduction
A memristor (with pinched hysteresis loop characteristic) is an electrical element conjectured to complete the lumped circuit theory in 1971 [1].It is a two-terminal passive nonlinear resistance element that exhibits the well-known pinched hysteresis loop at the origin of the voltage-current plane when it is applied across it, any bipolar-periodic zeromean excitatory voltage or current of any value.In 2008, Hewlett-Packard Co. reported the frst implemented memristor built with titanium oxide.Controversy arose during that period and continued till date centered around whether a memristor could be considered a fundamental element or not.Despite the controversy surrounding the technological realization, research into the properties of the pinched hysteresis loop continued to grow.Irrespective of whether it is implemented by emulating its behavior through circuitry composed of other active or passive components, the research and articles continue generating and sustaining optimistic expectations in the scientifc community about its usage and advantages.Te popularity in research into pinched hysteresis loop is motivated by its promising potential for building novel integrated circuits and computing systems as for instance proposed in [2][3][4][5].
Te main advantage of systems with pinched hysteresis loop lies in ofering the capability for memory and timevarying processing information through nonlinear transformations in a unique passive system.Te pinched hysteresis loop characteristic has started to receive formal treatment as research into its progress [6][7][8][9] and nowadays is defned as pinched hysteresis attractor.With the rising need for better understanding of pinched hysteresis attractor, researchers implement emulators through diodes and other passive elements.Such emulators enable the study (theoretically and numerically) of possible attractors and ways of implementing nonlinear transformations for novel computing paradigms.Te frst pinched hysteresis loop emulator was proposed in [10], based on a diode-bridge with a parallel R − C flter as load.Other studies, such as [11][12][13], used pinched hysteresis loop emulators as part of other circuits for in-depth study of their bifurcations and chaotic behaviors.
In this work, we found a system that also exhibits pinched hysteresis loop, which, consequently following the trend in the literature, can be called memristors; however, their dynamics do not match the equation of memristor that is widely spread and used in the literature.To accommodate this gap in defnitions and provide a comprehensive circuit taxonomy, we proposed a new defnition: the currentcontrolled or voltage-controlled hysteristor of order n.
In Section 2, we defne the memristor (with pinched hysteresis loop characteristic) according to terminology used in the literature.In Section 3, we touch upon the misconceptions and limitations associated with the defnition of the memristor and consequently introduce the motivation for the core results and aim of this article in Section 4, showing the concept of the hysteristor element of order n.It provides a more comprehensive taxonomy and completeness of the theory and defnition.Section 5 shows simulated results of a hysteristor emulator of order 1, and lastly, fnal conclusions are given in Section 6.

Definitions according to the State of the Art
According to [14], memristors exhibit three characteristics for any bipolar periodic signal excitation, including the following: (i) they show pinched hysteresis loop in the voltage-current plane, (ii) the area of the hysteresis loop decreases and shrinks to a single-valued V-I function as the signal excitation frequency tends to infnity, and (iii) the following equations apply: (a) For a current-controlled memristive time-invariant system, (b) For a voltage-controlled memristive time-invariant system, where i m is the current across the memristor and V m is the voltage across the terminals, R(x, i m ) is bounded, and f(x, i m ) is the equation of state which must be also bounded to guarantee the existence of a solution x(t).Te expression ) guarantees the zero-crossings at the origin of the V − i plane for all amplitudes, frequencies, and initial states.Te area of the lobes, shapes, and orientation of the hysteresis loop evolve with frequency.In particular, it is reported that the areas of lobes decrease (it shrinks or collapses) monotonically as the excitation i m (or V m ) frequency increases.A concise concept was presented in [14].

Pinched Loops and Hysteristors
In our study of memristors and pinched hysteresis loop, we faced the following dilemma: on the one hand, (a) as seen from ( 1), the expression V m � R(x, i m )i m shows the zerocrossing at the origin of the V − i plane for all amplitudes which defnes memristive dynamics.On other hand, (b) there is a school of thought or scientifc tradition, supported by a lot of literature that generalizes the concept ". ..Any 2terminal device exhibiting a pinched hysteresis loop which always passes through the origin in the voltage-current plane when driven by any periodic input current source, or voltage source, with zero DC component is called a memristor. ...." (text from [14]).It is noteworthy this concept was found in all literature we surveyed.Terefore, the following question arises: could it be possible to have a 2-terminal passive device or system that exhibits pinched hysteresis loop (situation where unequivocally it sufces to defne it as memristor in accordance with the literature), but without being governed by expression equations ( 1) and ( 2)? Te surprising answer is yes.In fact, while we studied the properties of the circuit proposed in Figure 1, we had to construct a dedicated classifcation and framework to give an interpretation to the results obtained: it exhibits pinched hysteresis loop and cannot be classifed as a memristor emulator in accordance with (1), neither as inverse-memristor as proposed in [15].Tis ambiguity formed our motivation for this work, conjecturing the concept of hysteristor of order n.It is conceptualized as a two-terminal passive nonlinear dynamical system or device that exhibits hysteresis loops with zerocrossing at the origin in their current-voltage characteristics for all applied current or voltage amplitudes, frequencies, and initial states, but for which the system of equations ( 1) or (2) does not hold, but which are instead subject to the following system of equations: (a) For a current-controlled hysteristor time-variant system, (b) For a voltage-controlled hysteristor time-variant system, where i hz is the current across the hysteristor device and V hz is the voltage across the terminals, H z (Dx, i hz , t) is bounded, and f(.) is the system of  (hysteristor of order n).In absence of the variable t (time), the hysteristor is said to be autonomous or time-invariant (readers can see in advance (18) as example of hysteristor of order n � 1).
Te hysteristor model encapsulates and generalizes the concept of a memristor.Memristors belong to a subset of hysteristor for which their system of equations (equations ( 3) and ( 4)) can be expressed in the form of equations ( 1) and (2).

Current-Controlled Hysteristor Emulator
Te frst-ever circuit realization of a current-controlled hysteristor (of order n � 1) passive emulator is established in Figure 1.It uses two diodes, two resistances, and one inductor instead of diode bridge and inductors as seen in [13,16] to emulate a memristor behavior.
We construct the equations of the proposed hysteristor by beginning with the diode equation without considering intrinsic parasitic and high-frequency efects that produce unwanted dynamic efects (interested readers can see in advance the fnal obtained equation ( 17) of the hysteristor); then: where 2α � 1/nV T , I s denotes the reverse saturation current, n is the emission coefcient, and V T is the thermal voltage.
According to the voltage drop, Te hysteristor current i hz corresponds to i hz � i D 1 − i D 2 .Using equations ( 5) and ( 6), we achieve the following relation: Using equation (7), it becomes convenient to express However, from equation ( 6), Ten, the V hz − i hz   relation is provided.
Next, we turn to the state equation.By taking we hence achieve from equation ( 12) the following: To complete the equations, we have to calculate V D 1 .It is quite straightforward.
Terefore, from equation ( 14), Finally, this hysteristor dynamic can be written as follows: Tat can be expressed as In accordance with equation ( 1), i L represents the inner state variable; however, it should be noted that the form V hz � R(V c , i hz )i hz is not accomplished (i.e. it does not contain i hz proportionality), neither the state equation.Instead, equation (3) contains proportionality with n � 1.It is surprising that the proposed current-controlled hysteristor has pinched loop hysteresis (zero-crossing hysteresis loop) for any bipolar periodic signal excitation with zero DC component, without being a memristor as defned by equation ( 1), neither as in [15].
It sufces to prove that the well-accepted and widely used concept in the literature, for example, in [14] is a conjecture that leads to incorrect conclusions.We have demonstrated that there could be systems with pinched loop hysteresis that are not memristors in the sense of the equation (1) for any value of bipolar-periodic zero-mean excitatory currents.We defne them as hysteristor and next we present some representative simulations to validate the theory.

Validation by Simulation
Te following parameters were used for the simulation of the current-controlled hysteristor of order n � 1 circuit emulator proposed in Figure 1: R � 200 Ω, L � 100 mH.Te assigned diode was 1N4148 with PSpice-model card available from [17].A state initial condition i L � 0 was selected and the simulator used was LTspice [18].For such diode models, the magnitudes of the circuit parameters as well as the values of the input signal amplitudes and frequencies are maintained similar so as to compare the lobe shapes obtained in other works such as [10,13,16,19] as well as for ease of illustration (interested readers can verify that the simulated loops exhibit similar trajectories as shown in the cited literature).Te current-voltage characteristics obtained from simulations are shown in Figure 2. Te loci in the V-I plane have hysteresis loops pinched at zero and the hysteresis loop shrinks to a single-valued function as the frequency inand the shape of the hysteristor depends on the circuit parameters.Te transient behavior while reaching steady state is due to the inductor state.It quickly achieves its steady-state internal magnetic feld.

Conclusion
We found a passive system with pinched hysteresis loop, which consequently could be named memristors; however, their dynamics do not match the equations of memristors widely spread and used in the literature.To accommodate this gap in defnitions and provide a comprehensive circuit taxonomy, we proposed a new defnition: the currentcontrolled or voltage-controlled hysteristor of order n.In this paper, the implemented current-controlled hysteristor of order n � 1 has a simple topology and comprises only one inductor, two diodes, and two resistances.Agreement between the theoretical proof and simulations results validates the existence of this kind of systems we refer to as hysteristors; memristors can be classifed as a subset of hysteristors.