A S-Type Bistable Locally-Active Memristor and Its Application in Oscillator Circuit


 In this paper, a S-type memristor with tangent nonlinearity is proposed. The introduced memristor can generate two kinds of stable pinched hysteresis loops with initial conditions from two flanks of the initial critical point. The power-off plot verifies that the memristor is nonvolatile, and the DC V-I plot shows that the memristor is locally active with the locally-active region symmetrical about the origin. The equivalent circuit of the memristor, derived by small-signal analysis method, is used to study the dynamics near the operating point in the locally-active region. Owing to the bistable and locally-active properties and S-type DC V-I curve, this memristor is called S-type BLAM for short. Then, a new Wien-bridge oscillator circuit is designed by substituting one of its resistances with S-type BLAM. It find that the circuit system can produce chaotic oscillation and complex dynamic behavior, which is further confirmed by analog circuit experiment.


Introduction
In 1971, Chua proposed the fourth basic circuit component called memristor in view of the symmetric logical relation of circuit theory [1], and the physical fabrication by Hewlett-Packard laboratory made the theoretical assumption of memristor come true [2]. Due to the special properties of nanometer size, low power consumption, inherent nonlinearity and nonvolatile, memristor possesses the great potentials of inducing new dynamical mode of electronic oscillation, enhancing the security of chaotic communication, increasing the reliability of chaotic encryption and improving the efficiency of neural network in searching optimal solution [3][4][5][6][7][8][9]. And it can be predicted that memristor will play a key role in the development of next-generation memory system with ultra-low energy consumption and high density memory [10]. Nevertheless, the characteristics of memristor need to be further explored and revealed.
The memristor with multistable characteristics can produce different types of pinched hysteresis loops under different initial conditions. In 2016, Ascoli proposed a bistable memristor endowed with a stable pinched hysteresis loop-pair, stimulated by DC as well as AC periodic signal [11]. Mannan found the coexisting pinched hysteresis loops in Chua corsage memristor [12]. In Ref [13], Chang proposed a bistable memristor model containing cubic term and found that its dynamics was governed by cubic term. Wang reported a three stable pinched hysteresis loops memristor by adding a polynomial characteristic function into the original Chua corsage memristor [14]. Wang also reported a multi-stable memristor by introducing periodic function and multiple equilibria [15]. It's inevitable that a multistable device would give rise to the multistability and complexity in a dynamical system. Therefore, it is of great significance to study the multistability of memristor.
Locally-active memristor (LAM) is the memristor with negative memristance under a certain voltage crossed or the memristor with negative memductance under a certain current crossed [16][17][18]. It is uncovered that local activity is essential for nonlinear system to keep oscillation and amplify weak fluctuation signal. Therefore, local activity is considered to be the origin of all complexities in dynamical system [19][20][21]. In general, locally-active memristor can be divided into two categories: the first one is not passive for which all points in the DC V-I plot lie in each quadrant, the second one is passive but locally-active whose points in the DC V-I plot only lie in the first and third quadrant with VI ≥ 0. For example, Chua proposed a passive but locally-active memristor based on piecewise linear function [22], and it was connected with an inductor and a battery to generate oscillating behavior with particular initial conditions and DC bias [12]. In addition, it has been found that some particular memristor, such as vanadium dioxide (VO2) and niobium oxide (NbOx) devices, are passive but locally-active memristors [23,24]. Pickett revealed that the NbOx devices have current-controlled negative differential resistance [25]. And the mathematical model for NbOx LAM was presented based on the Chua's unfolding theorem and parameter optimization method [26]. Unlike nonvolatile memristor, most of the nanoscale LAMs have the characteristics of volatile resistance switching. Recently, a new type of nanoscale devices of locally-active memristor, called S-Type LAM, was proposed [27][28][29]. And a S-Type LAM model with volatile resistance was then proposed [30]. S-Type LAM is a nonlinear local active device, which is simpler in concept than the passive memristor with negative resistance, thus it can form an oscillation circuit without pure negative resistor. However, the S-Type LAMs are difficult to commercially access due to the technology and cost of manufacturing nano-scale electronic component [31]. Therefore, in order to enrich the theoretical knowledge of S-Type LAM and explore its practical application in various fields, it is necessary to further study the emulator and simulation model in the area of negative differential resistance (NDR).
Memristor has been widely used in chaotic oscillator for the unique characteristics of storage and inherent nonlinear. For example, Sah proposed an oscillator made with only a memristor and a battery, which is distinct from the traditional electronic oscillator including at least two energy-storage elements and a locally-active nonlinearity [32]. Wang modeled the neural network by utilizing flux-controlled memristor to describe the influence of electromagnetic radiation on neuron, and found that the simple neural network can induce infinite number of coexisting hidden attractors [33]. Zeng introduced an inductor-free two-memristor-based chaotic circuit, which is developed from a current feedback op amp-based sinusoidal oscillator. The proposed circuit has three line equilibria and can perform the dynamics of extreme multistability, amplitude death and transient transition behavior [34]. As a nonlinear locally-active device, S-type LAM is conceptually simple for building oscillating circuit without pure negative resistor, and the local active region renders the oscillating system capable to amplify extermely small fluctuation in energy. Therefore, S-type LAM is preferred over other memristor in the design of chaotic oscillator [35][36][37]. In Ref [30], Wang designed a S-type LAM-based chaotic oscillator by using a resistor, a capacitor and an inductor. However, it is not clear whether S-type LAM could be used to other oscillating circuits such as Wien-bridge circuit. Therefore, it would be interesting and potentially valuable if S-type LAM could be successfully applied.
In this paper, we introduce a S-type bistable locally-active memristor with tangent nonlinearity and study its associated memristor oscillator circuit. The main contribution of this work is summarized as follows: (1) The memristor can generate two kinds of stable pinched hysteresis loops induced from two flanks of the initial critical point of the initial condition. (2) The memristor is locally active and the locally-active region is symmetrical about the origin. (3) The memristor has a S-type DC V-I plot and a nonvolatile power-off plot, which is different to the nanoscale LAMs and the S-type LAM in Ref [30] with volatile resistance. The rest of this paper is organized as below: In Section 2, the mathematical model of S-type bistable locally-active memristor is presented and the voltage-current characteristics are analyzed by coexisting Pinched hysteresis loops, power-off plot and DC V-I plot. In Section 3, the small-signal analysis method is used to study the equivalent circuit near the operating point in the locally-active region and the frequency response of the impedance function is also revealed. Then, in Section 4, we design a new Wien-bridge oscillator circuit by using the proposed S-type BLAM and investigate its complex dynamics. In Section 5, an analog circuit is designed to experimentally confirm the proposed oscillator circuit. Finally, a brief conclusion including some concluding remarks is drawn in Section 6.

Model of S-type BLAM
Memristor can be categorized into ideal memristor, generic memristor and extended memristor, according to the mathematical definition. The ideal memristor is the simplest model and the extended memristor is developed on the ideal one. The generic memristor is mathematically defined as where u(t), y(t) and x(t) are the input, output and internal state variable of the memristor; f(•) and g(•) are the functions that can be customized and related to the internal state variable x(t).
The memristor is the voltage-controlled when the input is voltage and the output is current, while the memristor is the current-controlled if the input is current and the output is voltage. Now, a current-controlled generic memristor model is proposed, as below where i and v are the input current and output voltage; a0, a1, a2, b1, b2 and c are the model parameters, which are set to a0= −1, a1= −1, a2=3, b1= −1, b2=3, c=0.5.
Next, we analyze the voltage-current characteristics of the memristor on the basis of coexisting pinched hysteresis loops, power-off plot and DC V-I plot, and consequently prove that it is bistable, nonvolatile and locally active.

Coexisting pinched hysteresis loops
When driven by a zero-mean input source with a certain amplitude and frequency, the input-output curve of memristor will pass through the origin in the voltage-current plane and exhibit a pinched hysteresis loop. A memristor with two coexisting pinched hysteresis loops is called bistable memristor. When two unequal initial values are located on both sides of the critical initial value XC, the dynamic behavior of the bistable memristor appears to be two distinctly different and stable pinched hysteresis loops; when two unequal initial values are located on one side of the critical initial value, the dynamic behavior of the bistable memristor displays one monostable pinched hysteresis loop.

The sinusoidal voltage signal v(t)=Asin(2πFt) with the amplitude A=2V and frequency
F=0.1Hz is chosen as the driving source. When the initial values of X0=0.526 and X0=0.527 are located on both sides of the critical initial value XC=0.52677, two completely different stable pinched hysteresis loops are drawn in Fig.1 (a), where the red curve stands for X0=0.526 and blue curve stands for X0=0.527. However, when the two initial values X0=0.527 and X0=0.528 are both greater than the critical initial value, the two pinched hysteresis loops are completely coincident, as shown in Fig.1 (b). And when the two initial values X0=0.525 and X0=0.526 are both smaller than the critical initial value, the two pinched hysteresis loops are also completely identical, as shown in Fig.1 (c).
The coexisting pinched hysteresis loops depend not only on the initial value of memristor but also on the amplitude and frequency of the driving signal. When the initial values and frequency of the sinusoidal voltage signal are fixed as X0= −1, X0=1 and F=0.1Hz, the lower and upper limits of the critical amplitude are determined as AC1=1.305 and AC2=2.049227. Fig.1 (d) shows two completely coincident pinched hysteresis loops with amplitude A=1V (A<AC1). While Fig.1 (e) displays the coexisting hysteresis loop with amplitude A=1.9V (AC1<A<AC2). When the amplitude increases to greater than the upper limit AC2, such as A=2.1V, the two hysteresis loops coincide again, as shown in Fig.1 The critical frequency value FC is approximately equal to 0.0826 when X0= −1, X0=1 and A=2V. Fig.1

Power-off plot and nonvolatility
When signal source is turned off, the non-volatile memristor can remember its most recent state. According to Chua's memristor theorem, the memristor is non-volatile if the power-off plot (POP) intersects the 0 dx dt  axis at least twice with a negative slope. Therefore, the POP technology can not only judge the nonvolatility of memristor, but also reveal the changing process of the scalar state variable.
The POP of S-type BLAM can be obtained by setting i=0 in g(•) of Eq. (2), as below ) tanh( It's known from the POP in Fig.2

DC V-I plot and local activity
As a powerful visualization tool to understand the intrinsic property of memristor, the DC V-I plot is a smooth curve composed of enough test points. A group of DC voltage v=Vk (k=1,2,3,...,n) can be obtained by adding a group of continuous DC current i=Ik (k=1,2,3,...,n) to S-type BLAM. Then, the state variable x=Xk (k=1,2,3,...,n) is an equilibrium state satisfying It's derived the functional relationship between the applied DC current I 2 and the equilibrium state X, expressed as And the function between I and X can be deduced as We obtain the DC voltage V when It's further deduced the functional relationship between the output DC voltage V and the equilibrium state X by substituting equation (7) into equation (8), as denoted by Similarly, the green and blue curve segments in the X-I diagram are related to the green and blue curve segments in the POP diagram respectively. On the one hand, the slopes of the red and blue curve segments in the X-I diagram are both negative since they are caused by the stable equilibrium state, so the entire X-I curve is locally active. On the other hand, the curve segment in the lower half plane of X-I 2 diagram causes the discontinuity of the X-I and the X-V diagram. Moreover, each color segment in the X-V diagram has the same source with the corresponding color segment in the X-I diagram.
It can be seen that the DC V-I plot has a continuous S-shaped behavior. And the slope of the green curve segment is negative, indicating that the introduced memristor is locally active.

Small-signal impedance function
We can analyze the system response in NDR region by applying a small signal to the DC operating point of the nonlinear dynamical system. Also we can derive the small-signal equivalent circuit of the memristor by using the small-signal analysis method. Thus, a small-signal input current ∆i is applied to the DC operating point (V, I) of the memristor, and it gives rise to the responses v and x Eq. (2) can be expanded at equilibrium point Q(X, I) by Taylor expansion. Because the input signal working at the Q point is small enough, the higher order term can be ignored.
Accordingly, the resulting equations are given below 11 12 21 22 ( , ) where g(X, I)=0 and The increments of voltage and state variable derivative can be expressed as 11 12 21 22 v a x a i x a x a i And we obtain from (12) Re ( , )

Small-signal equivalent circuit
The frequency responses of S-type BLAM with I=0.8mA and −100rad/s ≤ ω ≤100rad/s are drawn in Fig.4. It can be found that the real part ReZ(jω,Q) always remains a negative value in the entire frequency range. And the imaginary part ImZ(jω,Q) is negative when ω<0 while it remains positive for ω>0. What's more, the imaginary part ImZ(jω,Q) first increases and then decreases as the frequency increases from ω=0.
The frequency response ImZ(jω,Q) with ω belonging to [−100rad/s, 100rad/s] is depicted in Fig.5 (a), under some positive DC currents I within the local active area (I=0mA, 0.2mA, 0.4mA, 0.6mA, 0.7mA and 0.8165mA). It find that the imaginary parts of the impedance functions for all the operating points are located in the first and third quadrants.
When ω ≥ 0rad/s, the value of ImZ(jω,Q) is within the range of [0,0.4) and decreases with the decrease of the DC input current I. When the opposite DC input current I is selected as 0mA, −0.2mA, −0.4mA, −0.6mA, −0.7mA and −0.8165mA, the frequency response ImZ(jω,Q) is completely consistent with that of positive DC current I. This is probably induced by the symmetry of the local active region relative to the origin.

Fig.4. Frequency responses of S-type BLAM with I=0.8mA
Let ω vary in the range of [−100rad/s, 100rad/s], the frequency response ReZ(jω,Q) with some positive DC currents I in the local active area is shown in Fig.5 (b). It's observed from  Based on the above analysis, the small-signal equivalent circuit of S-type BLAM at the operating point Q is depicted in Fig.6, which can be regarded as the series connection of a negative resistance and an inductance And the equivalent resistance R(ω) and inductance L(ω) are given by 2

Circuit model
Wien-bridge circuit has the advantages of stable oscillation, high quality waveform and adjustable oscillation frequency [38,39]. Thus, an active oscillator circuit, manipulated by substituting one resistance of Wien-bridge circuit with S-type BLAM, is built in Fig.7. When taking the internal variable x of memristor, voltage VC1 of capacitor C1, voltage VC2 of capacitor C2 and current iL of inductor L as the state variables, the mathematical expression of S-type BLAM-based Wien-bridge oscillator circuit is described by For the convenience of analysis, we introduce the following dimensionless variables and normalized circuit parameters Thus, the corresponding dimensionless circuit equation is deduced to be For system (20), d, e, f, g are the control parameters; a0, a1, a2, b1, b2, c are the internal parameters of memristor. Therefore, only some of the control parameters will be considered for the bifurcation analysis of system (20). As an inherent property of nonlinear dynamical system, symmetry may help to explain the appearance of symmetrical attractors with different shapes. We can easily notice that system (20) is invariant under the coordinate transformation (x, y, z, w)→(x, −y, −z, −w), which indicates the symmetry about x-coordinate axis.

Dissipation and existence of attractor
It is well known that chaotic flow can be divided into either conservative or dissipative one [40,41]. For the conservative chaotic system, the phase space trajectory occupies an unchanged volume and there is no state space attribute, thus its divergence is zero. However, the phase orbit of dissipative system will shrink to a bounded subset with a zero-measured volume, which leads to the emergence of strange attractor and negative divergence [42,43].
Accordingly, we can obtain the preliminary information related to the existence of attractive sets in the introduced S-type BLAM by calculating the volume shrinkage ∆v. The divergence curve on the system orbit x is drawn in Fig.8. It find that the time period for ∆v>0 is sufficiently small before the point M (0.49, 0). And it can be considered that the divergence is always negative. Therefore, the system orbit in the phase space will converge to a subset of measure zero volume with an exponential rate, and there exists chaotic attractor in the introduced S-type BLAM.

Dynamics evolution with control parameter
The dynamics evolution influenced by the control parameters is intuitively studied, as a rule, executed with the detecting technologies of bifurcation diagram and Lyapunov exponent.     Fig. 13 (b) correspond to the red, yellow and blue regions in Fig.12 (b). It is obvious that the red region corresponds to the double-scroll chaotic attractor, the yellow region corresponds to stable point attractor, and the blue region corresponds to the single-scroll chaotic attractor. The numerical results show that the disconnected coexisting attractors emerge in the S-type BLAM-based Wien-bridge system. Therefore, the dynamical behavior not only depends on the memristor initial condition x0 but also on the other initial conditions z0, y0 and w0.

Transient dynamics
The transient dynamics will emerge when there is a non-attracting chaotic saddle in the phase space, for which the orbit is always characterized by chaotic behavior before another motion mode appears.

Implementation of S-type BLAM-based circuit
Circuit implementation is of great significance for the practical application of chaotic system. In addition, the results of theoretical analysis and numerical simulation also need to be further verified by circuit observation. In this section, an analog electronic circuit, for physically realizing system (20) in different cases, is constructed based on the improved module-based technique. The circuit schematic diagram of the S-type BLAM-based Wien-bridge system is designed as Fig.15, and the circuit diagrams of hyperbolic tangent functions tanh(•) and −tanh(•) are depicted in Fig.16. In the circuit design, the chip model of multiplication, operational amplifier and bipolar junction transistor are selected as AD633JN, TL082 and 2N1711 to achieve low-cost execution. In addition, different resistors and ceramic capacitors are used. The main circuit in Fig.15 consists of four integrators, three inverters, one tanh module and one inverting tanh module. And in Fig.15, A and B are the input and output ports of the hyperbolic tangent function module, C and D are the input and output ports of the inverting hyperbolic tangent function module.
Based on the topology of Fig.15 and circuit theory, we establish the circuit state equation For comparative analysis, the parameter condition P mentioned above is considered excepting for f=1.75, 1.753, 1.78 to obtain period-2 attractor, single scroll chaotic attractor and double scroll chaotic attractor, as shown by the numerical results in the left part of Fig.17.
When f equals to 1.75, 1.753 and 1.78, the resistances R11 and R13 are calculated to be 5.7143kΩ, 5.7045kΩ and 5.618kΩ, the resistance R12 is calculated to be 6.0606kΩ, 6.0496kΩ, and 5.952kΩ, and the resistance R14 is calculated to be 1.9048kΩ, 1.9015kΩ, and 1.873kΩ.
The corresponding phase diagrams are experimentally observed in the right part of Fig.17, which agree well with the numerical simulation.

Conclusion
This paper presented a S-type locally-active memristor and explored its application in oscillator circuit. The introduced S-type memristor possesses a symmetric locally-active domain and two different bistable pinched hysteresis loops. Especially, the bistable pinched hysteresis loops are not only affected by the initial value, but also by the amplitude and frequency of the applied sinusoidal signal. The DC V-I plot and POP plot have been carried out to verify the locally active and nonvolatile characteristics of the memristor. Compared with the reported S-Type memristor, the obvious feature of S-type BLAM is the nonvolatility and the origin symmetry of the local active region. Moreover, a new Wien-bridge oscillator circuit is designed based on S-type BLAM. It find that the circuit system can produce chaotic oscillation and complex dynamic behavior, which is further confirmed by analog circuit experiment.

Conflict of Interest:
The authors declare that they have no conflict of interest.

Data Availability:
The data used to support the findings of this study are included within the article.