Parallel-Type Asymmetric Memristive Diode-Bridge Emulator and Its Induced Asymmetric Attractor

Symmetry brings beauty, while asymmetry is the general law of nature. This paper reports a novel parallel-type asymmetric memristive diode-bridge (AMDB) emulator, which is implemented by an unbalance diode-bridge linked with a RC filter. Following the voltage constraints of the unbalance diode-bridge, the mathematical model of the parallel-type AMDB emulator is established. Thereafter, the asymmetric property of the hysteresis loops is demonstrated by the numerical simulations and confirmed by the hardware experiments. Furthermore, by importing the parallel-type AMDB emulator into the classical Chua’s circuit, a novel memristive Chua’s circuit is proposed, so that the asymmetric double-scroll chaotic attractor, asymmetric coexisting single-scroll chaotic attractors, and asymmetric coexisting limit cycles can be revealed herein. The parallel-type AMDB emulator enriches the types of memristor emulators and it can mimic the asymmetric property of the physical memristor device.


I. INTRODUCTION
Memristor, known as the fourth circuit element, is a dazzling star in electronic circuit fundamentals [1], [2]. Its nano-scale property makes the device occupy exceedingly small layout area in IC application [3], the non-volatile property makes it be well received in neuromorphic circuit and informatics processing [4]- [7], and the nonlinearity property makes it contribute to the generation of abundant and complex dynamical behaviors in memristive chaotic circuits [8]- [12]. Unfortunately, for the difficulty of its fabrication, the physical memristor device is inconvenient to acquire through regular purchasing channels. For the convenience of scientific research, numerous mathematical models [13]- [15], PSpice models [16] and analog circuit emulators [17]- [20] were reported for equivalently implementing the characteristics of various memristors in the past few years. Among those, the memristive diode-bridge emulator [8], [10] is greatly The associate editor coordinating the review of this manuscript and approving it for publication was Norbert Herencsar . welcomed because of the simple structure, no grounded limitation, easy circuit access and so on.
In general, the diode-bridge circuit has a symmetrical structure with four diode bridge arms. Thus, the pre-existing memristive diode-bridge emulators just exhibit the symmetric hysteresis loops pinched at the origin [8], [20], [21]. However, the physical memristor device usually possesses the asymmetric hysteresis loops [2], [22]. The influence of symmetricbreaking phenomenon on the dynamical systems has attracted much attention [23]- [27]. Recently, Kengne et al took the antiparallel semiconductor diodes pair as an asymmetric nonlinear emulator, upon which various asymmetry-induced dynamical behaviors were revealed in several asymmetric chaotic circuits [28]- [31]. Inspired by this, a series-type asymmetric memristive diode-bridge (AMDB) emulator was newly proposed by inserting an extra diode into the first bridge arm of diode-bridge circuit [32]. Based on the seriestype AMDB emulator, two asymmetric memristor-based jerk circuits were constructed. Since there is only one zero equilibrium, these two asymmetric memristor-based jerk circuits only generated the single-scroll attractors with the relatively simple dynamics, resulting in that the asymmetry of attractor topologies could not be presented perfectly. Meanwhile, it can be found that the asymmetry property can be enhanced when inserting more diodes into two symmetric bridge arms in series. However, the forward voltage required for the bridge arm will be enlarged. In this way, a much larger input voltage should be applied in its application circuit, which extremely limits the circuit applications of series-type AMDB emulator.
To solve this issue, a novel parallel-type AMDB emulator is proposed in this paper. The rest of the paper is arranged as follows. In Section II, the circuit structure and mathematical model of the parallel-type AMDB emulator is proposed, and the voltage constrains of the unbalance diode-bridge are verified. Afterwards, the volt-ampere curve of the proposed parallel-type AMDB emulator is demonstrated by the numerical simulations and confirmed by the hardware experiments. In Section III, a parallel-type AMDB emulator-based Chua's circuit is established and some asymmetric chaotic attractors and limit cycles are perfectly embodied by the numerical simulations and the hardware experiments. The end is some discussions and conclusions.

II. CIRCUIT STRUCTURE AND MATHEMATICAL MODEL
The memristive diode-bridge emulators, consisting of a symmetric diode-bridge and a RC filter, can exhibit the symmetric hysteresis loops pinched at the origin [21]. To mimic the asymmetric hysteresis loops appeared in physical memristor devices [2], [22], two diode parallel arrays are introduced into the first and third branches to obtain asymmetric property, as shown in Fig. 1. Each diode parallel array has m diodes connected in parallel, which is marked as DPA in Fig. 1. The parameter m is a positive integer. Thus, the branches B 1 and B 3 are different from the branches B 2 and B 4 , rusting in the construction of unbalance diode-bridge. In this paper, such an analog discrete components-based memristor emulator is called as a parallel-type AMDB emulator.

A. MATHEMATICAL MODEL
Denote i Dk , i Dl , and i as the currents flowing through the diode D k , diode parallel array DPA l , and parallel-type AMDB emulator, respectively. And denote v Dk , v Dl , and v as the voltages across D k , DPA l , and AMDB emulator, respectively. All the diodes have identical model parameters I S (reverse saturation current), n (emission coefficient), and V T (thermal voltage). Then, the constitutive relations of the diode D k and diode parallel array DPA l can be unified as and respectively, where k = 11, . . . , 1m, 2, 31, . . . , 3m, and 4, l = 1, 3, ξ = 1/(2nV T ). There exist two conditional identities for the voltages across two pairs of the parallel bridge arms (the branches The two voltage constraints are the key to derive the mathematical model of the parallel-type AMDB emulator, which will be confirmed in the next part. According to Kirchhoff's voltage law, for the circuit loop of DPA 1 , C 0 , DPA 3

and input voltage source, one can obtain
and for another circuit loop of D 2 , C 0 , D 4 and input voltage source, there yields Based on Kirchhoff's current law, for the nodes linked with the branches B 1 , B 4 , and the branches B 2 , B 3 , we can get the following equations and for the node linked with the branches B 1 , B 2 , we can get Substituting (3) into (4) and (5), there yields According to the constitutive relations of D k and DPA l in (1) and (2), the equations given in (6) and (7) can be rewritten as Substituting (8) and (9) into (10) and (11), the above equations can be organized as Therefore, the proposed parallel-type AMDB emulator in Fig. 1 can be described by (12), which accords with the definition of an extended memristor in [33].   In fact, once the symmetry of the diode-bridge is broken by connecting some diodes in parallel to one or two bridge arms, an asymmetric memristive diode-bridge emulator is established. Here, we just present a typical case that an equal number of diodes are connected in parallel to the first and third bridge arms. Thus, the yielded mathematical model can be relatively simple.

B. CONFIRMATION OF VOLTAGE CONSTRAIN
The voltage constraints in (3) are the key fundamental for constructing mathematical model (12). Take the parallel-type AMDB emulator with m = 4, R 0 = 1.3 k and C 0 = 100 nF as an example. Multisim simulations and hardware experiments are used to verify the correctness of the voltage constrains in (3).
Firstly, Multisim simulation circuit of the parallel-type AMDB emulator is built, consisting of ten 1N4148 diodes, one capacitor and one resistor. The AC voltage source is used to provide the periodic stimulus, and its peak voltage and frequency are set as 3 V and 20 kHz, respectively. Ocilloscope XSC1 set as X-Y mode is utilized to capture the electrical signals. The screenshots of the simulation circuit and oscilloscope interactive interface are shown in Fig. 2. The observation objects in Fig. 2(a) are the terminal voltages of branches B 1 and B 3 , whereas those in Fig. 2(b) are the terminal voltages of branches B 2 and B 4 . The simulated  synchronous lines indicate that the two pairs of observed electrical signals are in complete synchronization.
Secondly, the physical hardware circuit is also welded and tested. Tektronix AFG3022 function generator is employed to provide the AC voltage source and Tektronix TDS 3034C is used to capture the experimental plots. Different from the Multisim simulation, two additional subtraction circuits are needed to detect the branch voltages, which can reduce the influence of voltage probes on the hardware circuit. Each subtraction circuit is composed of four 2 M resistors and one AD711JN operational amplifier. The experimental results of the synchronous lines are shown in Fig. 3, which are consistent with the Multisim simulation plots in Fig. 2. VOLUME 8, 2020 After the Multisim simulations and the hardware experiments, one can draw a conclusion that the voltage constrains in (3) are correct. Thus, the derived mathematical model in (12) is credible. Besides, from Figs. 2(a) and 3(a), one can notice that the forward voltages of the diode-bridge arms B 1 and B 3 are about 0.7 V. Addtionally, these voltages remain unchanged when m changes. As a result, the DPA in the parallel-type AMDB emulator can be constructed by much more diodes, without the limitation of forward voltage. However, for the series-type AMDB emulator reported in [32], when increasing the numerber of diodes, the forward voltage increases by multiplier. From this aspect, the parallel-type AMDB emulator is much better than the series-type AMDB emulator.

C. ASYMMETRIC HYSTERESIS LOOP
Pinched hysteresis loop is the fingerprint of a memristor under periodic stimulus [33]. When the parallel-type AMDB emulator is excited by an AC voltage source V = 3 sin(2πft), its volt-ampere curves in different parameters are plotted in Fig. 4. The RC filter with R 0 = 1.3 k , C 0 = 100 nF is selected, and the 1N4148 diode with I S = 5.84 nA, n = 1.94, V T = 26 mV is used.
In Fig. 4(a), the parameter m is fixed as 4, and the frequency f is set to 2 kHz, 10 kHz, and 20 kHz, respectively. It can be seen that each volt-ampere curve is pinched at the origin, implying the current will vanish when the applied AC voltage vanishes. When increasing the frequency, the lobe area of the volt-ampere curve decreases, but the difference between the peak current and valley current increases. In short, with the increase of f , the asymmetry of hysteresis loop becomes remarkable. In Fig. 4(b), f is fixed as 20 kHz, and m is set to 1, 4, 8 and 16, respectively. With the increase of m, the left lobe area becomes smaller and smaller, whereas the right lobe area gets bigger and bigger. That is to say the differences between the peak currents and valley currents are gradually enhanced when increasing the parameter m. As can be seen, with the frequency evolution, the parallel-type AMDB emulator can exhibit the asymmetric hysteresis loops pinched at the origin (G(v 0 , 0) = ∞). This implies that the parallel-type AMDB emulator belongs to the extended memristor but without nonvolatile property [33].
Also, the hardware experiments for the pinched hysteresis loops are completed according to Fig. 1. The corresponding results are plotted in Fig. 5. Tektronix TCP213A current probe is used to detect the port currents of the parallel-type AMDB emulator. For better visual effect, the test wire is wound around the current probe ten turns, i.e., the output signal in Ch4 (40 mA /div) is enlarged by 10 times of the test current i (4 mA/div). The experimental results in Fig. 5 match well with the numerical results in Fig. 4.

III. PARALLEL-TYPE AMDB EMULATOR-BASED CHUA'S CIRCUIT
Chua's diode is a nonlinear resistor. It is the key element for achieving chaotic oscillations in Chua's circuit. Generally,  Chua's circuit can generate symmetric double-scroll attractors, symmetric coexisting single-scroll attractors, or symmetric multi-scroll attractors [34]- [38]. In this part, the parallel-type AMDB emulator-based Chua's circuit is taken as an example to explore the dynamical effect of asymmetric nonlinearity.

A. MEMRISTIVE CIRCUIT AND ITS ATTRACTORS
To demonstrate the dynamical effect of asymmetric nonlinearity, an asymmetric memristor-based Chua's circuit is constructed using a parallel-type AMDB emulator to couple a passive LC network and an active RC filter, as shown in Fig. 6. The voltages v 1 , v 2 across the capacitors C 1 and C 2 , and the current i L flowing through the inductor L are chosen as the state variables. Together with the inner state variable v 0 of the parallel-type AMDB emulator G M , there are four state variables, namely, v 1 , v 2 , i L and v 0 .
Based on the mathematical model of the parallel-type AMDB emulator, when applying Kirchhoff's law to the asymmetric memristor-based Chua's circuit in Fig. 6, the circuit state equations can be described as 10. Experimental results of the phase plots with different C 2 (a) double-scroll chaotic attractors, C 2 = 15.5 nF, (b) coexisting single-scroll chaotic attractors, C 2 = 20.2 nF; (c) coexisting period-2 limit cycles, C 2 = 29.5 nF; (d) coexisting period-1 limit cycles, C 2 = 42.7 nF.
where v = v 2 − v 1 . Seen from (13), all the nonlinear terms are related to the parallel-type AMDB emulator. It means that the parallel-type AMDB emulator has a great influence on the attractor topologies of system (13). This influence can be revealed by MATLAB numerical simulations based on (13), in which the circuit parameters are fixed as C 1 = 10 nF, L = 20 mH, R = 2 k , R 0 = 1.3 k , and C 0 = 100 nF, and the other two parameters, C 2 and m, are taken as the controllable parameters.
Obviously, from the phase plots in the first column (m = 1) in Fig. 7, it can be seen that the left-and right-scroll attractors are symmetric about the origin. By contrast, from the second column (m = 4), third column (m = 8), and fourth column (m = 16), it can be found that the right-scroll attractors become smaller and smaller. Therefore, the difference between the right-and left-scroll attractors is becoming more and more apparent. One can also find that this evolution is consistent with that of the asymmetric hysteresis loops exhibited by the parallel-type AMDB emulator.

B. EQUILBRIUM POINT ANSLYSIS
For the parallel-type AMDB emulator-based Chua's system, the equilibrium point is expressed as E = (0,ṽ 2 ,ṽ 0 ,ĩ L ), in whichĩ L = I S [me ξ (ṽ 2 −ṽ 0 ) − e −ξ (ṽ 2 +ṽ 0 ) − m + 1], andṽ 2 and v 0 can be obtained by solving the following equations As can be seen, the equilibrium point E has no connection with the capacitance C 2 . Take the asymmetric coexisting limit cycles at C 2 = 34 nF as examples to explain the equilibrium point evolutions with m increasing. By using the graphical method, the equilibrium points fixed by the function curve intersections and phase portraits with different m in the v 2 -v 0 plane are depicted in Fig.8. MATLAB function 'ezplot' is employed to plot the curves of f 1 and f 2 . As can be clearly seen, system (13) has one zero equilibrium point E 0 and two non-zero equilibrium points E 1 and E 2 . The coexisting period-2 limit cycles are generated around E 1 or E 2 . With m increasing, E 0 and E 1 keep unchanged, while E 2 gradually glides along the f 2 in the first quadrant. As a result, the asymmetric limit cycle pairs are thereby coexisted.

C. EXPERIMENTAL RESULTS
Based on one chip of AD711JN operational amplifier, one inductance coil and some other discrete components, the hardware breadboard of the asymmetric memristor-based Chua's circuit is fabricated, as shown in Fig. 9. The inductance coil is measured as 18.3 mH with the parasitic resistance 2 . An AD711JN operational amplifier is employed to realize the negative resistor −R. And the non-standard capacitances of C 2 are obtained byparalleling several tantalum capacitors, as listed in Tab. 1. Tektronix TDS 3034C with Tektronix TCP213A current probe is used to capture the experimental results of the phase plots. The results are displayed in Fig. 10, which are in good agreement with the numerical ones given in Fig. 7. It is noticed that the coexisting left-or right-attractor is emerged by switching the power on and off repeatedly.

IV. CONCLUSION
This paper reported a novel parallel-type AMDB emulator implemented by an asymmetric diode-bridge cascaded with a RC filter. The mathematical modeling, Multisim circuit analyses, MATLAB numerical simulations, and breadboard hardware experiments were executed. The parallel-type AMDB emulator, inexpensive and easy to be physically fabricated with the on-the-shelf components, was confirmed to behave the pinched property of physical memristor device. In addition, the parallel-type AMDB emulator-based Chua's circuit was taken as an example. Due to the existence of asymmetric nonlinearity, the equilibrium points of memristive Chua's circuit are asymmetrically distributed in the phase space, resulting in the appearance of different types of asymmetric attractors. Of course, the dynamical mechanism is more complex and interesting, which will be studied in our next work.

DATA AVAILABILITY
The data used to support the findings of this study are available from the corresponding author upon request.

CONFLICTS OF INTEREST
The authors declare that they have no conflicts of interest.