Memory Circuit Elements: Complexity, Complex Systems, and Applications
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MemristorBased Canonical Chua’s Circuit: Extreme Multistability in VoltageCurrent Domain and Its Controllability in FluxCharge Domain
Abstract
This paper investigates extreme multistability and its controllability for an ideal voltagecontrolled memristor emulatorbased canonical Chua’s circuit. With the voltagecurrent model, the initial conditiondependent 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 fluxcharge describing equations for the memristorbased canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established. As a result, the initial conditiondependent dynamics in the voltagecurrent domain is converted into the system parameterassociated dynamics in the fluxcharge 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 chaosbased engineering applications of multistable memristive circuits.
1. Introduction
Initial conditiondependent extreme multistability, first encountered in several coupled nonlinear dynamical systems [1–3], is a coexisting phenomenon of infinitely many attractors for a given set of system parameters. More recently, due to the existence of infinitely many equilibrium points, for example, line equilibrium point or plane equilibrium point, this special dynamical phenomenon of extreme multistability is naturally exhibited in a class of ideal flux/voltagecontrolled memristorbased chaotic circuits/systems [4–9], thereby leading to the emergence of infinitely many disconnected attractors.
Extreme multistability is a fantastic kind of multistability, which makes a nonlinear dynamical circuit or system supply great flexibility for its potential uses in chaosbased engineering applications [10–12], but also raises new challenges for its control of the existing multiple stable states [11–14]. Generally, multistability is confirmed in hardware experiments by randomly switching on and off experimental circuit supplies [9, 15–21] or by MATLAB numerical or PSPICE/PSIM circuit simulations [4–8, 22–28]. Consequently, to direct the nonlinear dynamical circuit or system to a desired oscillating mode, an effective control approach should be proposed [12]. To this end, this paper takes an ideal voltagecontrolled memristor emulatorbased canonical Chua’s circuit as an example; a controllable strategy for extreme multistability is achieved through converting the initial conditiondependent dynamics in the voltagecurrent domain into the system parameterassociated dynamics in the fluxcharge domain [29, 30].
Besides, for a memristorbased circuit or system with line equilibrium point or plane equilibrium point, its stability at the equilibrium point is very difficult to be determined due to the existence of one or two zero eigenvalues [5–9], which results in the fact that the coexisting infinitely many attractors’ behaviors can not be precisely interpreted from the stabilities of the nonzero eigenvalues. As a matter of fact, the memristor initial condition and other initial conditions all have dynamical effects on the memristorbased circuit or system [8, 9]. However, the dynamical effects are implied, which can not be explicitly expressed in the voltagecurrent domain. How about the memristorbased circuit or system in the fluxcharge domain?
Fluxcharge analysis method was first postulated as a tool of dimensionality reduction [31–36], in which the initial conditions of the memristorbased circuit or system are not precisely formulated, thereby leading to the absence of the initial conditiondependent dynamical behaviors [34–36]. In the last two years, a new fluxcharge analysis method is reported in [29, 30], which judiciously utilizes the incremental flux and charge to substitute the conventional flux and charge and efficaciously solves the issue of the original fluxcharge analysis method. Accordingly, based on the voltagecurrent relation, an accurate fluxcharge relation of the ideal voltagecontrolled memristor emulator is established. With the accurate constitutive relation, an incremental fluxcharge model for the memristorbased canonical Chua’s circuit is constructed, upon which all the initial conditions in the voltagecurrent model can be explicitly formulated by the system parameters in the fluxcharge model and the multiple stable states can be consequently controlled by changing the initial conditionrelated system parameters.
The rest of the paper is structured as follows. In Section 2, an ideal voltagecontrolled memristor emulatorbased canonical Chua’s circuit is presented. With the voltagecurrent model, the initial conditiondependent extreme multistability is explored and then the coexisting infinitely many attractors are numerically uncovered. In Section 3, based on the accurate constitutive relation of the memristor emulator, a set of incremental fluxcharge equations for the memristorbased canonical Chua’s circuit are formulated and a dimensionality reduction model is thus established, upon which the feasibility of the fluxcharge analysis method is verified by MATLAB numerical simulations. In Section 4, an equivalent circuit of the incremental fluxcharge model is designed and circuit simulations for the initial conditiondependent behaviors are executed, from which the controllability of extreme multistability is physically confirmed. The conclusions are drawn in the last section.
2. Extreme Multistability in the VoltageCurrent Domain
Based on a canonical Chua’s circuit and an ideal voltagecontrolled memristor emulator, a new memristorbased canonical Chua’s circuit is constructed, as shown in Figure 1(a), which is simple and physically realizable. The ideal voltagecontrolled memristor emulator is equivalently implemented with an electronic circuit via opamp integrators and analog multipliers [5, 6, 34], as shown in Figure 1(b). In our next work, the considered circuit parameters remained unchanged and are listed in Table 1, where is the total gain of two multipliers and .

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2.1. Conventional VoltageCurrent Model
For the ideal voltagecontrolled memristor emulator in Figure 1(b), the relationships of the input voltage , the input current , and the voltage of the capacitor can be mathematically described in the voltagecurrent domain asThus, for the state variables of , , , and in Figure 1, the describing circuit equations are easily given in the voltagecurrent domain aswhere .
Introduce four new state variables and scale the circuit parameters asModel (2) can be reexpressed aswhich indicates that there are only four parameters in the normalized system model.
With the circuit parameters in Table 1, the normalized parameters for model (4) are obtained by (3) asIn the following works, the memristorbased canonical Chua’s circuit modeled by (4) and the typical system parameters given by (5) are utilized.
2.2. Stability Distribution of Line Equilibrium Point
Similar to the memristive Chua’s circuit containing an ideal voltagecontrolled memristor emulator [5, 6], the memristorbased canonical Chua’s circuit has a line equilibrium point, which is expressed bywhere the constant is uncertain.
At the line equilibrium point , the Jacobian matrix is given asFor the Jacobian matrix given in (7), the normalized parameters determined in (5), and the constant that increased in the region , four eigenvalues with a zero root, a real root, and a pair of conjugated complex roots can be calculated by MATLAB numerical simulations. The real parts of these four eigenvalues are drawn in Figure 2, which can be used to classify the stability distributions. It can be seen from Figure 2 that the sign of the real parts of three nonzero eigenvalues varies with the increase of , leading to the occurrence of three kinds of unstable regions marked with I, II, and IV and a kind of stable region marked with III. Additionally, it can also be observed that the stability distributions of the nonzero eigenvalues in the negative region of are symmetrical to those in the positive region of .
Due to the existence of the zero eigenvalue, the stability of the memristorbased canonical Chua’s circuit can not be simply determined by the three nonzero eigenvalues of the line equilibrium point. The following numerical simulations demonstrate that the zero eigenvalue also has influence on the dynamics of the circuit under some circuit parameters [6–9].
The initial conditions for numerical simulations of the coexisting attractors’ behaviors are taken as [, 10^{−9}, 0, 0]; that is, only the memristor initial condition is variable. For some different values of the memristor initial condition , nonzero eigenvalues, stability regions given in Figure 2, and the related attractor types numerically solved by (4) are summarized in Table 2. It is demonstrated that, for the different values of located in different stability regions, there exist various attractor types with different topologies or different periodicities or different locations. Consequently, coexisting infinitely many attractors’ behavior or extreme multistability occurs in the memristorbased canonical Chua’s circuit.

2.3. Coexisting Infinitely Many Attractors
With reference to the stability distributions in Figure 2 and the initial conditiondependent attractor types in Table 2, the proposed memristorbased canonical Chua’s circuit has various stable states under different initial conditions; that is, its longterm behavior closely relies on the initial conditions therefore leading to the emergence of coexisting infinitely many attractors.
For the normalized parameters in (5) and the initial conditions of = 0 and = 0, the attraction basin in the plane of the initial conditions of and is depicted in Figure 3. It should be illustrated that many more diverse attractor types for different initial conditions can be certainly observed in the memristorbased canonical Chua’s circuit; however, for visual effects, only fourteen kinds of color areas are classified by the basin of attraction in the  plane, among which the largest black area represents the unbounded zone.
Corresponding to different color areas, different types of coexisting attractors are listed in Table 3. Spontaneously, for the initial conditions in the different color areas of the attraction basin, the phase portraits of typical coexisting attractors are obtained, as shown in Figure 4, where for the sake of observations two point attractors in Figure 4(a) are marked by two fivepointed stars. Of course, these generated coexisting attractors intersect the neighborhood of the line equilibrium point, implying that the initialsdependent dynamical system (4) [37] always oscillates in selfexcited states, rather than hidden states [38–40].

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It should be mentioned that just like the ideal flux/voltagecontrolled memristorbased chaotic circuits [4–9], the proposed memristorbased canonical Chua’s circuit has a line equilibrium point with complicated stability distributions already depicted in Figures 2–4, whereas most of conventionally nonlinear dynamical systems with no equilibrium point [10], with only several determined equilibrium points [15–21], or with curves of equilibrium points [41–43] have relatively simple stability distributions with some divinable nonlinear dynamical behaviors.
3. Controllability of Extreme Multistability in the FluxCharge Domain
Due to the existence of the line equilibrium point, the memristorbased canonical Chua’s circuit can exhibit the special phenomenon of extreme multistability under different initial conditions. For seeking the potential uses of the multistable memristive circuit in chaosbased engineering applications [10–12], an effective control method should be applied to direct the memristive circuit to the desired oscillation mode [12]. For this purpose, an incremental fluxcharge model is newly constructed, in which the initial conditiondependent dynamics in the voltagecurrent domain is converted into the system parameterassociated dynamics in the fluxcharge domain.
3.1. Newly Constructed Incremental FluxCharge Model
The accurate constitutive relation of the ideal voltagecontrolled memristor emulator in Figure 1(b) should be firstly established in the fluxcharge domain. Define and as the incremental charge and incremental flux of the ideal memristor emulator, respectively. According to (1), the incremental charge within the time interval is deduced aswhere , stands for the memristor initial state, and the memristor inner state variable . Thus, the initial state can be explicitly represented in the fluxcharge domain.
Suppose that , , and are the incremental fluxes of the capacitors and and the incremental charge of the inductor , respectively, and set , , and as the initial voltages of the capacitors and and the initial current of the inductor L, respectively. While connecting the power supply at t = 0, the incremental fluxcharge model of the memristorbased canonical Chua’s circuit can be yielded by integrating (2) from 0 to .
Integrating the second, third, and fourth equations of (2) from 0 to givesConsider the fact that the fluxes and of the capacitors and and the charge of the inductor can be expressed asrespectively, where represent three arbitrary real constants. Therefore, the state variables , , and in the voltagecurrent domain can be signified asSubstituting (11) into (9), the circuit equations of Figure 1 can be thereby modeled in the fluxcharge domain asEquation (12) is the incremental fluxcharge model of the proposed memristorbased canonical Chua’s circuit.
Analogously, introduce three new state variables and scale the circuit parameters asModel (12) can be rewritten aswhere the normalized memristor constitutive relation is turned as .
It should be emphasized that the initial conditions of (14) are ensured as . The memristor inner parameter denotes the initial capacitor voltage of the memristor emulator in the voltagecurrent domain, and the system parameters , , and reflect the three initial voltages of the canonical Chua’s circuit in the voltagecurrent domain.
For the circuit parameters given in Table 1, the normalized parameters , , , and in (14) are the same as those given in (5). With these determined parameters, the initial conditionsdependent extreme multistability in the memristorbased canonical Chua’s circuit can be effectively controlled by adjusting the system parameters , , , and .
3.2. System ParameterRelated Stability Distribution
For the normalized model (14), the equilibrium points are obviously obtained as in which can be numerically solved by
Define and to be According to the classical Cardan discriminant , when , there are three real roots in (16), which can be given asindicating that the model (14) has three equilibrium points.
The Jacobian matrix at equilibrium point is deduced aswhere . Consequently, three eigenvalues of the model (14) at are yielded by solving the following characteristic polynomial:in which
Based on (15)–(20), it can be known that the line equilibrium point described by (6) in the voltagecurrent domain is converted into several determined equilibrium points represented by (15) in the fluxcharge domain, whose locations and stabilities are decided by the initial conditionrelated system parameters , , , and . Therefore, the extreme multistability in the voltagecurrent domain can readily be controlled by the system parameters in the fluxcharge domain.
Take as an example. When the normalized system parameters a, b, c, and for the model (14) are fixed as given in (5) and the relationship of is satisfied, three equilibrium points consisting of one zero equilibrium point and two nonzero equilibrium points are solved from (16) aswhere , which means that the nonzero equilibrium points depended on the initial condition of the memristor emulator.
For the zero equilibrium point , there exist . In this way, the Jacobian matrix of (19) with the increase of is identical with the Jacobian submatrix obtained from (7) by deleting row 1 and column 1 with the increase of . As a consequence, the Jacobian matrix of (19) has three eigenvalues; their stability distributions are the same as those of the nonzero eigenvalues of the Jacobian matrix of (7). Whereas for two nonzero equilibrium points , the complicated stability distributions can be numerically found as the memristor initial condition is tuned.
3.3. Controllability of Extreme Multistability
With model (14), it is demonstrated that the initial conditiondependent extreme multistability in the memristorbased canonical Chua’s circuit is transformed into the system parameterassociated dynamics, therefore leading to the controllability of extreme multistability through directly adjusting the system parameters.
The normalized system parameters a, b, c, and are given in (5) and the initial conditions for (14) are ensured as . Referring to the initial conditions in Figure 4, and in (14) remained unchanged, whereas and are assigned as some different values in the regions [−6, 6] and [−4, 4], respectively. For different locations of the parameter space constructed by and , various types of disconnected attractors are numerically simulated, as shown in Figure 5, where for the sake of observations two point attractors overlapped together in Figure 5(a) are marked by two fivepointed stars and two limit cycles in Figure 5(b) are marked with bold lines. Obviously, the dynamical behaviors featured by Figure 5 are consistent with those featured by Figure 4, ignoring the computational errors in MATLAB simulation [44], which verify the feasibility of the incremental fluxcharge model of the memristorbased canonical Chua’s circuit.
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4. Controlling Multiple Stable States in Physical Circuit
With model (14), the equivalent circuit using analog multipliers and opamps linked with resistors and/or capacitors [7, 8, 45] can be handily designed, as shown in Figure 6, which is composed of the linear calculating circuit with three integrating channels and the memristor constitutive relation circuit. In Figure 6, , , and represent three state variables of the capacitor voltages, respectively, and stands for the time constant of the integrators.
According to the fundamental theory of circuit, the circuit state equations of Figure 6 are expressed aswhere . Thus, the circuit parameters can be chosen as
The opamps OP07CP and multipliers AD633JNZ with ±15 V power supplies are utilized. The integrating time constant is selected as kΩ × 10 nF = 100 μs. Thus, for the system parameters given in (5), the circuit parameters in Figure 6 are calculated as = 1.4242 kΩ, = 4.0404 kΩ, = 0.9795 kΩ, = 4.0404 kΩ, and = 50 kΩ. Additionally, the gains of the multipliers in Figure 6 are all fixed as 1 and the values of and are maintained as 0.
To better present the control effect of the multistable states generated from the equivalent circuit in Figure 6, the NI Multisim 12.0 simulation and circuit design software is utilized, in which the default initial values of three capacitors are assigned as 0. For several different values of and , the Multisim intercepted phase portraits are displayed, as shown in Figure 7. Note that the initial value of 1 × 10^{−9 }V can be achieved by a slightly induced voltage in the equivalent circuit, so the value of is set as 0. Comparing the results of Figure 7 with those of Figure 5, it is concluded that the circuit simulations agree with the numerical simulations, further confirming the feasibility of the controllable strategy for extreme multistability.
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5. Conclusion
By replacing Chua’s diode in the canonical Chua’s circuit with an ideal voltagecontrolled memristor emulator, a memristorbased canonical Chua’s circuit is presented in this paper. Because of the existence of a line equilibrium point, the initial conditiondependent extreme multistability easily emerged in such a memristive circuit, resulting in the coexistence of infinitely many attractors. To implement the controllability of the extreme multistability, an incremental fluxcharge model for the memristive circuit is formulated through deriving the accurate constitutive relation of the memristor emulator. Thus, the initial conditiondependent dynamics in the voltagecurrent domain is converted into the system parameterassociated dynamics in the fluxcharge domain, that is, the implicit expression of the initial conditions in the voltagecurrent model can be transformed into the explicit representation of the system parameters in the fluxcharge model, leading to the fact that the multiple steady states emerging in the memristive circuit can be consequently controlled by changing the initial conditionrelated system parameters. The feasibility of the controllable strategy for extreme multistability is confirmed by numerical simulations and circuit simulations, which is greatly significant for seeking the potential uses of the multistable memristive circuits in chaosbased engineering applications.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundations of China under Grants nos. 51777016, 61601062, 51607013, and 11602035 and the Natural Science Foundations of Jiangsu Province, China, under Grant no. BK20160282.
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Copyright © 2018 Han Bao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.