Complex Dynamics in a Memcapacitor-Based Circuit Yuan, Fang

Complex dynamics in a memcapacitor-based circuit

Yuan, Fang; Li, Yuxia; Wang, Guangyi; Dou, Gang; Chen, Guanrong

Published in: Entropy

Published: 01/02/2019

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Published version (DOI): 10.3390/e21020188

Publication details: Yuan, F., Li, Y., Wang, G., Dou, G., & Chen, G. (2019). Complex dynamics in a memcapacitor-based circuit. Entropy, 21(2), [188]. https://doi.org/10.3390/e21020188

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Article Complex Dynamics in a Memcapacitor-Based Circuit

Fang Yuan 1,* , Yuxia Li 1,*, Guangyi Wang 2, Gang Dou 1 and Guanrong Chen 3 1 College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China; [email protected] 2 Institute of Modern Circuits and Intelligent Information, Hangzhou Dianzi University, Hangzhou 310018, China; [email protected] 3 Department of Electronic Engineering, City University Hong Kong, Hong Kong 999077, China; [email protected] * Correspondence: [email protected] (F.Y.); [email protected] (Y.L.)

Received: 11 January 2019; Accepted: 14 February 2019; Published: 16 February 2019

Abstract: In this paper, a new memcapacitor model and its corresponding circuit emulator are proposed, based on which, a chaotic oscillator is designed and the system dynamic characteristics are investigated, both analytically and experimentally. Extreme multistability and coexisting attractors are observed in this complex system. The basins of attraction, multistability, bifurcations, Lyapunov exponents, and initial-condition-triggered similar bifurcation are analyzed. Finally, the memcapacitor-based chaotic oscillator is realized via circuit implementation with experimental results presented.

Keywords: chaos; memcapacitor; extreme multistability; circuit emulator; complex dynamics

1. Introduction Memcapacitor is a type of memory device composed of a memristor and meminductor [1], which emerged after the realization of a real memristor prototype [2]. A memcapacitor is actually a nonlinear capacitor with instantaneous responses depending on the internal states and the input signals. Although several possible realizations of the memcapacitor were attempted, e.g., using a micro-electro-mechanical system [2], ionic transport [3], or electronic effect [4], a memcapacitor is not yet available commercially in any form. Therefore, building functional analog memcapacitor models and emulators for computer simulations and laboratory experiments has become urgent, attracting immense interest from both academia and industry. There have been many papers about the memristor [5–9]. In [6], a novel complex Lorenz system with a flux-controlled memristor is introduced and investigated. An active controller is designed to achieve modified projective synchronization (MPS) based on Lyapunov stability theory. In [7], a new memristor-based hyperchaotic complex Lu system is investigated, where an adaptive controller and a parameter estimator are proposed to realize complex generalized synchronization. Furthermore, the complex dynamics of fractional-order and diode bridge-based memristive circuits are studied in [8] and [9]„ respectively. Compared with the reports concerning memristors, references of memcapacitors are relatively fewer. Existing research involves designing a memcapacitor SPICE (Simulation program with integrated circuit emphasis) simulator [10–12] and mutator that can transform a memristor to a memcapacitor [13,14]. In [15], the boundary dynamics of a charge-controlled memcapacitor is investigated, where Joglekar’s window function is used to describe the nonlinearities of memcapacitor’s boundaries. In [16], a mathematical memcapacitor model is introduced and a memcapacitor oscillator is designed, with theoretical and experimental analyses on their basic dynamic characteristics given. In [17], a floating emulator circuit is built, using common off-the-shelf active devices, to mimic the dynamic behaviors of flux-coupled memcapacitors.

Entropy 2019, 21, 188; doi:10.3390/e21020188 www.mdpi.com/journal/entropy Entropy 2019, 21, 188 2 of 14

Memcapacitors are also employed to construct chaotic circuits. A smooth-curve memcapacitor model and a memcapacitive chaotic circuit are presented in [18], where different kinds of coexisting attractors are shown and their corresponding conditions are given. In [19], a Hewlett–Packard memristor model and charge-controlled memcapacitor model are presented, based on which, a new chaotic oscillator is designed to explore characteristics of memristors and memcapacitors in nonlinear circuits. In [20], a chaotic oscillator composed of a meminductor and a memcapacitor is proposed, and in [21] a chaotic memcapacitor-based oscillator with two unstable equilibrium points are studied, along with its fractional form, for potential engineering applications. In the latest study [22], a buckled membrane is used as the plate of a capacitor with memory to realize the function of a memcapacitor. Along the same line as the above extensive investigations, in this paper, a new memcapacitor model is proposed and its corresponding emulator is designed. The instantaneous responses of the emulator to internal states and input signals are investigated experimentally by applying different voltage excitations. Furthermore, a memcapacitor-based chaotic oscillator is constructed. The system dynamic behaviors are analyzed theoretically, including coexisting attractors, basins of attraction, extreme multistability, and so on. Compared with the previously published papers, a memcapacitor with absolute value relation is ﬁrst proposed and the corresponding emulator can be directly used in application circuits. Besides, we further investigate the similar bifurcation structures triggered by different initial conditions, which has not been reported in memcapacitive systems yet. Finally, the proposed oscillator is realized via analog circuits, veriﬁed by laboratory experiments.

2. The Memcapacitor Model and Its Emulator The concept of memcapacitor was presented by Chua et al. [1], where a charge-controlled memcapacitor is described using:

( −1 vc(t) = C (σM(t))qM(t) . (1) σM(t) = qM(t)

−1 where vc(t) is the voltage across the memcapacitor and C is the inverse memcapacitance. The symbol qM(t) is the charge going through the memcapacitor at time t, and σM(t) is the integral of qM(t). To describe the proposed charge-controlled memcapacitor model, Equation (2) is used, where the memcapacitance depends on the device charge and is changing nonlinearly. Since the meminductor has not been fabricated, in this paper we assume the inverse memcapacitance to be precisely deﬁned −1 as C = a + b|σM(t)|, so as to obtain: ( vc(t) = (a + b|σM(t)|)qM(t) . (2) σM(t) = qM(t)

An emulating circuit is designed, as shown in Figure1, to realize the above charge-controlled memcapacitor. In this circuit, it is assumed that all components are ideal without losses and the output limitations are set as ±15 V. The circuit resistors are set as R1 = R2, R3 = R4, and R5 = R6. The operational amplifier U1 is used to reverse the sign of i, which is the current going into the floating terminal of the memcapacitor. Then, the charge q going through capacitor C1 can be calculated by integrating i in the time domain. The operational amplifier U2 constitutes a subtraction circuit, used to extract the voltage across the capacitor C1, with output voltage:

R6 R6 qM(t) vu2(t) = − vc1 = (3) R4 R4 C1 Entropy 2019, 21, 188 3 of 14

The operational ampliﬁer U3 is an integral circuit and its output is: Z Z 1 R6 R6 vu3(t) = − vu2(t)dt = − qM(t)dt = − σM(t) (4) R7C2 C1C2R4R7 C1C2R4R7 where σM(t) is the integral of qM(t). Operational ampliﬁers U4 and U5 construct an absolute-valued circuit with output voltage:

R6 vu4(t) = |vu3(t)| = |σM(t)| (5) C1C2R4R7

Now, based on Equations (3) and (5), the output voltage of the multiplier M1 can be calculated using: R2 ( ) = 6 | ( )| ( ) vM1 t 2 2 σM t qM t (6) C1C2R4R7 Then, the voltage across this grounded memcapacitor can be written as: ! 1 R2 v (t) = v (t) + v (t) = − + 6 | (t)| q (t) C C1 M1 2 2 σM M (7) C1 C1C2R4R7 Entropy 2019, 21, 188 3 of 14

input

+ R1 vM1 1kΩ R3 R5 vc - 10kΩ 1kΩ U1 C1 M1 vc1 10nF - Y X R2 + R4 U2 vu2 1kΩ 10kΩ R6 1kΩ C2 R8 R10 10nF 10kΩ 10kΩ R7 100kΩ U3 U4 vu3 vu4 D2

R9 10kΩ R11 U5 10kΩ Figure 1. Equivalent circuit of the charge-controlled memcapacitor. Figure 1. Equivalent circuit of the charge-controlled memcapacitor. If the memcapacitor-based emulator have voltage signals applied, the instantaneous responses are The circuit resistors are set as R1 = R2, R3 = R4, and R5 = R6. The operational amplifier U1 is used to obtained as shown in Figure2. The voltage signal is set as vinput = 5sin(2πf ), with f being tested at 50 Hz,reverse 80 Hz, the andsign 200 of i Hz., which The is simulations the current of going pinched into hysteresis the floating loops, terminal referred of the to qmemcapacitor.M–vC characteristics, Then, canthe charge be got fromq going Multisim through 12 capacitor software, C which1 can arebe calculated shown in Figureby integrating2. i in the time domain. The

operational amplifier U2 constitutes a subtraction circuit, used to extract the voltage across the

capacitor C1, with output voltage:

() () = − = (3)

The operational amplifier U3 is an integral circuit and its output is:

1 () = − () = − () = − () (4)

where σM(t) is the integral of qM(t).

Operational amplifiers U4 and U5 construct an absolute-valued circuit with output voltage:

() = |()| = |()| (5)

Now, based on Equations (3) and (5), the output voltage of the multiplier M1 can be calculated using: () = |()|() (6) Then, the voltage across this grounded memcapacitor can be written as: 1 | | () = () + () = (− + () )() (7) If the memcapacitor-based emulator have voltage signals applied, the instantaneous responses are obtained as shown in Figure 2. The voltage signal is set as vinput = 5sin(2πf ), with f being tested at 50 Hz, 80 Hz, and 200 Hz. The simulations of pinched hysteresis loops, referred to qM–vC characteristics, can be got from Multisim 12 software, which are shown in Figure 2.

Entropy 2019, 21, 188 4 of 14 Entropy 2019, 21, 188 4 of 14

Figure 2. qM–vC characteristic curves of the memcapacitor. Figure 2.2. qM–v–vCC characteristiccharacteristic curves curves of of the the memcapacitor. memcapacitor. 3. Memcapacitor-Based Chaotic Oscillator and Its Dynamics 3. Memcapacitor-Based Chaotic Oscillator and Its Dynamics 3.1. Memcapacitor-Based Chaotic Oscillator 3.1. Memcapacitor-Based Chaotic Oscillator 3.1. Memcapacitor-Based Chaotic Oscillator Based on the memcapacitor model in Equation (7), a chaotic oscillator is designed as shown in Based on the memcapacitor model in Equation (7), a chaotic oscillator is designed as shown in FigureBased 3, which on the contains memcapacitor conductances model G in1 and Equation G2, capacitor (7), a chaotic C1, inductor oscillator L, and is designedmemcapacitor as shown CM. in Figure3, which contains conductances G1 and G2, capacitor C1, inductor L, and memcapacitor CM. Figure 3, which contains conductances G1 and G2, capacitor C1, inductor L, and memcapacitor CM.

FigureFigure 3. 3. TheThe memcapacitor-based memcapacitor-based chaotic chaotic oscillator. Figure 3. The memcapacitor-based chaotic oscillator. Taking the current iL through the inductor, the voltage v2 across the capacitor and the charge Taking the current iL through the inductor, the voltage v2 across the capacitor and the charge qM qM on the memcapacitor as state variables, a set of four ﬁrst-order state equations can be obtained, on theTaking memcapacitor the current as i Lstate through variables, the inductor, a set of thefour voltage first-order v2 across state the equations capacitor can and be the obtained, charge qasM as follows: on the memcapacitor as state variables, a set of four first-order state equations can be obtained, as follows:  diL = −  L dt v2 v1 follows:  dv  C 2 = −i − G v 1 dt L 2 2 (8) dqM= −  dt = iL + G1v1  = −  dσM = q dt=− M− where v1 = (a + b|σM(t)|)qM. Let the circuit parameters=− − be chosen as shown in Table1 with initial(8) conditions (0.02, 0, 0, 0). Then, System (8) is chaotic with chaotic attractors as shown in Figure4. (8) = + = + Table 1. Circuit parameters for simulations and experiments. = Parameters = Meanings Values where v1 = (a + b|σM(t)|)qM. Let Lthe circuit parametersInductance be chosen 0.3 mHas shown in Table 1 with initial conditionswhere v1 = (0.02,(a + b|0,σ 0,M (0).t)| Then,)qM. Let SystemC the1 circuit (8) is chaoticparametersCapacitor with bechaotic chosen attractors 7.8 as nF shown as shownin Table in Figure1 with 4.initial conditions (0.02, 0, 0, 0). Then, SystemG1 (8) is chaoticConductance with chaotic 0.42attractors mS as shown in Figure 4. G2 Conductance 2.2 mS a Variable −0.7 b Variable 0.5

EntropyEntropy2019 2019, 21, 21, 188, 188 55 of of 14 14

(a) (b)

FigureFigure 4. 4.Chaotic Chaotic attractors attractors and and chaotic chaoticv 1v–1q–MqMhysteresis hysteresis loops loops of of the the memcapacitor-based memcapacitor-based oscillator: oscillator: (a(–ac,b), chaoticc) chaotic attractors; attractors; (d) ( chaoticd) chaoticv1– vq1M–qhysteresisM hysteresis loops. loops.

The system LyapunovTable exponents 1. Circuit parameters were LE1 for= 0.0382,simulationsLE2 and= 0.0067, experiments.LE3 = −0.0097, and LE4 = −0.2142. The Lyapunov dimension was DL = 3.1643. Figure5a shows the Poincar é map on z = 0 and Parameters Meanings Values Figure5b shows the Poincar é maps with initial condition v2(0) varying in the range of (−0.08, 0.08), L Inductance 0.3 mH showingEntropy 2019 that, 21 the, 188 system’s dynamic behaviors were affected by initial conditions. All the above results6 of 14 veriﬁed that System (8)C is1 chaotic [23]. Capacitor 7.8 nF G1 Conductance 0.42 mS G2 Conductance 2.2 mS a Variable –0.7 b Variable 0.5

The system Lyapunov exponents were LE1 = 0.0382, LE2 = 0.0067, LE3 = –0.0097, and LE4 = –0.2142. The Lyapunov dimension was DL = 3.1643. Figure 5a shows the Poincaré map on z = 0 and Figure 5b shows the Poincaré maps with initial condition v2(0) varying in the range of (–0.08, 0.08), showing that the system’s dynamic behaviors were affected by initial conditions. All the above results verified that System (8) is chaotic [23].

(a) (b)

FigureFigure 5. 5.Poincar Poincaréé maps maps on onz =z = 0: 0: (a ()a projection) projection on on the thex–y x–yplane, plane, and and (b ()b projection) projection on on the thex–y x–yplane plane

withwith initial initial condition conditionv2 v(0)2(0) varying varying in in the the range range of of (− (–0.08, 0.08, 0.08). 0.08). 3.2. Equilibrium Points 3.2. Equilibrium Points The system equilibrium set can be calculated using E = {(x, y, z, w) | iL = v2 = qM = 0, σM = c} by The system equilibrium. . set can. be. calculated using E = {(x, y, z, w) | iL = v2 = qM = 0, σM = c} by solving the equations of iL = v2 = q = σM = 0, in which there is a real constant parameter c. solving the equations of = =M = =0, in which there is a real constant parameter c. The Jacobian matrix J at this equilibrium set E is: 1 +|| 0 − 0 1 =− − 00 (9) 10( + ||)0 00 1 0 The corresponding characteristic equation is: ( )( ) ( )( ) − +|| + 1 − +|| – (−/ + ||) + − (10) It is obvious that the characteristic equation has one zero value and three nonzero values. Let a1 = –(G1a − G2/C1 + G1b|c|), a2 = [(C1 − G1G2L)(a + b|c|) + 1]/C1L, a3 = –[(G1 − G2)(a + b|c|)]/C1L. Then, according to the Routh–Hurwitz condition, the system is stable if:

= >0 = − >0 (11) =( −)>0 When the parameters are set as in Table 1, one can find that: 1.4002 < |c| < 2.7431 (12) To make the equilibrium set E unstable, which is a necessary condition for the possible existence of chaos, the constant c should satisfy: |c| < 1.4002 or |c| > 2.7431 (13) Equation (13) demonstrates that the dynamical behaviors of the chaotic circuit (8) were strongly dependent on the memcapacitor internal state variable σM. For example, if the system parameters were set as in Table 1, with c = 1, then four eigenvalues at the equilibrium set E were obtained as:

λ1 = 0, λ2 = 0.54411, λ3, 4 = – 0.45508 ± 0.26927i (14) In this case, the equilibrium set E was unstable with one zero root, two complex conjugate roots with negative real parts, and one positive real root. Thus, a self-excited attractor could be generated via excitation from the unstable focal point in E.

Entropy 2019, 21, 188 6 of 14

The Jacobian matrix J at this equilibrium set E is:

 1 a+b|c|  0 L − L 0  − 1 − G2   C C 0 0  J =  1 1  (9)  1 0 G1(a + b|c|) 0  0 0 1 0

The corresponding characteristic equation is:

4 3 (C1 − G1G2L)(a + b|c|) + 1 2 (G1 − G2)(a + b|c|) λ − (G1a − G2/C1 + G1b|c|)λ + λ − λ (10) C1L C1L

It is obvious that the characteristic equation has one zero value and three nonzero values. Let a1 = −(G1a − G2/C1 + G1b|c|), a2 = [(C1 − G1G2L)(a + b|c|) + 1]/C1L, a3 = −[(G1 − G2)(a + b|c|)]/C1L. Then, according to the Routh–Hurwitz condition, the system is stable if:  ∆ = a > 0  1 1 ∆2 = a1a2 − a3 > 0 (11)   ∆1 = a3(a1a2 − a3) > 0

When the parameters are set as in Table1, one can ﬁnd that:

1.4002 < |c| < 2.7431 (12)

To make the equilibrium set E unstable, which is a necessary condition for the possible existence of chaos, the constant c should satisfy:

|c| < 1.4002 or |c| > 2.7431 (13)

Equation (13) demonstrates that the dynamical behaviors of the chaotic circuit (8) were strongly dependent on the memcapacitor internal state variable σM. For example, if the system parameters were set as in Table1, with c = 1, then four eigenvalues at the equilibrium set E were obtained as:

λ1 = 0, λ2 = 0.54411, λ3, 4 = −0.45508 ± 0.26927i (14)

In this case, the equilibrium set E was unstable with one zero root, two complex conjugate roots with negative real parts, and one positive real root. Thus, a self-excited attractor could be generated via excitation from the unstable focal point in E.

3.3. Parameters Region When the inductor L increased gradually with other circuit parameters fixed as in Table1, the bifurcation diagram of the state variable iL is shown in Figure6a, where the orbits of the system started from chaotic behavior and then entered into periodic behavior via the reverse period-doubling bifurcation route. After that, the orbits returned to chaotic behavior through the forward period-doubling bifurcation route, then jumped into chaotic behavior, and finally approached infinity in the range of L > 1.25. The corresponding Lyapunov exponent spectra are presented in Figure6b, where the maximum Lyapunov exponent was positive within chaotic regions and equalled zero within periodic regions. Several periodic windows can be observed in Figure6b, which match well with that of the bifurcation diagram shown in Figure6a. Entropy 2019, 21, 188 7 of 14 Entropy 2019, 21, 188 7 of 14 3.3. Parameters Region 3.3. Parameters Region When the inductor L increased gradually with other circuit parameters fixed as in Table 1, the bifurcationWhen thediagram inductor of theL increased state variable gradually iL is withshown other in Figurecircuit 6a,parameters where the fixed orbits as inof Table the system 1, the bifurcationstarted from diagram chaotic of behavior the state and variable then ienteredL is shown into in periodic Figure 6a, behavior where thevia orbitsthe reverse of the period-system starteddoubling from bifurcation chaotic route.behavior After and that, then the enteredorbits returned into periodic to chaotic behavior behavior via throughthe reverse the forwardperiod- doublingperiod-doubling bifurcation bifurcation route. After route, that, then the jumpedorbits returned into chaotic to chaotic behavior, behavior and through finally theapproached forward period-doublinginfinity in the range bifurcation of L > 1.25. route, The then corresponding jumped into Lyapunov chaotic behavior,exponent spectraand finally are presentedapproached in infinityFigure 6b, in wherethe range the maximumof L > 1.25. Lyapunov The corresponding exponent wa Lyapunovs positive withinexponent chaotic spectra regions are andpresented equalled in Figurezero within 6b, where periodic the maximumregions. Several Lyapunov periodic exponent windows was positivecan be observed within chaotic in Figure regions 6b, whichand equalled match zerowellEntropy withwithin2019 that, 21 periodic, 188of the bifurcationregions. Several diagram periodic shown windows in Figure can 6a. be observed in Figure 6b, which match7 of 14 well with that of the bifurcation diagram shown in Figure 6a.

(a) (b) (a) (b) Figure 6. Dynamic characters with respect to L: (a) bifurcation diagram, and (b) lyapunov exponent Figurespectrum.Figure 6. 6. Dynamic Dynamic characters characters with withrespect respect to L: ( toa) bifurcationL:(a) bifurcation diagram, diagram, and (b) lyapunov and (b) exponent lyapunov spectrum.exponent spectrum. The new system also had coexisting bifurcations. When the inductor G1 varied with other circuit parametersTheThe new new fixed system system as in also also Table had had 1, coexisting coexistingthe bifurcation bifurcations. bifurcations. diagram When When of the the the state inductor inductor variable GG11 variedivariedL is shown with with inother other Figure circuit circuit 7a, parameterswhereparameters the red fixed ﬁxed orbit as as startedin in Table Table from 11,, thethe initial bifurcationbifurcation conditions diagramdiagram (0, –0.03, ofof thethe 0, 0), statestate and variablevariable the blue iiL Lone isis shown shownstarts fromin in Figure Figure (0, 0.03, 7a,7a, where0,where 0). The the the symmetricallyred red orbit orbit started started coexisting from initial bifurcation conditions orbits (0, –0.03,− were0.03, 0,generated 0, 0), 0), and and the theby blue bluethe onesymmetrical one starts starts from from coexisting (0, (0, 0.03, 0.03, 0,attractors,0, 0). 0). The The symmetrically symmetricallywhich are shown coexisting coexisting in Figure bifurcation bifurcation 9. The corre orbits orbitssponding were were generated generatedLyapunov by by exponent the the symmetrical symmetrical spectrum coexisting coexisting is shown attractors,inattractors, Figure 7b. which which From are the shown bifurcation inin FigureFigure diagram 9.9. TheThe and correspondingcorre thesponding Lyapunov Lyapunov Lyapunov exponent exponent exponent spectrum, spectrum spectrum it can be is isseen shown shown that in intheFigure Figure system7b. 7b. From orbit From the started the bifurcation bifurcation from a diagram limit diagram cycle and and and the the Lyapunovthen Lyapunov turned exponent intoexponent the spectrum, chaotic spectrum, state it can it throughcan be seen be seen thatperiod- that the thedoublingsystem system orbit bifurcations orbit started started from in fromthe a limit rang a limit cyclee of (0,cycle and 0.6). then and When turned then the turned into parameter the into chaotic the G1 chaotic> state 0.6, throughthe state bifurcation through period-doubling diagram period- doublingandbifurcations the Lyapunov bifurcations in the rangeexponent in the of (0,rang spectrum 0.6).e of When (0, had0.6). the blankWhen parameter areasthe parameter sinceG1 > 0.6,the G thesystem1 > bifurcation0.6, thewas bifurcation diverging diagram diagramwith and theno andsolution.Lyapunov the LyapunovBetween exponent the exponent spectrumtwo blank spectrum hadareas, blank th hade system areas blank since stayed areas the insince systema periodic the wassystem state. diverging was diverging with no with solution. no solution.Between Between the two blankthe two areas, blank the areas, system the stayed system in stayed a periodic in a state.periodic state.

(a) (b) (a) (b) FigureFigure 7. 7. DynamicDynamic characteristics characteristics due dueto G1 to: (aG) bifurcation1:(a) bifurcation diagram, diagram, and (b) Lyapunov and (b) Lyapunovexponent Figurespectrum.exponent 7. Dynamic spectrum. characteristics due to G1: (a) bifurcation diagram, and (b) Lyapunov exponent spectrum. 3.4. Similar Bifurcation Structures with Initial Conditions

Different bifurcation parameters usually lead to different bifurcation structures. However, there exist similar bifurcation structures with different initial conditions, which is rare compared with other chaotic systems. When we set the circuit parameters as given in Table1, similar bifurcation diagrams with respect to iL(0), v2(0), and qM(0) were discovered, as presented in Figure8. Although these diagrams depended on the various bifurcation parameters, it is remarkable that the bifurcation structures were almost the same and all symmetrical about the origin. As the bifurcation parameters increased gradually, the system orbit went to a chaotic status via period-doubling bifurcations. Then, the dynamics of the system settled down to periodic behaviors via reverse period-doubling bifurcations. The origin was the boundary of the two processes. The corresponding Lyapunov exponent spectra are given in Figure8d–f, where the graphs also have similar shapes. Entropy 2019, 21, 188 8 of 14

3.4. Similar Bifurcation Structures with Initial Conditions Different bifurcation parameters usually lead to different bifurcation structures. However, there exist similar bifurcation structures with different initial conditions, which is rare compared with other chaotic systems. When we set the circuit parameters as given in Table 1, similar bifurcation diagrams with respect to iL(0), v2(0), and qM(0) were discovered, as presented in Figure 8. Although these diagrams depended on the various bifurcation parameters, it is remarkable that the bifurcation structures were almost the same and all symmetrical about the origin. As the bifurcation parameters increased gradually, the system orbit went to a chaotic status via period-doubling bifurcations. Then, the dynamics of the system settled down to periodic behaviors via reverse period-doubling Entropybifurcations.2019, 21, 188The origin was the boundary of the two processes. The corresponding Lyapunov8 of 14 exponent spectra are given in Figure 8d–f, where the graphs also have similar shapes.

(a) (b) (c)

(d) (e) (f)

FigureFigure 8. 8.Dynamic Dynamic characteristics: characteristics: ( a(–ac,)b bifurcation,c) bifurcation diagrams; diagrams; and and (d– f)(d Lyapunov,e,f) Lyapunov exponent exponent spectra. spectra. Furthermore, the system’s dynamic maps are used to illustrate the similar initial-condition- triggeredFurthermore, bifurcation the structures, system’s whichdynamic are maps shown are in Figureused to9 .illustrate These dynamic the similar maps initial-condition- describe different dynamicaltriggered bifurcation regions with structures, respect which to the are bifurcation shown in parametersFigure 9. TheseiL(0), dynamicv2(0), andmapsqM describe(0), where different the red areasdynamical indicate regions a chaotic with ﬁeld,respect the to blue the bifurcation regions represent parameters a periodic iL(0), v2 status,(0), and and qM(0), yellow where areas the red show unboundedareas indicate zones. a chaotic It is obvious field, the the blue two regions dynamic represent maps own a periodic a similar status, distribution and yellow structure. areas Thus, show we unbounded zones. It is obvious the twoi dynamicv mapsq own a similar distribution structure. Thus, Entropycan conclude 2019, 21, 188 that the initial conditions L(0), 2(0), and M(0) had a similar dynamic inﬂuence for9 of the 14 presentedwe can conclude system inthat the the especial initial conditions parameter i spaces,L(0), v2(0), which and isqM not(0) had common a similar in the dynamic otherchaotic influence systems. for the presented system in the especial parameter spaces, which is not common in the other chaotic

systems. 0.6 P 0.05 0.4 P 0.2

C (0) C

(0) 0 0 2 M v q -0.2

-0.05 U -0.4 U -0.6 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 i (0) i (0) L L (a) (b)

FigureFigure 9.9. DynamicDynamic mapsmaps ofof thethe system:system: ((aa)) iiLL(0)–v2(0)(0) plane, plane, and and ( (b)) iiL(0)(0)–q–qMM(0)(0) plane. plane.

3.5 Extreme Multistability and Coexisting Attractors Multistability is a common phenomenon in many nonlinear dynamical systems, corresponding to the coexistence of more than one stable attractor for the same set of system parameters [24]. When infinitely many attractors coexist for the same set of system parameters, multistability is referred to as extreme multistability [25]. The previously published literature have reported that the dynamical stability of memristive systems are heavily dependent on the initial conditions, which easily leads the system to generate multistability or even extreme multistability [26–29]. One of the main features of extreme multistability is that the system track can present bifurcation without varying any system parameter. When the circuit parameters were set as in Table 1, with initial condition iL(0) varying, the resulting bifurcation diagram of the state variable iL is shown in Figure 10a, where the system presents extreme multistability. It is remarkable to see that the bifurcation diagram is symmetrical about the origin, which came from symmetrical coexisting attractors. As initial condition iL(0) increased gradually within the region of [−0.94, 0], the system orbit started from a limit cycle and turned into a chaotic state through period-doubling bifurcations, showing several periodic windows. Within the region of [0, 0.94], the system orbit settled down to periodic behavior from the chaotic state through reverse period-doubling bifurcations, which is the reverse evolutionary process of that in the region [−0.94, 0]. The corresponding Lyapunov exponent spectra are shown in Figure 10b, where the maximum Lyapunov exponents stay zero with limit cycles but were positive in chaotic states, consistent with the bifurcation diagram.

(a) (b)

Figure 10. Dynamic characters due to iL(0): (a) bifurcation diagram, and (b) Lyapunov exponent spectrum.

To show more details, various typical coexisting attractors are displayed in Figure 11. When the circuit parameters were set as in Table 1, with different initial conditions, the main coexisting regimes were symmetric pairs of limit cycles, chaotic attractors, and point attractors. Figure 11a shows limit

Entropy 2019, 21, 188 9 of 14

0.6 P 0.05 0.4 P 0.2

C (0) C

(0) 0 0 2 M v q -0.2

-0.05 U -0.4 U -0.6 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 i (0) i (0) L L (a) (b) Entropy 2019, 21, 188 9 of 14 Figure 9. Dynamic maps of the system: (a) iL(0)–v2(0) plane, and (b) iL(0)–qM(0) plane.

3.53.5. Extreme Extreme Multistability Multistability an andd Coexisting Coexisting Attractors Attractors Multistability is a common phenomenon in many nonlinear dynamical systems, corresponding toto the the coexistence coexistence of of more more than than one one stable stable attracto attractorr for the for same the same set of setsyst ofem system parameters parameters [24]. When [24]. infinitelyWhen inﬁnitely many attractors many attractors coexist for coexist the same for theset of same system set ofparamete systemrs, parameters, multistability multistability is referred to is asreferred extreme to asmultistability extreme multistability [25]. The previously [25]. The published previously literature published have literature reported have that reported the dynamical that the stabilitydynamical of stabilitymemristive of memristive systems are systems heavily are dependent heavily dependent on the initial on the conditions, initial conditions, which whicheasily easilyleads theleads system the system to generate to generate multistability multistability or even or extreme even extreme multistability multistability [26–29]. [26 –29]. One of of the the main main features features of of extreme extreme multistability multistability is that that the the system system track track can can present present bifurcation bifurcation without varyingvarying anyany system system parameter. parameter. When When the the circuit circuit parameters parameters were were set asse int as Table in Table1, with 1, initial with initialcondition conditioniL(0) varying, iL(0) varying, the resulting the resulting bifurcation bifurcation diagram diagram of the state of variablethe statei Lvariableis shown iL in is Figureshown 10 ina, Figurewhere the10a, system where presents the system extreme presents multistability. extreme It ismult remarkableistability. to It see is thatremarkable the bifurcation to see diagram that the is bifurcationsymmetrical diagram about the is origin, symmetrical which came about from the symmetrical origin, which coexisting came from attractors. symmetrical As initial coexisting condition attractors.iL(0) increased As initial gradually condition within iL(0) the increased region of gradually [−0.94, 0], within the system the region orbit startedof [−0.94, from 0], athe limit system cycle orbit and startedturned intofrom a chaotica limit statecycle throughand turned period-doubling into a chaotic bifurcations, state through showing period-doubling several periodic bifurcations, windows. showingWithin the several region periodic of [0, 0.94], windows. the system Within orbit the settled region down of [0, to 0.94], periodic the behavior system orbit from settled the chaotic down state to periodicthrough behavior reverse period-doubling from the chaotic bifurcations, state through which reverse is theperiod-doubling reverse evolutionary bifurcations, process which of that is the in reversethe region evolutionary [−0.94, 0]. process The corresponding of that in the Lyapunov region [− exponent0.94, 0]. The spectra corresponding are shown Lyapunov in Figure 10 exponentb, where spectrathe maximum are shown Lyapunov in Figure exponents 10b, where stay the zeromaximum with limitLyapunov cycles exponents but were stay positive zero inwith chaotic limit cycles states, butconsistent were positive with the in bifurcation chaotic states, diagram. consistent with the bifurcation diagram.

(a) (b)

Figure 10. 10. DynamicDynamic characters characters due dueto iL(0): to i(La(0):) bifurcation (a) bifurcation diagram, diagram, and (b) Lyapunov and (b) Lyapunovexponent spectrum.exponent spectrum.

To show more details, various typical coexisting attractors are displayed in Figure 11.11. When the circuit parameters were set as in in Table Table 11,, withwith differentdifferent initialinitial conditions,conditions, thethe mainmain coexistingcoexisting regimesregimes were symmetric pairs of limit cycles, chaotic attrac attractors,tors, and point attractors. Figure 11a11a shows limit cycles coexisting with attractors for initial conditions (0.02, ±0.06, 0, 0). Figure 11b,c shows coexisting chaotic attractors with initial values (0, ±0.04, 0, 0) and (0.02, 0, 0, 0), respectively. Figure 11d displays a point attractor with initial conditions (−1, 0.18, 0, 0). Since the system track presented bifurcations with initial conditions and had inﬁnitely many coexisting attractors for the same set of system parameters, the system presented typical extreme multistability. Entropy 2019, 21, 188 10 of 14

Basins of attraction can clearly display the distribution of different coexisting attractors. As shown in Figure 11, the system had six kinds of coexisting attractors. When initial conditions iL(0) and v2(0) were set as variables, with qM(0) = 0 and σM(0) = 0, the corresponding basin of attraction is displayed in Figure 12a, where coexisting attractor distributions are painted with different colors. As shown in Figure 12a, the system was divergent with no attractor in most blank areas and generated a narrow Entropydistribution 2019, 21, of 188 each kind of the coexisting attractors in the middle areas. The evolution process10 of of14 coexisting attractors is shown in Figure 12b, which contains a complete process of period-doubling cyclesbifurcations coexisting and with reverse attractors period-doubling for initial conditions bifurcations (0.02, as v±0.06,2(0) decreases. 0, 0). Figure The 11b,c system shows orbit coexisting started chaoticfrom a pointattractors attractor with (Typeinitial 6) values to a limit (0, ±0.04, cycle 0, (Type 0) and 1), (0.02, and then 0, 0, turned0), respectively. to a chaotic Figure attractor 11d displays (Type 5) avia point period-doubling attractor with bifurcations initial conditions (Type (–1, 3). As0.18,v2 (0)0, 0). decreased Since the further, system thetrack chaotic presented attractor bifurcations (Type 5) withreturned initial to aconditions limit cycle and (Type had 2) viainfinitely reverse many period-doubling coexisting bifurcationsattractors for (Type the 4)same and set ﬁnally of system settled parameters,into a point attractorthe system (Type presented 6). typical extreme multistability.

(a) (b)

Figure 11. PhasePhase portraits portraits of of coexisting coexisting attractors: attractors: (a) (a Ty) Typepe 1 and 1 and Type Type 2; 2;(b) ( bType) Type 3 and 3 and Type Type 4; 4; (c) (c)Type Type 5; ( d5;) (d) Type Type 6. 6.

BasinsSimilarly, of theattraction basin of can attraction clearly withdisplay respect the todist initialribution conditions of differentqM(0) andcoexistingσM(0) isattractors. displayed As in shownFigure 12in c,Figure which 11, is roughlythe system symmetric had six aboutkinds theof coexisting origin and attractors. each attractor When distribution initial conditions appears i inL(0) a andnarrow v2(0) reverse were set ‘S’ as shape. variables, The correspondingwith qM(0) = 0 and evolution σM(0) = process 0, the corresponding of coexisting attractors basin of attraction is shown inis displayedFigure 12d. in Figure 12a, where coexisting attractor distributions are painted with different colors. As shown in Figure 12a, the system was divergent with no attractor in most blank areas and generated a narrow distribution of each kind of the coexisting attractors in the middle areas. The evolution process of coexisting attractors is shown in Figure 12b, which contains a complete process of period- doubling bifurcations and reverse period-doubling bifurcations as v2(0) decreases. The system orbit started from a point attractor (Type 6) to a limit cycle (Type 1), and then turned to a chaotic attractor (Type 5) via period-doubling bifurcations (Type 3). As v2(0) decreased further, the chaotic attractor (Type 5) returned to a limit cycle (Type 2) via reverse period-doubling bifurcations (Type 4) and finally settled into a point attractor (Type 6). Similarly, the basin of attraction with respect to initial conditions qM(0) and σM(0) is displayed in Figure 12c, which is roughly symmetric about the origin and each attractor distribution appears in a narrow reverse ‘S’ shape. The corresponding evolution process of coexisting attractors is shown in Figure 12d.

Entropy 2019,, 21,, 188188 11 of 14

(a) (b)

Figure 12. Basins of attraction of coexisting attractorsattractorsin in( (aa)) the thei LiL(0)(0)–v–v22(0)(0) planeplane andand ((cc)) thetheq qMM(0)–σM(0)(0) plane; the corresponding evolution processprocess ofof coexistingcoexisting attractorsattractors inin ((bb)) iiLL(0)(0)–v2(0) plane and ( d) the qM(0)(0)––σσMM(0)(0) plane. plane.

4. Experimental Results An analog electronic circuit was built to physically realize the above-presented chaotic oscillator to verify the basic dynamic behaviors ofof thethe newnew system.system. Since the inductor and the capacitor in the systemsystem were not standard, which made the circuit difficultdifﬁcult toto designdesign and and implement, implement, an an equivalent equivalent circuit circuit was was designed designed to realize to realize the system. the system. The circuit The circuitschematic schematic of the experimental of the experimental circuit is circuit shown is in shown Figure in 13 Figure, and the 13, corresponding and the corresponding chaotic attractors chaotic Entropy 2019, 21, 188 12 of 14 attractorsobtained byobtained Multisim by simulationMultisim simulation are displayed are displayed in Figure 14 in. Figure 14. In practical experiments, since there are tolerances in resistors and capacitors, it is necessary to adjust the actual resistance values in the analog circuit, which will lead to some deviations between experimental results and the simulation ones. The experimental results shown on the oscilloscope are depicted in Figure 15. The chip AD633JN was chosen as the analog multiplier and LF347N as the operational amplifier with reference voltages of ±15 V. It is clear that the dynamical behaviors observed from the experimental circuit were generally similar with those displayed via numerical simulations.

Figure 13. The equivalent circuit of the new system. Figure 13. The equivalent circuit of the new system.

(a) (b)

Figure 14. Chaotic attractors observed via Multisim simulations: (a,b,c) chaotic attractors; and (d) chaotic v1–qM hysteresis loops.

(a) (b)

Entropy 2019, 21, 188 12 of 14

Entropy 2019, 21, 188 Figure 13. The equivalent circuit of the new system. 12 of 14

Figure 13. The equivalent circuit of the new system.

(a) (b)

Figure 14. Chaotic attractors observed via Multisim simulations: (a–c) chaotic attractors; and (d)

chaoticFigurev1 –14.qM Chaotichysteresis attractors loops. observed via Multisim simulations: (a,b,c) chaotic attractors; and (d) chaotic v1–qM hysteresis loops. In practical experiments, since there are tolerances in resistors and capacitors, it is necessary to adjust the actual resistance(c) values in the analog circuit, which will lead to(d some) deviations between experimental results and the simulation ones. The experimental results shown on the oscilloscope are depicted in Figure 15. The chip AD633JN was chosen as the analog multiplier and LF347N as the Figure 14. Chaotic attractors observed via Multisim simulations: (a,b,c) chaotic attractors; and (d) operational ampliﬁer with reference voltages of ±15 V. It is clear that the dynamical behaviors observed chaotic v1–qM hysteresis loops. from the experimental circuit were generally similar with those displayed via numerical simulations.

(a) (b)

Entropy 2019, 21, 188 13 of 14 (a) (b)

FigureFigure 15. 15.Chaotic Chaotic attractors attractors observed observed from thefrom oscilloscope the oscilloscope in the experiment: in the experiment: (a–c) chaotic (a, attractors;b,c) chaotic andattractors; (d) chaotic andv1 –(dqM) chaotichysteresis v1–q loops.M hysteresis loops.

5. Conclusions In this paper, a new memcapacitor model and a novel memcapacitor-based chaotic circuit are presented. The system extreme multistability was analyzed, including bifurcation diagrams, Lyapunov spectra, coexisting attractors, coexisting bifurcations, and basins of attraction of various attractors. Moreover, the new memcapacitor-based system was realized using an experimental circuit, which agreed well with the numerical simulations and verified the theoretical analysis results. Due to the rich and unusual complex dynamical characteristics of the proposed memcapacitor system, it was deemed that it would find some novel and non-traditional applications in engineering and technology in the future. In our future works, we will continue to try to build physical memcapacitors and explore special dynamics in memcapacitive circuits.

Author Contributions: Investigation, Gang Dou; Methodology, Fang Yuan; Project administration, Yuxia Li; Validation, Yuxia Li, Guangyi Wang and Guanrong Chen; Writing—original draft, Fang Yuan; Writing—review and editing, Fang Yuan.

Acknowledgments: The work was supported by the National Natural Science Foundation of China under Grants (61801271, 61473177, 61771176 and 61703247).

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Ventra, M.D.; Pershin, Y.V.; Chua, L.O. Circuit elements with memory: Memristors, memcapacitors and meminductors. Proc. IEEE 2009, 97, 1717–1724. 2. Strukov, D.B.; Snider, G.S.; Stewart, D.R.; Williams, R.S. The missing memristor found. Nature 2008, 453, 80–83. 3. Martinez-Rincon, J.; Pershin, Y.V. Bistable non-volatile elastic membrane memcapacitor exhibiting chaotic behavior. IEEE Trans. Electron Dev. 2011, 58, 1809–1812. 4. Lai, Q.; Zhang, L.; Li, Z.; Stickle, W.F.; Williams, R.S.; Chen, Y. Analog memory capacitor based on field- configurable ion-doped polymers. Appl. Phys. Lett. 2009, 95, 213503. 5. Guo, M.; Xue, Y.B.; Gao, Z.H.; Zhang, Y.M.; Dou, G.; Li, Y.X. Dynamic analysis of a physical SBT memristor- based chaotic circuit. Int. J. Bifurc. Chaos 2017, 27, 1730047. 6. Wang, S.B.; Wang, X.Y.; Zhou, Y.F. A memristor-based complex lorenz system and its modified projective synchronization. Entropy 2015, 17, 7628–7644. 7. Wang, S.B.; Wang, X.Y.; Zhou, Y.F.; Han, B. A memristor-based hyperchaotic complex Lu system and its adaptive complex generalized synchronization. Entropy 2016, 18, 58. 8. Xi, H.L.; Li, Y.X.; Huang, X. Generation and nonlinear dynamical analyses of fractional-order memristor- based Lorenz systems. Entropy 2014, 16, 6240–6253. 9. Chen, M.; Yu, J.J.; Yu, Q.; Li, C.D.; Bao, B.C. A memristive diode bridge-based canonical chua’s circuit. Entropy 2014, 16, 6464–6476. 10. Martinez-Rincon, J.; Di Ventra, M.; Pershin, Y.V. Solid-state memcapacitive system with negative and diverging capacitance. Phys. Rev. B 2010, 81, 195430.

Entropy 2019, 21, 188 13 of 14

5. Conclusions In this paper, a new memcapacitor model and a novel memcapacitor-based chaotic circuit are presented. The system extreme multistability was analyzed, including bifurcation diagrams, Lyapunov spectra, coexisting attractors, coexisting bifurcations, and basins of attraction of various attractors. Moreover, the new memcapacitor-based system was realized using an experimental circuit, which agreed well with the numerical simulations and veriﬁed the theoretical analysis results. Due to the rich and unusual complex dynamical characteristics of the proposed memcapacitor system, it was deemed that it would ﬁnd some novel and non-traditional applications in engineering and technology in the future. In our future works, we will continue to try to build physical memcapacitors and explore special dynamics in memcapacitive circuits.

Author Contributions: Investigation, G.D.; Methodology, F.Y.; Project administration, Y.L.; Validation, Y.L., G.W. and G.C.; Writing—original draft, F.Y.; Writing—review and editing, F.Y. Acknowledgments: The work was supported by the National Natural Science Foundation of China under Grants (61801271, 61473177, 61771176 and 61703247). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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