# Coexisting Oscillation and Extreme Multistability for a Memcapacitor-Based Circuit

Authors: Guangyi Wang; Chuanbao Shi; Xiaowei Wang; Fang Yuan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The coexisting oscillations are observed with a memcapacitor-based circuit that consists of two linear inductors, two linear resistors, and an active nonlinear charge-controlled memcapacitor. We analyze the dynamics of this circuit and find that it owns an infinite number of equilibrium points and coexisting attractors, which means extreme multistability arises. Furthermore, we also show the stability of the infinite many equilibria and analyze the coexistence of fix point, limit cycle, and chaotic attractor in detail. Finally, an experimental result of the proposed oscillator via an analog electronic circuit is given.


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## Body

## 1. Introduction

Memristor is known as the fourth basic circuit element that was firstly postulated by Chua in 1971 [1] and has attracted worldwide immense attention from both theory and applications even since a solid-state device called the memristor was fabricated by the HP Lab in 2008 [2]. Then, Di Ventra et al. extend the notion of memristor to memcapacitor and meminductor [3], whose properties depend on the state and history of them in circuit. These nanoscale elements, that store information without need for an internal power supply, can be used for nonvolatile memories, nonlinear circuits, neural networks, and so on.In the field of nonlinear dynamics, memristors can be as fundamental elements for the designs of new nonlinear circuits substituting other nonlinear devices, whereupon many memristor-based chaotic oscillators were presented by using HP memristor and the other memristor models of piecewise linear, quadric, and cubicφ-q (flux-charge) functions. Itoh and Chua firstly derived several nonlinear oscillators based on Chua’s oscillators by replacing Chua’s diodes with piecewise linear models of memristor [4]. The authors in [5] designed a periodically forced memristive Chua’s circuit using the flux-controlled memductance model W(φ)=-a+b|φ(t)|. HP memristor model is a first model of actual memristive device, which was implemented by a memristor emulator in [6]. The paper [7] introduced a complete mathematical model for the HP memristor which takes into consideration all boundary situations providing the interrelation between memristance charge and flux. Based on HP memristor and other memristor models, some chaotic oscillators were presented [8–10]. Recently, a new memristor-based chaotic system was designed based on HP memristor in [11] and was realized using FPGA (Field Programmable Gate Array) technology. A novel digital-analog hybrid chaotic system with generalized memristor was constructed for the production of random number [12]. However, the memcapacitor and meminductor have received a little attention since the actual solid-state devices of them have not been successfully achieved. Because of their potential application values, memcapacitor and meminductor have still attracted more and more attentions.The first meminductor emulator was designed in [13] whose inductance can be varied by an external current source without employing any memristive system. In [14], a flux-controlled memristive emulator using light-dependent resistor (LDR) was proposed and the mutator for transferring memristor into a flux-controlled meminductor is described. A mathematical model of a flux-controlled meminductor and its equivalent circuit model for exploring the properties of the meminductor in a nonlinear circuit were presented in [15].It would be worthwhile to prospectively study effective memcapacitor models and its applications. So several memcapacitor models, including piecewise linear, quadric and cubic function models, memristor-based memcapacitor models, and memcapacitor emulators, were proposed in [16–21], and a mathematical memcapacitor model and a corresponding circuit model are established in [22]. Some special phenomena such as hidden attractors, coexistence attractors, and extreme multistability were found in memcapacitor-based chaotic oscillators [23, 24] and memristor-based chaotic oscillators [25–27]. In fact, multistability and coexisting attractors have caught the attention of researcher in general chaotic systems [28–30].This paper introduces a new chaotic oscillator based on a charge-controlled memcapacitor model and its dynamical behaviors are analyzed. The most important properties of the memcapacitor-based circuit are that it possesses an infinite number of equilibrium points and coexisting attractors and displays the coexisting attractors and stability of the infinite many equilibria, called extreme multistability. The rest of this paper is organized as follows. In Section2, the chaotic oscillator is described and the typical chaotic attractors are given. In Section 3, the dissipativity and equilibrium stability of the system are studied. In Section 4, coexisting attractors with different initial x(0) and u(0) are described. In Section 5, an analog circuit is designed to realize the memcapacitor-based oscillator. Finally, some conclusions of this paper are given in Section 6.


## 2. Memcapacitor-Based Chaotic Oscillator Circuit

The chaotic oscillator circuit based on a charge-controlled memcapacitor is shown in Figure1, which contains two linear inductors, L1 and L2, two linear resistors, R1 and R2, and an active memcapacitor. The active memcapacitor is an active 2-terminal circuit consisting of a linear active resistor -R0 and a nonlinear memcapacitor in series, in which the active resistor -R0 provides energy for the circuit. Hence, the series circuit, that is, the active 2-terminal circuit as a whole, exhibits an active feature, so called the active memcapacitor.Figure 1
Memcapacitor-based circuit.The circuit dynamics can be described by(1)L1di1dt=i1+i2R0-vm-i1R1,L2di2dt=i1+i2R0-vm-i2R2,dqmdt=i1+i2,where qm and vm are the charge through the memcapacitor and the voltage across the memcapacitor, respectively. The relationship between the charge qm and voltage vm of the memcapacitor is defined as [19](2)vm=α+βσm2qm,where α+βσm2 is the inverse memcapacitance and σm is the time integral of qm; namely, σ=∫t0tq(τ)dτ.Equation (2) has a new variable σm, so we should add an equation of σm in (1). If we define x=i1, y=i2, z=qm, u=σm, a=α/L1, b=β/L1, c=R0-R1/L1, d=R0/L1, e=R0-R2/L2, f=R0/L2, m=β/L2, and n=α/L2, (1) becomes(3)dxdt=cx+dy-az-bzu2,dydt=fx+ey-nz-mzu2,dzdt=x+y,dudt=z.Typical chaotic attractors obtained by simulating (3) are shown in Figure 2 in terms of the parameters listed in Table 1, with the initial condition (0.01, 0.01, 0.01, 0.01).Table 1
Parameters of system (3).
Parameter Value a = α / L 1 5.8 b = β / L 1 1.5 c = ( R 0 - R 1 ) / L 1 2.6 d = R 0 / L 1 0.1 e = ( R 0 - R 2 ) / L 2 −3.4 f = R 0 / L 2 0.2 m = β / L 2 6.8 n = α / L 2 2.8Figure 2
Chaotic attractors obtained from (3). (a) x-y plane. (b) x-z plane. (c) y-z plane. (d) y-u plane.
(a) (b) (c) (d)The corresponding Lyapunov exponents are calculated as LE1 = 0.3787, LE2 = 0.0092, LE3 = −0.0094, and LE4 = −0.9786. The Poincare mapping on z=0 and the time-domain waveforms of system (3) are described in Figure 3. Obviously, time-domain waveforms are nonperiodic continuous chaotic sequences, and the curve of Poincare map is a continuous curve, and the time-domain waveform is not periodic, which indicate that these attractors are chaotic attractors.Figure 3
Poincare mapping and time-domain waveforms. (a) Poincare mapping. (b) Time-domain waveforms.
(a) (b)


## 3. Dynamical Characteristic Analysis

### 3.1. Dissipativity and Stability of Equilibrium Set

Dissipative systems will shrink to a phase space of a limited area, called the attractor. The dissipativity of the system is described as(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u=c+e.When the parameters are set as c=2.6 and e=-3.4, the exponential constraint rate satisfies(5)∇V=-0.8<0.It means that the volume of phase space will be contracted to zero and all trajectories of the system are confined to a subset of zero volume.The equilibrium points of system (3) can be calculated as(6)s1=x,y,z,u∣z=0,x=k1,y=-k1,u=k2,c=d,f=e,s2=x,y,z,u∣x=y=z=0,u=k3,c≠d,f≠e,where k1, k2, and k3 are arbitrary constants. Obviously, the system has two equilibrium sets, and every set has an infinite number of equilibria.If the parameters arec=d, f=e or c≠d, f≠e, the Jacobian matrix of the circuit at equilibrium sets S1 and S2 is given by (7)J1,2=cd-a-bk2,320fe-n-mk2,32011000010,where k2 and k3 have the same effect on the system, so they can be replaced by k. The characteristic equation is written as(8)λ4-c+eλ3+a+bk2+n+mk2+ce-fdλ2+af+bk2f+dn+dmk2-ae-bk2e-nc-cmk2λ=0.According to the Routh–Hurwitz criterion, the necessary and sufficient condition for stability of the system is that all the principal minors are greater than zero; that is,(9)Δ1=a1>0,Δ2=a1a0a3a2>0,Δ3=a1a00a3a2a10a4a3>0,Δ4=a1a000a3a2a1a00a4a3a2000a4>0,where (10)a0=1,a1=-c+e,a2=a+bk2+n+mk2+ce-fd,a3=af+bk2f+dn+dmk2-ae-bk2e-nc-cmk2,a4=0,whereas it is unstable and has the possibility of generating chaos under the proper parameters. If we fix a=5.8, b=1.5, c=2.6, d=0.1, e=-3.4, f=0.2, m=6.8, and n=2.8, the stability of equilibrium sets S1 and S2 depends on the constant k. Figure 4 shows the change rule of Δ1, Δ2, Δ3, and Δ4 with the constant k. Here, Δ1 is slightly greater than zero and Δ4 is equal to zero, so the black curve overlaps so much with the green curve in Figure 4. Note that the stability of an infinite number of equilibria is called extreme multistability. The following analysis will show that the stability of the all equilibrium points depends on the parameter k, resulting in chaos when k∈[-1.5,1.5].Figure 4
Curves ofΔ1, Δ2, Δ3, and Δ4 with k.


### 3.2. Coexisting Oscillations

In this section, we show that the dynamics of the system depend on not only its parameters but also its initial conditions, which leads to a special characteristic, called the coexisting oscillation in this paper. In the coexisting oscillation, many coexisting attractors appear with different initial conditions, such as coexistence of limit circle or fixed point and chaotic attractor, coexistence of chaotic and hyperchaotic attractors, and coexistence of two limits or two fixed points.Multistability is a common phenomenon in many nonlinear systems, which corresponds to the coexistence of more than one stable attractor for the same set of system parameters [31]. If infinitely many attractors coexist for the same set of system parameters, the phenomenon is called extreme multistability [32], showing that various coexisting attractors have essentially extreme multistability.


#### 3.2.1. Coexisting Attractors with Initial Variablex(0)

If we fix the parameters as Table1 and vary initial value x(0) in the initial conditions (x0, 0.01, 0.01, 0.01), the bifurcation diagram and corresponding Lyapunov exponent spectrum with respect to x(0) can be obtained as shown in Figure 5. System (3), with the increase of initial state x(0) in Figure 5, undergoes a complex bifurcation from chaos to period, or from period to chaos, implying the coexistence of periodic and chaotic attractors.Figure 5
Bifurcation diagram and Lyapunov exponent spectrum of coexisting oscillation with respect to initial valuex(0). (a) Bifurcation diagram. (b) Lyapunov exponent spectrum.
(a) (b)Whenx(0) is varied, many coexisting attractors appear, as shown in Figure 6, in which every subdiagram represents a pair of coexisting attractors; the blue attractor and the red attractor start from different initial values of x(0). The blue attractors start from x(0)∈(-0.85,0) and the red attractors start from x(0)∈(0,0.84). From Figure 6, we can see that there exist different coexistence attractors, including coexistence of a pair of chaotic attractors, coexistence of a pair of quasiperiodic limit cycles, and coexistence of a pair of limit cycles. Various coexisting attractors and their corresponding initial values are listed in Table 2.Table 2
Coexisting attractors for various conditions(x(0),0.01,0.01,0).
Regimes x ( 0 ) Diagrams A symmetric pair of chaotic attractors −0.800 Figure6(a), blue 0.800 Figure6(a), red A symmetric pair of quasiperiodic limit cycles −0.779 Figure6(b), blue 0.764 Figure6(b), red A symmetric pair of limit cycles −0.490 Figure6(c), blue 0.470 Figure6(c), red A symmetric pair of chaotic attractors −0.268 Figure6(d), blue 0.252 Figure6(d), red A symmetric pair of chaotic attractors −0.250 Figure6(e), blue 0.190 Figure6(e), redFigure 6
Coexisting attractors onx-y plane. (a) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.800 and x(0)=0.800. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with x(0)=-0.779 and x(0)=0.764. (c) Coexistence of a symmetric pair of limit cycles with x(0)=-0.490 and x(0)=0.470. (d) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.268 and x(0)=0.252. (e) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.250 and x(0)=0.190.
(a) (b) (c) (d) (e)


#### 3.2.2. Coexisting Attractors with Initial Variableu(0)

Similar to the abovementioned, the bifurcation diagrams and Lyapunov exponent spectrums with respect to the initial valueu(0) are described in Figure 7, in which we also fix parameters as Table 1 and vary parameter u(0) in the initial condition: (0.01, 0.01, 0.01, u0). Figures 7(a) and 7(b) indicate that the system undergoes a period-doubling bifurcation in the region u(0)∈[-1,-0.65] and then enters chaos in the region u(0)∈[-0.65,-0.59]. With the further increase of u(0), Figures 7(c) and 7(d) show that this system possesses the symmetric dynamics characteristic with respect to u(0) = 1.05 at the region u(0)∈[0.6,1.5].Figure 7
Bifurcation and Lyapunov exponents of coexisting oscillation with respect to initialu(0). ((a) and (b)) Bifurcation diagrams. ((c) and (d)) Corresponding Lyapunov exponent spectrum.
(a) (b) (c) (d)Coexisting attractors with respect to initial valueu(0) are shown in Figure 8, and the corresponding regimes and initial conditions for all attractors are listed in Table 3. Note that Figure 8(a) shows the coexistence of the three attractors, that is, fixed point, two different limited circles.Table 3
Coexisting attractors for initial condition(0.01,0.01,0.01,u(0)).
Regimes u ( 0 ) Diagrams Point attractor −0.865 and 0.865 Figure8(a), black A symmetric pair of limit cycles −0.740 Figure8(a), blue 0.760 Figure8(a), red A symmetric pair of quasiperiodic limit cycles −0.650 Figure8(b), blue 0.667 Figure8(b), red A symmetric pair of chaotic attractors −0.635 Figure8(c), blue 0.650 Figure8(c), red A symmetric pair of chaotic attractors −0.610 Figure8(d), blue 0.630 Figure8(d), redFigure 8
Coexisting attractors in thex-y plane with respect to u(0). (a) Coexistence of a point attractor and a symmetric pair of limit cycles with u(0) = −0.865 (black), u(0) = −0.740 (blue), and u(0) = 0.760 (red), respectively. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with u(0) = −0.650 (blue) and u(0) = 0.667 (red). (c) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.635 (blue) and u(0) = 0.650 (red). (d) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.610 (blue) and u(0) = 0.630 (red).
(a) (b) (c) (d)Note that the phenomenon of coexisting oscillation is different with the initial value sensitivity of usual chaotic systems. For the usual chaotic system, trajectories starting from different initial values evolve along with the time and eventually converge to the same chaotic attractor. For the proposed oscillator, however, trajectories starting from different initial values can converge eventually to different attractors, including chaotic attractors, periodic attractors, or fixed points. Hence, it seems that trajectories in the coexisting attractors are more sensitive to initial values not only for chaotic orbits but also for periodic orbits.


## 3.1. Dissipativity and Stability of Equilibrium Set

Dissipative systems will shrink to a phase space of a limited area, called the attractor. The dissipativity of the system is described as(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂u˙∂u=c+e.When the parameters are set as c=2.6 and e=-3.4, the exponential constraint rate satisfies(5)∇V=-0.8<0.It means that the volume of phase space will be contracted to zero and all trajectories of the system are confined to a subset of zero volume.The equilibrium points of system (3) can be calculated as(6)s1=x,y,z,u∣z=0,x=k1,y=-k1,u=k2,c=d,f=e,s2=x,y,z,u∣x=y=z=0,u=k3,c≠d,f≠e,where k1, k2, and k3 are arbitrary constants. Obviously, the system has two equilibrium sets, and every set has an infinite number of equilibria.If the parameters arec=d, f=e or c≠d, f≠e, the Jacobian matrix of the circuit at equilibrium sets S1 and S2 is given by (7)J1,2=cd-a-bk2,320fe-n-mk2,32011000010,where k2 and k3 have the same effect on the system, so they can be replaced by k. The characteristic equation is written as(8)λ4-c+eλ3+a+bk2+n+mk2+ce-fdλ2+af+bk2f+dn+dmk2-ae-bk2e-nc-cmk2λ=0.According to the Routh–Hurwitz criterion, the necessary and sufficient condition for stability of the system is that all the principal minors are greater than zero; that is,(9)Δ1=a1>0,Δ2=a1a0a3a2>0,Δ3=a1a00a3a2a10a4a3>0,Δ4=a1a000a3a2a1a00a4a3a2000a4>0,where (10)a0=1,a1=-c+e,a2=a+bk2+n+mk2+ce-fd,a3=af+bk2f+dn+dmk2-ae-bk2e-nc-cmk2,a4=0,whereas it is unstable and has the possibility of generating chaos under the proper parameters. If we fix a=5.8, b=1.5, c=2.6, d=0.1, e=-3.4, f=0.2, m=6.8, and n=2.8, the stability of equilibrium sets S1 and S2 depends on the constant k. Figure 4 shows the change rule of Δ1, Δ2, Δ3, and Δ4 with the constant k. Here, Δ1 is slightly greater than zero and Δ4 is equal to zero, so the black curve overlaps so much with the green curve in Figure 4. Note that the stability of an infinite number of equilibria is called extreme multistability. The following analysis will show that the stability of the all equilibrium points depends on the parameter k, resulting in chaos when k∈[-1.5,1.5].Figure 4
Curves ofΔ1, Δ2, Δ3, and Δ4 with k.


## 3.2. Coexisting Oscillations

In this section, we show that the dynamics of the system depend on not only its parameters but also its initial conditions, which leads to a special characteristic, called the coexisting oscillation in this paper. In the coexisting oscillation, many coexisting attractors appear with different initial conditions, such as coexistence of limit circle or fixed point and chaotic attractor, coexistence of chaotic and hyperchaotic attractors, and coexistence of two limits or two fixed points.Multistability is a common phenomenon in many nonlinear systems, which corresponds to the coexistence of more than one stable attractor for the same set of system parameters [31]. If infinitely many attractors coexist for the same set of system parameters, the phenomenon is called extreme multistability [32], showing that various coexisting attractors have essentially extreme multistability.


### 3.2.1. Coexisting Attractors with Initial Variablex(0)

If we fix the parameters as Table1 and vary initial value x(0) in the initial conditions (x0, 0.01, 0.01, 0.01), the bifurcation diagram and corresponding Lyapunov exponent spectrum with respect to x(0) can be obtained as shown in Figure 5. System (3), with the increase of initial state x(0) in Figure 5, undergoes a complex bifurcation from chaos to period, or from period to chaos, implying the coexistence of periodic and chaotic attractors.Figure 5
Bifurcation diagram and Lyapunov exponent spectrum of coexisting oscillation with respect to initial valuex(0). (a) Bifurcation diagram. (b) Lyapunov exponent spectrum.
(a) (b)Whenx(0) is varied, many coexisting attractors appear, as shown in Figure 6, in which every subdiagram represents a pair of coexisting attractors; the blue attractor and the red attractor start from different initial values of x(0). The blue attractors start from x(0)∈(-0.85,0) and the red attractors start from x(0)∈(0,0.84). From Figure 6, we can see that there exist different coexistence attractors, including coexistence of a pair of chaotic attractors, coexistence of a pair of quasiperiodic limit cycles, and coexistence of a pair of limit cycles. Various coexisting attractors and their corresponding initial values are listed in Table 2.Table 2
Coexisting attractors for various conditions(x(0),0.01,0.01,0).
Regimes x ( 0 ) Diagrams A symmetric pair of chaotic attractors −0.800 Figure6(a), blue 0.800 Figure6(a), red A symmetric pair of quasiperiodic limit cycles −0.779 Figure6(b), blue 0.764 Figure6(b), red A symmetric pair of limit cycles −0.490 Figure6(c), blue 0.470 Figure6(c), red A symmetric pair of chaotic attractors −0.268 Figure6(d), blue 0.252 Figure6(d), red A symmetric pair of chaotic attractors −0.250 Figure6(e), blue 0.190 Figure6(e), redFigure 6
Coexisting attractors onx-y plane. (a) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.800 and x(0)=0.800. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with x(0)=-0.779 and x(0)=0.764. (c) Coexistence of a symmetric pair of limit cycles with x(0)=-0.490 and x(0)=0.470. (d) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.268 and x(0)=0.252. (e) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.250 and x(0)=0.190.
(a) (b) (c) (d) (e)


### 3.2.2. Coexisting Attractors with Initial Variableu(0)

Similar to the abovementioned, the bifurcation diagrams and Lyapunov exponent spectrums with respect to the initial valueu(0) are described in Figure 7, in which we also fix parameters as Table 1 and vary parameter u(0) in the initial condition: (0.01, 0.01, 0.01, u0). Figures 7(a) and 7(b) indicate that the system undergoes a period-doubling bifurcation in the region u(0)∈[-1,-0.65] and then enters chaos in the region u(0)∈[-0.65,-0.59]. With the further increase of u(0), Figures 7(c) and 7(d) show that this system possesses the symmetric dynamics characteristic with respect to u(0) = 1.05 at the region u(0)∈[0.6,1.5].Figure 7
Bifurcation and Lyapunov exponents of coexisting oscillation with respect to initialu(0). ((a) and (b)) Bifurcation diagrams. ((c) and (d)) Corresponding Lyapunov exponent spectrum.
(a) (b) (c) (d)Coexisting attractors with respect to initial valueu(0) are shown in Figure 8, and the corresponding regimes and initial conditions for all attractors are listed in Table 3. Note that Figure 8(a) shows the coexistence of the three attractors, that is, fixed point, two different limited circles.Table 3
Coexisting attractors for initial condition(0.01,0.01,0.01,u(0)).
Regimes u ( 0 ) Diagrams Point attractor −0.865 and 0.865 Figure8(a), black A symmetric pair of limit cycles −0.740 Figure8(a), blue 0.760 Figure8(a), red A symmetric pair of quasiperiodic limit cycles −0.650 Figure8(b), blue 0.667 Figure8(b), red A symmetric pair of chaotic attractors −0.635 Figure8(c), blue 0.650 Figure8(c), red A symmetric pair of chaotic attractors −0.610 Figure8(d), blue 0.630 Figure8(d), redFigure 8
Coexisting attractors in thex-y plane with respect to u(0). (a) Coexistence of a point attractor and a symmetric pair of limit cycles with u(0) = −0.865 (black), u(0) = −0.740 (blue), and u(0) = 0.760 (red), respectively. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with u(0) = −0.650 (blue) and u(0) = 0.667 (red). (c) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.635 (blue) and u(0) = 0.650 (red). (d) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.610 (blue) and u(0) = 0.630 (red).
(a) (b) (c) (d)Note that the phenomenon of coexisting oscillation is different with the initial value sensitivity of usual chaotic systems. For the usual chaotic system, trajectories starting from different initial values evolve along with the time and eventually converge to the same chaotic attractor. For the proposed oscillator, however, trajectories starting from different initial values can converge eventually to different attractors, including chaotic attractors, periodic attractors, or fixed points. Hence, it seems that trajectories in the coexisting attractors are more sensitive to initial values not only for chaotic orbits but also for periodic orbits.


## 3.2.1. Coexisting Attractors with Initial Variablex(0)

If we fix the parameters as Table1 and vary initial value x(0) in the initial conditions (x0, 0.01, 0.01, 0.01), the bifurcation diagram and corresponding Lyapunov exponent spectrum with respect to x(0) can be obtained as shown in Figure 5. System (3), with the increase of initial state x(0) in Figure 5, undergoes a complex bifurcation from chaos to period, or from period to chaos, implying the coexistence of periodic and chaotic attractors.Figure 5
Bifurcation diagram and Lyapunov exponent spectrum of coexisting oscillation with respect to initial valuex(0). (a) Bifurcation diagram. (b) Lyapunov exponent spectrum.
(a) (b)Whenx(0) is varied, many coexisting attractors appear, as shown in Figure 6, in which every subdiagram represents a pair of coexisting attractors; the blue attractor and the red attractor start from different initial values of x(0). The blue attractors start from x(0)∈(-0.85,0) and the red attractors start from x(0)∈(0,0.84). From Figure 6, we can see that there exist different coexistence attractors, including coexistence of a pair of chaotic attractors, coexistence of a pair of quasiperiodic limit cycles, and coexistence of a pair of limit cycles. Various coexisting attractors and their corresponding initial values are listed in Table 2.Table 2
Coexisting attractors for various conditions(x(0),0.01,0.01,0).
Regimes x ( 0 ) Diagrams A symmetric pair of chaotic attractors −0.800 Figure6(a), blue 0.800 Figure6(a), red A symmetric pair of quasiperiodic limit cycles −0.779 Figure6(b), blue 0.764 Figure6(b), red A symmetric pair of limit cycles −0.490 Figure6(c), blue 0.470 Figure6(c), red A symmetric pair of chaotic attractors −0.268 Figure6(d), blue 0.252 Figure6(d), red A symmetric pair of chaotic attractors −0.250 Figure6(e), blue 0.190 Figure6(e), redFigure 6
Coexisting attractors onx-y plane. (a) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.800 and x(0)=0.800. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with x(0)=-0.779 and x(0)=0.764. (c) Coexistence of a symmetric pair of limit cycles with x(0)=-0.490 and x(0)=0.470. (d) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.268 and x(0)=0.252. (e) Coexistence of a symmetric pair of chaotic attractors with x(0)=-0.250 and x(0)=0.190.
(a) (b) (c) (d) (e)


## 3.2.2. Coexisting Attractors with Initial Variableu(0)

Similar to the abovementioned, the bifurcation diagrams and Lyapunov exponent spectrums with respect to the initial valueu(0) are described in Figure 7, in which we also fix parameters as Table 1 and vary parameter u(0) in the initial condition: (0.01, 0.01, 0.01, u0). Figures 7(a) and 7(b) indicate that the system undergoes a period-doubling bifurcation in the region u(0)∈[-1,-0.65] and then enters chaos in the region u(0)∈[-0.65,-0.59]. With the further increase of u(0), Figures 7(c) and 7(d) show that this system possesses the symmetric dynamics characteristic with respect to u(0) = 1.05 at the region u(0)∈[0.6,1.5].Figure 7
Bifurcation and Lyapunov exponents of coexisting oscillation with respect to initialu(0). ((a) and (b)) Bifurcation diagrams. ((c) and (d)) Corresponding Lyapunov exponent spectrum.
(a) (b) (c) (d)Coexisting attractors with respect to initial valueu(0) are shown in Figure 8, and the corresponding regimes and initial conditions for all attractors are listed in Table 3. Note that Figure 8(a) shows the coexistence of the three attractors, that is, fixed point, two different limited circles.Table 3
Coexisting attractors for initial condition(0.01,0.01,0.01,u(0)).
Regimes u ( 0 ) Diagrams Point attractor −0.865 and 0.865 Figure8(a), black A symmetric pair of limit cycles −0.740 Figure8(a), blue 0.760 Figure8(a), red A symmetric pair of quasiperiodic limit cycles −0.650 Figure8(b), blue 0.667 Figure8(b), red A symmetric pair of chaotic attractors −0.635 Figure8(c), blue 0.650 Figure8(c), red A symmetric pair of chaotic attractors −0.610 Figure8(d), blue 0.630 Figure8(d), redFigure 8
Coexisting attractors in thex-y plane with respect to u(0). (a) Coexistence of a point attractor and a symmetric pair of limit cycles with u(0) = −0.865 (black), u(0) = −0.740 (blue), and u(0) = 0.760 (red), respectively. (b) Coexistence of a symmetric pair of quasiperiodic limit cycles with u(0) = −0.650 (blue) and u(0) = 0.667 (red). (c) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.635 (blue) and u(0) = 0.650 (red). (d) Coexistence of a symmetric pair of chaotic attractors with u(0) = −0.610 (blue) and u(0) = 0.630 (red).
(a) (b) (c) (d)Note that the phenomenon of coexisting oscillation is different with the initial value sensitivity of usual chaotic systems. For the usual chaotic system, trajectories starting from different initial values evolve along with the time and eventually converge to the same chaotic attractor. For the proposed oscillator, however, trajectories starting from different initial values can converge eventually to different attractors, including chaotic attractors, periodic attractors, or fixed points. Hence, it seems that trajectories in the coexisting attractors are more sensitive to initial values not only for chaotic orbits but also for periodic orbits.


## 4. Dimension Reduction Analysis

In order to solve the 3D equations (1), we must add an equation of σm which is an internal variable of the memcapacitor and obtain the 4D equations (3). However, this makes it difficult to solve the 4D equations. In general, if there is one memory element increased into a circuit, there will be two state equations added to the system equations, in which one is the equation corresponding to a circuit state variable and the other corresponds to the internal variable for the memory element.In order to reduce the complexity of solving (3), we integrate directly (1) on both sides and then the 3D equations can be obtained:(11)L1dq1dt=q1+q2R0-φm-q1R1,L2dq2dt=q1+q2R0-φm-q2R2,dσmdt=q1+q2+q0,where q0 is the initial charge of the memcapacitor and q1, q2, φm, and σm are the time integrals of i1, i2, vm, and qm, respectively.For convenience, letx=q1, y=q2, z=σm, a=a/L1, b=β/L1, c=R0-R1/L1, d=R0/L1, e=R0-R2/L2, f=R0/L2, m=β/L2, n=α/L2, and q0=0; (11) can be transformed as follows:(12)dxdt=cx+dy-az-bz3,dydt=fx+ey-nz-mz3,dzdt=x+y.We now present the attracting basins shown in Figure 9 in the x(0)-y(0) plane and the x(0)-z(0) plane for observing the dynamics of the memcapacitor-based circuit. Here, we fix a=6.6, b=1, c=2.5, d=0.25, e=-3.41, f=0.25, m=5.2, and n=2. And the initial conditions are (x(0),y(0),0.01) and (x(0),0.01,z(0)) for Figures 9(a) and 9(b), respectively. Each attracting basin is divided into blue region and red region, corresponding to two types of coexisting attractors, which are shown in Figure 10.Figure 9
Cross sections of attracting basins for two chaotic attractors (blue and red). (a)x(0)-y(0) plane. (b) x(0)-z(0) plane.
(a) (b)Figure 10
Coexisting chaotic attractors with initial conditions (0.01, 0.01, 0.01) (blue) and (0.01, 0.01, −0.01) (red). (a)x-y plane, (b) x-z plane cycles, and (c) y-z plane.
(a) (b) (c)The initial chargeq0 of the memcapacitor plays an important role in the circuit. The bifurcation diagram and corresponding Lyapunov exponent spectrum with respect to q0 are presented in Figure 11 with the initial condition (0.01, 0.01, 0.01) and the parameters a=6.6, b=1, c=2.5, d=0.25, e=-3.41, f=0.25, m=5.2, and n=2.Figure 11
Bifurcation Lyapunov exponents with respect toq0. (a) Bifurcation diagram. (b) Lyapunov exponent spectrum.
(a) (b)Figure11(a) shows a complex bifurcation process, which has typical period 3 window, and also possesses novel period 1 and period 2 windows. Furthermore, there exist period-doubling and inverse period-doubling bifurcations in both sides of the period 2 window. Figures 12(a)–12(d) give the typical attractors in the y-z plane with q0=0.01, q0=0.05, q0=0.11, and q0=0.29.Figure 12
Evolved attractors in they-z plane with respect to q0. (a) Chaotic attractor with q0 = 0.01. (b) Period 2 attractor with q0 = 0.05. (c) Chaotic attractor with q0 = 0.11. (d) Period 2 attractor with q0 = 0.29.
(a) (b) (c) (d)


## 5. Analog Circuit Experiment

An experimental circuit is designed to realize the memcapacitor-based chaotic oscillator. The designed circuit for realizing (12) is shown in Figure 13, and its corresponding state equations are written as(13)dxdτ=-1R1C1-x-1R2C1-y-1R3C1z-1100R4C1z3,dydτ=-1R5C2-x-1R6C2y-1R7C2z-1100R8C2z3,dzdτ=-1R9C3-x-1R10C3-y.Figure 13
Circuit diagram of implementing (3).Circuit parameters can be obtained asR1=40kΩ, R2=400kΩ, R3=15kΩ, R4=1kΩ, R5=400kΩ, R6≈30kΩ, R7=50kΩ, R8≈180Ω, and R9=R10=100kΩ, when we set C1=C2=C3=C4=10nF, and R11=R12=R13=R14=100kΩ. In this circuit, the operational amplifier and the analog multiplier are selected as LF347N and AD633, respectively. The voltage references are ±VCC = ±15 V. The experimental chaotic attractors observed via the digital oscilloscope (DSO3102A) are shown in Figure 14.Figure 14
Experimental attractors and setup. (a)x-y plane. (b) x-z plane. (c) y-z plane. (d) Experimental setup.
(a) (b) (c) (d)Note that the circuit shown in Figure13 and used in the experiment is designed based on (12), which is a set of dimension reduction (three-dimensional) equations which is obtained from five-dimensional equation (3). Compared with other memcapacitor-based chaotic oscillators, the advantages of this experiment are that the experimental circuit is simple and the attractors or time-domain waveforms with respect to charge (q1 and q2) through the inductor (L1 and L2) can be observed directly from the experimental oscilloscope.


## 6. Conclusion

In this paper, a new chaotic oscillator based on charge-controlled memcapacitor is designed. The dynamical characteristics with respect to the variations of initial conditions are investigated, and then the complex dynamic characteristics, including equilibrium set or an infinite number of equilibrium points, extreme multistability of the equilibrium set, and coexisting attractors, are found. Finally, chaotic attractors are captured experimentally by the analog circuit of implementing this oscillator. The most prominent feature of this system is the coexisting oscillation and coexisting attractors. It is concluded that one coexisting oscillation system can be as multi-random signal seeds to design different chaotic pseudorandom number generators, which can be used in secret communications, information encryptions, and various pseudorandom number generation.



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