# Heterogeneous and Homogenous Multistabilities in a Novel 4D Memristor-Based Chaotic System with Discrete Bifurcation Diagrams

Authors: Lilian Huang; Wenju Yao; Jianhong Xiang; Zefeng Zhang
Journal: Complexity
 (2020)

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## Abstract

In this paper, a new 4D memristor-based chaotic system is constructed by using a smooth flux-controlled memristor to replace a resistor in the realization circuit of a 3D chaotic system. Compared with general chaotic systems, the chaotic system can generate coexisting infinitely many attractors. The proposed chaotic system not only possesses heterogeneous multistability but also possesses homogenous multistability. When the parameters of system are fixed, the chaotic system only generates two kinds of chaotic attractors with different positions in a very large range of initial values. Different from other chaotic systems with continuous bifurcation diagrams, this system has discrete bifurcation diagrams when the initial values change. In addition, this paper reveals the relationship between the symmetry of coexisting attractors and the symmetry of initial values in the system. The dynamic behaviors of the new system are analyzed by equilibrium point and stability, bifurcation diagrams, Lyapunov exponents, and phase orbit diagrams. Finally, the chaotic attractors are captured through circuit simulation, which verifies numerical simulation.


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## Body

## 1. Introduction

Memristor was first proposed by Chua [1] in 1971 and is the fourth basic electronic component manufactured by HP Labs in 2008 [2]. The discovery of memristors has caused an upsurge in studying and applying memristors. Due to the nonlinearity of memristor, it has been applied in many fields, such as flash memory [2, 3], neuromorphic computing [4, 5], neural network [6, 7], and chaotic system [8–11] based on chaos synchronization for encryption algorithms [12, 13] and secure communication [14, 15].Memristor is a nonlinear element, and its resistance depends on the voltage or current signal, so it has been widely used in the construction of chaotic circuits in recent years [16–19]. In 2008, Itoh and Chua proposed together a Chua’s chaotic circuit based on memristor. The dynamic analysis results show that Chua’s chaotic circuit based on memristors has more complex dynamic characteristics than classic Chua’s chaotic circuit [17]. In 2010, a Chua’s chaotic circuit based on memristance was proposed by replacing Chua’s diode with a smooth flux-controlled memristor and a negative conductance [18]. In 2017, a multiscroll hyperchaotic system was proposed by introducing the memristor into the jerk multiscroll system, and the numbers of scrolls can be controlled by adjusting the coefficient before the term related to memristor [19].In recent years, multistability [20–25] and extreme multistability [26–32] have become research hotspots in the field of chaotic systems. Multistability means that when the system parameters remain unchanged, the system can generate more than one attractor with different initial values. When the number of attractors is infinite, this phenomenon is called extreme multistability. Coexisting attractors and hidden coexisting attractors are shown in a memristive system with many equilibrium points in reference [23]. A wing-variable chaotic system with coexisting twin-wing attractors is proposed by replacing one of the resistors of the pseudo-four-wing chaotic system with a memristor in reference [24]. In the same year, a memristor-based chaotic system is constructed by introducing an ideal flux-controlled memristor with absolute value nonlinearity into an existing hypogenetic chaotic jerk system, which can exhibit the extreme multistability phenomenon in reference [31]. A simplest third-order memristive chaotic system with hidden attractors is proposed, which exhibits the extreme multistability phenomenon of coexisting infinitely many attractors in reference [32].Although multistability and extreme multistability in memristive chaotic systems had been reported in many existing papers, most of them researched the heterogeneous multistability of chaotic systems, and homogenous multistability was rarely reported. Heterogeneous multistability means that under the same parameters, the chaotic system has some chaotic attractors with different structures, while homogenous multistability means the chaotic system can generate attractors with the same structure, but the amplitudes and positions of their attractors can be different. In this article, the mathematical model of a memristor is employed to construct the chaotic system owning heterogeneous and homogenous multistabilities. Besides, the presented memristor-based system displays other complex dynamic characteristics, including constant Lyapunov exponents, discrete bifurcation diagrams, the symmetry of coexisting attractors, and so on.The rest of this paper is organized as follows. InSection 2, a new chaotic system based on the model of a memristor is studied. And basic properties of the proposed system are investigated, including symmetry and dissipation, equilibrium, and stability. In Section 3, complex dynamic behaviors of the memristor-based chaotic system are analyzed. In Section 4, extreme multistability of the chaotic system is investigated by bifurcation diagrams and Lyapunov exponent spectra, and the dynamic analysis results show that the chaotic system possesses not only heterogeneous multistability but also homogenous multistability. In Section 5, the presented chaotic system is realized by analog circuit and the experimental results are given. Finally, some conclusions are drawn.


## 2. Basic Properties of the Memristive Chaotic System

A 3D pseudo-four-wing chaotic system was proposed by Liu and Chen [33, 34]. Actually, it is a coexisting two-wing system, and it can be described as follows:(1)x˙=ax−byz,y˙=−cy+xz,z˙=−dz+xy,where a, b, c, and d are all constants and x, y, and z are the state variables.By utilizing a smooth flux-controlled memristor to substitute a resistor in realization circuit of system (1), a novel 4D memristor-based chaotic system is given by(2)x˙=ax−byz,y˙=−cy+xz+eWwx,z˙=−dz+xy,w˙=hx,where e and h are positive parameters and Ww is a memductance function.The memductance functionWw is shown as(3)Ww=f+3gw2,where f and g are two positive constants and w is the state variable.


### 2.1. Symmetry and Dissipativity

The symmetry property of chaotic systems is an important property. The memristive chaotic system (2) is invariant if we do the transformation x,y,z,w⟶−x,−y,z,−w, which means system (2) has to be symmetric with respect to z axis in state space.The dissipativity of system (2) is expressed by the following formula:(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w=a−c−d.Whena, c, and d satisfy condition a−c−d<0, the system is dissipative. It means that the volume of phase space will be contracted to zero in exponential form e−a−c−d and all trajectories of the system are confined to zero volume.


### 2.2. Equilibria and Stability

Let the terms on left-hand side of system (2) be zero, and we can easily observe that the system has a line equilibrium(5)O=x,y,z,wx=y=z=0,w=k,where k is any real constant.By linearizing system (2) at point O, we can obtain the Jacobian matrix of the equation on O.(6)Jo=a000eWk−c0000−d0h000.According to the Jacobian matrix (6), the characteristic equation can be obtained as follows:(7)λλ−aλ+cλ+d=0.We can solve its eigenvalues easily, and they can be expressed by(8)λ1=0,λ2=a,λ3=−c,λ4=−d.The values ofa, c, and d are all positive, so λ3 and λ4 are always negative, and λ2 is always positive. Therefore, system (2) has an unstable saddle point.


## 2.1. Symmetry and Dissipativity

The symmetry property of chaotic systems is an important property. The memristive chaotic system (2) is invariant if we do the transformation x,y,z,w⟶−x,−y,z,−w, which means system (2) has to be symmetric with respect to z axis in state space.The dissipativity of system (2) is expressed by the following formula:(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w=a−c−d.Whena, c, and d satisfy condition a−c−d<0, the system is dissipative. It means that the volume of phase space will be contracted to zero in exponential form e−a−c−d and all trajectories of the system are confined to zero volume.


## 2.2. Equilibria and Stability

Let the terms on left-hand side of system (2) be zero, and we can easily observe that the system has a line equilibrium(5)O=x,y,z,wx=y=z=0,w=k,where k is any real constant.By linearizing system (2) at point O, we can obtain the Jacobian matrix of the equation on O.(6)Jo=a000eWk−c0000−d0h000.According to the Jacobian matrix (6), the characteristic equation can be obtained as follows:(7)λλ−aλ+cλ+d=0.We can solve its eigenvalues easily, and they can be expressed by(8)λ1=0,λ2=a,λ3=−c,λ4=−d.The values ofa, c, and d are all positive, so λ3 and λ4 are always negative, and λ2 is always positive. Therefore, system (2) has an unstable saddle point.


## 3. Dynamics of the Memristor-Based Chaotic System

### 3.1. Phase Portraits and Lyapunov Exponents

The Lyapunov exponent spectra are an effective way of judging whether the system has chaotic behavior. The main characteristics of nonlinear dynamical systems can be described by the number of positive Lyapunov exponents. When the system has only one Lyapunov exponent greater than zero, the system has chaotic dynamic behavior. When a nonlinear system has more than two or equal to two Lyapunov exponents greater than zero, the system has hyperchaotic dynamic behavior.When the parameters of the chaotic system are set asa=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1 and the initial conditions are set to 1,1,0,0, system (2) can generate chaotic attractor as shown in Figure 1. The corresponding Lyapunov exponents are computed as 0.5161, −0.0104, −0.0645, and −21.5665, and the Lyapunov dimension dL=3.0204, which indicates the system has chaotic behavior.Figure 1
Phase portraits of system (2) when a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1: (a) projection on x−y plane, (b) projection on x−z plane, (c) projection on y−z plane, and (d) 3D view in the x−y−z space.
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### 3.2. Poincaré Projection

The Poincaré projection is another effective way of judging whether the system has chaotic behaviour. Take projectionsx=0, y=0, z=6, and w=1.4, respectively, and the system obtains the Poincaré projections as shown in Figure 2. A large area of points can be observed in these pictures, which indicate that the system has chaotic behaviour.Figure 2
Poincaré sections of system (2): (a) projection on y−w plane when Poincaré section is x=0, (b) projection on x−w plane when Poincaré section is y=0, (c) projection on x−w plane when Poincaré section is z=6, and (d) projection on x−y plane when Poincaré section is w=1.4.
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## 3.1. Phase Portraits and Lyapunov Exponents

The Lyapunov exponent spectra are an effective way of judging whether the system has chaotic behavior. The main characteristics of nonlinear dynamical systems can be described by the number of positive Lyapunov exponents. When the system has only one Lyapunov exponent greater than zero, the system has chaotic dynamic behavior. When a nonlinear system has more than two or equal to two Lyapunov exponents greater than zero, the system has hyperchaotic dynamic behavior.When the parameters of the chaotic system are set asa=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1 and the initial conditions are set to 1,1,0,0, system (2) can generate chaotic attractor as shown in Figure 1. The corresponding Lyapunov exponents are computed as 0.5161, −0.0104, −0.0645, and −21.5665, and the Lyapunov dimension dL=3.0204, which indicates the system has chaotic behavior.Figure 1
Phase portraits of system (2) when a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1: (a) projection on x−y plane, (b) projection on x−z plane, (c) projection on y−z plane, and (d) 3D view in the x−y−z space.
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## 3.2. Poincaré Projection

The Poincaré projection is another effective way of judging whether the system has chaotic behaviour. Take projectionsx=0, y=0, z=6, and w=1.4, respectively, and the system obtains the Poincaré projections as shown in Figure 2. A large area of points can be observed in these pictures, which indicate that the system has chaotic behaviour.Figure 2
Poincaré sections of system (2): (a) projection on y−w plane when Poincaré section is x=0, (b) projection on x−w plane when Poincaré section is y=0, (c) projection on x−w plane when Poincaré section is z=6, and (d) projection on x−y plane when Poincaré section is w=1.4.
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## 4. Extreme Multistability in the Memristor-Based Chaotic System

### 4.1. Dynamic Analysis of Heterogeneous Multistability

Heterogeneous multistability means under the same parameters, a system can generate several or even infinitely many coexisting attractors with different structures, while homogenous multistability means that a chaotic system can generate the same structure coexisting attractors but with different positions or amplitudes.In system (2), when the parameters are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1 and initial conditions are set as 1,0,0,w0, the system can generate various coexisting attractors depending on w0. The typical chaotic attractors are shown in Figure 3. Besides, system (2) can generate other kinds of coexisting attractors, coexisting limit cycle attractors, and symmetric limit cycle attractor as shown in Figures 4, 5, and 6, respectively.Figure 3
Phase portraits of coexisting infinitely many attractors in (a)y−w and (b) planes (the blue one starts from initial conditions 1,0,0,1, the red one starts from 1,0,0,−1, the yellow one starts from 1,0,0,4, and the black one starts from 1,0,0,−4).
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(b)Figure 4
Phase portraits of coexisting attractors in (a)x−y and (b)x−z planes (the blue one starts from initial conditions 1,0,0,22 and the red one starts from 1,0,0,−22).
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(b)Figure 5
Phase portraits of coexisting limit cycle attractors in (a)x−y and (b)x−z planes (the blue one starts from initial conditions 1,0,0,26 and the red one starts from 1,0,0,−26).
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(b)Figure 6
Phase portraits of limit cycle in (a)x−y and (b)x−z planes for initial conditions set as 1,0,0,30.
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(b)Whenw0 is changed in the region −50,50, the bifurcation diagram of the state variable w and Lyapunov exponent spectra are shown in Figures 7(a) and 7(b), respectively. As shown in Figure 7(a), the bifurcation diagram of the state variable w is almost linear. It can be seen from Figure 7(b) that the chaotic attractor with a positive Lyapunov exponent is mainly located at the region −36,−32, −22,22, and 32,35 (the last Lyapunov exponent is not displayed because it is always a big negative number). And system (2) can also generate other kinds of attractors and limit cycle attractors, which means the system has heterogeneous multistability.Figure 7
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to memristor initial condition w0.
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### 4.2. Dynamic Analysis of Homogenous Multistability

The parameters of system (2) remain unchanged, and the initial conditions are set as x0,1,0,0. When x0 is varied in the region −104,104, the bifurcation diagrams of the state variable z, the state variable w, and its Lyapunov exponent spectra are plotted in Figures 8(a), 8(b), and 8(c), respectively. Besides, when x0 is varied in the region −0.1,0.1, Lyapunov exponent spectra are plotted in Figure 8(d).Figure 8
(a) Bifurcation diagram of the state variablez, (b) bifurcation diagram of the state variable w, (c) Lyapunov spectrum with respect to initial condition x0 in the region −104,104, and (d) Lyapunov spectrum with respect to initial condition x0 in the region −0.1,0.1.
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(d)It can be seen from Figure8(a) that when the initial condition x0 is varied in the region −104,104, the state variable z shows two kinds of steady chaotic states. When x0 is varied in the region −104,0, state variable z is located below or above the z axis, while x0 is varied in the region 0,104, the state variable z is all located above the z axis. From Figure 8(b), when the initial condition x0 is varied in the region −104,104, there are many discrete small line segments in bifurcation diagram of the state variable w, which means the state variable w exists in infinite steady chaotic states. And this phenomenon also indicates the system can generate coexisting infinitely many attractors, which means system (2) has the extreme multistability. Different from other systems possessing extreme multistability, system (2) only generates two kinds of chaotic attractors with different positions in a very large range of initial values, which are distributed along the w axis parallelly. As shown in Figure 8(c), it is obvious that the four Lyapunov exponents are always approximately constant and the largest Lyapunov exponent is always positive except for the zero point when x0 is varied in the region −104,104, which means system (2) can exhibit chaotic behavior except zero point. Figure 8(d) shows that the region cannot exhibit chaotic behavior which is very small.Considering the particularity of system (2), it is necessary to discuss the system when x0=1. The control parameters of system (2) remain unchanged, and the initial conditions are set as 0,1,0,0. The LEs are 4.0002, −0.2034, −5.0005, and −19.8272, which means system (2) cannot exhibit chaotic behavior under this circumstance.It has been confirmed that there are coexisting infinitely many attractors in chaotic system (2) according to the above analysis. When x0 is set to 10, −10, 50, +50, 100, −100, and 1, respectively, the phase portraits of coexisting infinitely many attractors in the x−w plane, the y−w plane, the z−w plane, and the w−z−x space are shown in Figures 9(a), 9(b), 9(c), and 9(d), respectively. Figure 9 clearly reveals the coexistence of a large number of same attractors with different positions, which implies the emergence of homogenous multistability. This result of the phase portraits is consistent with the bifurcation diagrams and Lyapunov spectrum with respect to initial condition x0.Figure 9
Phase portraits of coexisting infinitely many attractors on (a)x−w plane, (b)y−w plane, (c)z−w plane, and (d)w−z−x plane in chaotic system (2) (the blue one starts from initial conditions 10,1,0,0, the red one starts from −10,1,0,0, the pink one starts from 50,1,0,0, the cyan one starts from −50,1,0,0, the yellow one starts from 100,1,0,0, the black one starts from −100,1,0,0, and the green one starts from 1,1,0,0).
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(d)The parameters of system (2) are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the initial conditions are set as 1,y0,0,0, and the initial condition y0 is used as the independent variable of bifurcation diagram. When y0 is changed in the region −104,104, the bifurcation diagram of the state variable w and its Lyapunov exponent spectra are plotted in Figures 10(a) and 10(b), respectively. As can be seen from Figure 10(a), when the initial condition y0 is changed in the region −104,104, there are many discrete small line segments in bifurcation diagram of the state variable w, which implies that there are coexisting infinitely many attractors in system (2). It can be clearly seen from Figure 10(b) that four Lyapunov exponents are always approximately constant and largest Lyapunov exponent is always positive when is y0 changed in the region −104,104, which means system (2) only exhibits chaotic behavior.Figure 10
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to initial condition y0 in the region −104,104.
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(b)The parameters of system (2) are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the initial conditions are set as 1,0,z0,0, and the initial condition z0 is used as the independent variable of bifurcation diagram. When z0 is changed in the region −104,104, the bifurcation diagram of the state variable w and its Lyapunov exponent spectra are plotted in Figures 11(a) and 11(b), respectively. The bifurcation diagram of the state variable w and its Lyapunov exponent spectra are similar with those with the initial conditions set as 1,y0,0,0. Many discrete small line segments are more closely clustered in the bifurcation diagram of the state variable w for the initial conditions set as 1,0,z0,0 compared with the one for the initial conditions set as 1,y0,0,0. And Lyapunov exponent spectra for the initial conditions set as (1, 0, z (0), 0) are similar with the ones for the initial conditions set as 1,y0,0,0, and four Lyapunov exponents are always approximately constant and largest Lyapunov exponent is always positive when z0 is changed in the region −104,104.Figure 11
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to initial condition z0 in the region −104,104.
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### 4.3. Symmetry in Infinitely Many Coexisting Attractors

Obviously, system (2) is invariant if we do the transformation x,y,z,w⟶−x,−y,z,−w, which means that x,y,z,w and −x,−y,z,−w are all solutions to the equation of the system. This symmetry characteristic of system (2) could be served to explain the presence of symmetric coexisting attractors in state space.If we set parameters asa=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the system can generate many pairs of symmetric coexisting attractors for the corresponding conditions set as x0,y0,z0,w0 and −x0,−y0,z0,−w0. And two pairs of symmetric coexisting attractors of these are shown in Figure 12, where the blue one and the red one are a pair of symmetric attractors and the pink one and the cyan one are another pair of symmetric attractors. The projections of the symmetric attractors on the coordinate planes can be shown in the form of central symmetry or axial symmetry. In system (2), it is centrosymmetric on the x−y plane, the x−w plane, and the y−w plane and axisymmetric on the x−z plane, the y−z plane, and the z−w plane. It is worth noting that the structures of many coexisting attractors in Figure 9 are the same roughly, and there are differences in the details. But the structures of each pairs of coexisting attractors in Figure 12 are symmetric exactly.Figure 12
Phase portraits of symmetric coexisting infinitely many attractors in chaotic system (2) (the blue one starts from initial conditions 1,1,−10,1, the red one starts from −1,−1,−10,−1, the pink one starts from 10,20,15,2, and the cyan one starts from −10,−20,15,−2).
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(f)In order to further verify symmetry of coexisting infinitely many attractors in chaotic system (2), bifurcation diagrams of the state variable z and the state variable w are given. The parameters of system (2) are assigned as a=4, b=6, c=20, d=5, e=0.01, f=0.1, 3g=0.1, and h=0.1, and the initial conditions are set as 1,1,z0,1, and the initial condition z0 is used as the independent variable of bifurcation diagram. When z0 is varied in the region [–100, 100], the bifurcation diagrams of the state variable z and the state variable w are plotted in Figures 13(a) and 13(b), respectively. Similarly, the parameters of system (2) remain unchanged, the initial conditions are set as −1,−1,z0,−1, and the initial condition z0 is used as the independent variable. When z0 is varied in the region 100,−100, the bifurcation diagrams of the state variable z and the state variable w are plotted in Figures 14(a) and 14(b), respectively.Figure 13
(a) Bifurcation diagram of the state variablez and (b) bifurcation diagram of the state variable w with respect to initial conditions 1,1,z0,1.
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(a) Bifurcation diagram of the state variablez and (b) bifurcation diagram of the state variable w with respect to initial conditions −1,−1,z0,−1.
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(b)A comparison of the Figures13(a) and 14(a) indicates that bifurcation diagrams of the state variable z for the initial conditions set as 1,1,z0,1 and −1,−1,z0,−1 are the same exactly. And a comparison of the Figures 13(b) and 14(b) shows that they are symmetric about the horizontal axis. This phenomenon reflects the fact that when the parameters of system (2) set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, h=0.1, the structures of coexisting attractors with conditions set as 1,1,z0,1 and −1,−1,z0,−1 are symmetric with respect to z coordinate axis.We compared the advantages and disadvantages of this system with other improved Liu-and-Chen systems in the literature [11, 23, 24]. The system proposed in reference [11] is a three-dimensional chaotic system without memristor. And it can generate three-scroll and four-scroll chaotic attractors. The system in reference [23] is a four-dimensional chaotic system with two memristors, which can generate various kinds of attractors and hidden attractors. The system in reference [24] is a four-dimensional chaotic system with a memristor, which can generate three-wing, four-wing, and coexisting two-wing chaotic attractors. It has line equilibrium points. This system has a line equilibrium, within which the attractors generated are hidden. The new system is a four-dimensional chaotic system with a memristor, which can generate coexisting infinitely many attractors distributed along the w axis parallelly. It produces hidden attractors because it has a line equilibrium. In addition, the system has discrete bifurcation diagrams and many symmetrical coexisting attractors. Compared with above improved Liu-and-Chen systems, we can find that chaotic systems with memristor have more complex dynamic behaviors than chaotic systems without memristor and are more likely to have extreme multistability.


## 4.1. Dynamic Analysis of Heterogeneous Multistability

Heterogeneous multistability means under the same parameters, a system can generate several or even infinitely many coexisting attractors with different structures, while homogenous multistability means that a chaotic system can generate the same structure coexisting attractors but with different positions or amplitudes.In system (2), when the parameters are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1 and initial conditions are set as 1,0,0,w0, the system can generate various coexisting attractors depending on w0. The typical chaotic attractors are shown in Figure 3. Besides, system (2) can generate other kinds of coexisting attractors, coexisting limit cycle attractors, and symmetric limit cycle attractor as shown in Figures 4, 5, and 6, respectively.Figure 3
Phase portraits of coexisting infinitely many attractors in (a)y−w and (b) planes (the blue one starts from initial conditions 1,0,0,1, the red one starts from 1,0,0,−1, the yellow one starts from 1,0,0,4, and the black one starts from 1,0,0,−4).
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Phase portraits of coexisting attractors in (a)x−y and (b)x−z planes (the blue one starts from initial conditions 1,0,0,22 and the red one starts from 1,0,0,−22).
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(b)Figure 5
Phase portraits of coexisting limit cycle attractors in (a)x−y and (b)x−z planes (the blue one starts from initial conditions 1,0,0,26 and the red one starts from 1,0,0,−26).
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(b)Figure 6
Phase portraits of limit cycle in (a)x−y and (b)x−z planes for initial conditions set as 1,0,0,30.
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(b)Whenw0 is changed in the region −50,50, the bifurcation diagram of the state variable w and Lyapunov exponent spectra are shown in Figures 7(a) and 7(b), respectively. As shown in Figure 7(a), the bifurcation diagram of the state variable w is almost linear. It can be seen from Figure 7(b) that the chaotic attractor with a positive Lyapunov exponent is mainly located at the region −36,−32, −22,22, and 32,35 (the last Lyapunov exponent is not displayed because it is always a big negative number). And system (2) can also generate other kinds of attractors and limit cycle attractors, which means the system has heterogeneous multistability.Figure 7
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to memristor initial condition w0.
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## 4.2. Dynamic Analysis of Homogenous Multistability

The parameters of system (2) remain unchanged, and the initial conditions are set as x0,1,0,0. When x0 is varied in the region −104,104, the bifurcation diagrams of the state variable z, the state variable w, and its Lyapunov exponent spectra are plotted in Figures 8(a), 8(b), and 8(c), respectively. Besides, when x0 is varied in the region −0.1,0.1, Lyapunov exponent spectra are plotted in Figure 8(d).Figure 8
(a) Bifurcation diagram of the state variablez, (b) bifurcation diagram of the state variable w, (c) Lyapunov spectrum with respect to initial condition x0 in the region −104,104, and (d) Lyapunov spectrum with respect to initial condition x0 in the region −0.1,0.1.
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(d)It can be seen from Figure8(a) that when the initial condition x0 is varied in the region −104,104, the state variable z shows two kinds of steady chaotic states. When x0 is varied in the region −104,0, state variable z is located below or above the z axis, while x0 is varied in the region 0,104, the state variable z is all located above the z axis. From Figure 8(b), when the initial condition x0 is varied in the region −104,104, there are many discrete small line segments in bifurcation diagram of the state variable w, which means the state variable w exists in infinite steady chaotic states. And this phenomenon also indicates the system can generate coexisting infinitely many attractors, which means system (2) has the extreme multistability. Different from other systems possessing extreme multistability, system (2) only generates two kinds of chaotic attractors with different positions in a very large range of initial values, which are distributed along the w axis parallelly. As shown in Figure 8(c), it is obvious that the four Lyapunov exponents are always approximately constant and the largest Lyapunov exponent is always positive except for the zero point when x0 is varied in the region −104,104, which means system (2) can exhibit chaotic behavior except zero point. Figure 8(d) shows that the region cannot exhibit chaotic behavior which is very small.Considering the particularity of system (2), it is necessary to discuss the system when x0=1. The control parameters of system (2) remain unchanged, and the initial conditions are set as 0,1,0,0. The LEs are 4.0002, −0.2034, −5.0005, and −19.8272, which means system (2) cannot exhibit chaotic behavior under this circumstance.It has been confirmed that there are coexisting infinitely many attractors in chaotic system (2) according to the above analysis. When x0 is set to 10, −10, 50, +50, 100, −100, and 1, respectively, the phase portraits of coexisting infinitely many attractors in the x−w plane, the y−w plane, the z−w plane, and the w−z−x space are shown in Figures 9(a), 9(b), 9(c), and 9(d), respectively. Figure 9 clearly reveals the coexistence of a large number of same attractors with different positions, which implies the emergence of homogenous multistability. This result of the phase portraits is consistent with the bifurcation diagrams and Lyapunov spectrum with respect to initial condition x0.Figure 9
Phase portraits of coexisting infinitely many attractors on (a)x−w plane, (b)y−w plane, (c)z−w plane, and (d)w−z−x plane in chaotic system (2) (the blue one starts from initial conditions 10,1,0,0, the red one starts from −10,1,0,0, the pink one starts from 50,1,0,0, the cyan one starts from −50,1,0,0, the yellow one starts from 100,1,0,0, the black one starts from −100,1,0,0, and the green one starts from 1,1,0,0).
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(d)The parameters of system (2) are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the initial conditions are set as 1,y0,0,0, and the initial condition y0 is used as the independent variable of bifurcation diagram. When y0 is changed in the region −104,104, the bifurcation diagram of the state variable w and its Lyapunov exponent spectra are plotted in Figures 10(a) and 10(b), respectively. As can be seen from Figure 10(a), when the initial condition y0 is changed in the region −104,104, there are many discrete small line segments in bifurcation diagram of the state variable w, which implies that there are coexisting infinitely many attractors in system (2). It can be clearly seen from Figure 10(b) that four Lyapunov exponents are always approximately constant and largest Lyapunov exponent is always positive when is y0 changed in the region −104,104, which means system (2) only exhibits chaotic behavior.Figure 10
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to initial condition y0 in the region −104,104.
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(b)The parameters of system (2) are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the initial conditions are set as 1,0,z0,0, and the initial condition z0 is used as the independent variable of bifurcation diagram. When z0 is changed in the region −104,104, the bifurcation diagram of the state variable w and its Lyapunov exponent spectra are plotted in Figures 11(a) and 11(b), respectively. The bifurcation diagram of the state variable w and its Lyapunov exponent spectra are similar with those with the initial conditions set as 1,y0,0,0. Many discrete small line segments are more closely clustered in the bifurcation diagram of the state variable w for the initial conditions set as 1,0,z0,0 compared with the one for the initial conditions set as 1,y0,0,0. And Lyapunov exponent spectra for the initial conditions set as (1, 0, z (0), 0) are similar with the ones for the initial conditions set as 1,y0,0,0, and four Lyapunov exponents are always approximately constant and largest Lyapunov exponent is always positive when z0 is changed in the region −104,104.Figure 11
(a) Bifurcation diagram of the state variablew and (b) Lyapunov spectrum with respect to initial condition z0 in the region −104,104.
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## 4.3. Symmetry in Infinitely Many Coexisting Attractors

Obviously, system (2) is invariant if we do the transformation x,y,z,w⟶−x,−y,z,−w, which means that x,y,z,w and −x,−y,z,−w are all solutions to the equation of the system. This symmetry characteristic of system (2) could be served to explain the presence of symmetric coexisting attractors in state space.If we set parameters asa=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1, the system can generate many pairs of symmetric coexisting attractors for the corresponding conditions set as x0,y0,z0,w0 and −x0,−y0,z0,−w0. And two pairs of symmetric coexisting attractors of these are shown in Figure 12, where the blue one and the red one are a pair of symmetric attractors and the pink one and the cyan one are another pair of symmetric attractors. The projections of the symmetric attractors on the coordinate planes can be shown in the form of central symmetry or axial symmetry. In system (2), it is centrosymmetric on the x−y plane, the x−w plane, and the y−w plane and axisymmetric on the x−z plane, the y−z plane, and the z−w plane. It is worth noting that the structures of many coexisting attractors in Figure 9 are the same roughly, and there are differences in the details. But the structures of each pairs of coexisting attractors in Figure 12 are symmetric exactly.Figure 12
Phase portraits of symmetric coexisting infinitely many attractors in chaotic system (2) (the blue one starts from initial conditions 1,1,−10,1, the red one starts from −1,−1,−10,−1, the pink one starts from 10,20,15,2, and the cyan one starts from −10,−20,15,−2).
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(f)In order to further verify symmetry of coexisting infinitely many attractors in chaotic system (2), bifurcation diagrams of the state variable z and the state variable w are given. The parameters of system (2) are assigned as a=4, b=6, c=20, d=5, e=0.01, f=0.1, 3g=0.1, and h=0.1, and the initial conditions are set as 1,1,z0,1, and the initial condition z0 is used as the independent variable of bifurcation diagram. When z0 is varied in the region [–100, 100], the bifurcation diagrams of the state variable z and the state variable w are plotted in Figures 13(a) and 13(b), respectively. Similarly, the parameters of system (2) remain unchanged, the initial conditions are set as −1,−1,z0,−1, and the initial condition z0 is used as the independent variable. When z0 is varied in the region 100,−100, the bifurcation diagrams of the state variable z and the state variable w are plotted in Figures 14(a) and 14(b), respectively.Figure 13
(a) Bifurcation diagram of the state variablez and (b) bifurcation diagram of the state variable w with respect to initial conditions 1,1,z0,1.
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(a) Bifurcation diagram of the state variablez and (b) bifurcation diagram of the state variable w with respect to initial conditions −1,−1,z0,−1.
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(b)A comparison of the Figures13(a) and 14(a) indicates that bifurcation diagrams of the state variable z for the initial conditions set as 1,1,z0,1 and −1,−1,z0,−1 are the same exactly. And a comparison of the Figures 13(b) and 14(b) shows that they are symmetric about the horizontal axis. This phenomenon reflects the fact that when the parameters of system (2) set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, h=0.1, the structures of coexisting attractors with conditions set as 1,1,z0,1 and −1,−1,z0,−1 are symmetric with respect to z coordinate axis.We compared the advantages and disadvantages of this system with other improved Liu-and-Chen systems in the literature [11, 23, 24]. The system proposed in reference [11] is a three-dimensional chaotic system without memristor. And it can generate three-scroll and four-scroll chaotic attractors. The system in reference [23] is a four-dimensional chaotic system with two memristors, which can generate various kinds of attractors and hidden attractors. The system in reference [24] is a four-dimensional chaotic system with a memristor, which can generate three-wing, four-wing, and coexisting two-wing chaotic attractors. It has line equilibrium points. This system has a line equilibrium, within which the attractors generated are hidden. The new system is a four-dimensional chaotic system with a memristor, which can generate coexisting infinitely many attractors distributed along the w axis parallelly. It produces hidden attractors because it has a line equilibrium. In addition, the system has discrete bifurcation diagrams and many symmetrical coexisting attractors. Compared with above improved Liu-and-Chen systems, we can find that chaotic systems with memristor have more complex dynamic behaviors than chaotic systems without memristor and are more likely to have extreme multistability.


## 5. Circuit Implementation

In this section, complex dynamic behaviors of the proposed chaotic system can be observed by analog circuit, where integrated operational amplifiers and multipliers are used to construct the circuit for generating chaotic attractors. Supply voltages of operational amplifiers areE=±15V. The input and output range of all the multipliers is between −15V to 15V. However, the values of the state variables x, y, z, and w may be out of this range. Thus, it is necessary that state variables x, y, z, and w are compressed to the 1/10 of original system to be limited in the region of −15V,15V, which is the reference voltage of the operational amplifiers. At the same time, taking the time scale factor RC into account, system (2) after scale transformation can be represented as follows:(9)RCx˙=ax−10byz,RCy˙=−cy+10xz+ef+300gw2x,RCz˙=−dz+10xy,RCw˙=hx.A flux-controlled memristor depicted in the dashed box of Figure15 is applied to construct the chaotic circuit. The analog circuit of system (2) is shown in Figure 15. The state equations can be obtained as follows:(10)C1v˙x=vxR1−vyvzR2,C2v˙y=−vyR3+vxvzR4+vxRc+vxvw2Rb,C3v˙z=−vzR5+vxvyR6,C4v˙w=vxRa,where vx, vy, vz, and vw are the voltages on capacitors. Compared with (9) and (10), the parameters are taken as follows: C1=C2=C3=C4=C, R1=R/a, R2=R/10b, R3=R/c, R4=R/10, R5=R/d, R6=R/10, Ra=R/h, Rb=R/e∗3g∗100, and Rc=R/ef.Figure 15
Circuit diagram of memristive system (2).System (2) can generate coexisting infinitely many attractors when the parameters are set as a=4, b=6, c=20, d=5, e=0.01, f=1, 3g=0.1, and h=0.1. Let us take R=100kΩ and C=10000nF, and the resistance parameters can be obtained as R1=25kΩ, R2=1.67kΩ, R3=5kΩ, R4=10kΩ, R5=20kΩ, R6=10kΩ, Ra=1000kΩ, Rb=1000kΩ, and Rc=10000kΩ. Also, the initial voltages of all capacitors are set as 1V,0.1V,0V,0V (initial voltages of all capacitors should be compressed to the 1/10 of original values). According to the above parameter settings, circuit simulation of chaotic attractor can be obtained by PSpice as shown in Figure 16. Similarly, when the initial voltages of all capacitors are set as −1V,0.1V,0V,0V, circuit simulation of chaotic attractor can be obtained by PSpice as shown in Figure 17. Obviously, the chaotic attractors obtained in the analog circuit well verify those shown by numerical simulations.Figure 16
PSpice simulated phase portraits of chaotic attractors with initial voltages1V,0.1V,0V,0V.
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PSpice simulated phase portraits of chaotic attractors with initial voltages−1V,0.1V,0V,0V.
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## 6. Conclusion

A new memristor-based chaotic system with coexisting infinitely many attractors is proposed by using a smooth flux-controlled memristor to replace a resistor in the analog circuit of the three-dimensional chaotic system. The system has a line equilibrium and exhibits homogenous and heterogeneous multistabilities. The dynamical behaviors of the system are analyzed by equilibrium point and stability, phase portraits, bifurcation diagrams and Lyapunov exponent spectra, and so on. Compared with general chaotic systems, this chaotic system has some special properties. When the parameters of system are fixed, the chaotic system only generates two kinds of chaotic attractors with different positions in a very large range of initial values and has constant Lyapunov exponent spectra. In addition, the system has discrete bifurcation diagrams, which has not been found in existing chaotic systems. Further, the relationship between the symmetry of the coexisting attractors and the symmetry of initial values in the system is explored and verified by phase portraits and bifurcation diagrams. Finally, the chaotic system is realized by analog circuit, and the numerical simulation results are verified by the simulation results of the analog circuit. The complex dynamical behaviors of the proposed system are very useful for various chaos-based information encryption and secure communication applications.



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*Source: 2408460-2020-04-25.xml*