# A New Memristor-Based 5D Chaotic System and Circuit Implementation

Authors: Rui Wang; Mingjin Li; Zhaoling Gao; Hui Sun
Journal: Complexity
 (2018)

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## Abstract

This paper proposes a new 5D chaotic system with the flux-controlled memristor. The dynamics analysis of the new system can also demonstrate the hyperchaotic characteristics. The design and analysis of adaptive synchronization for the new memristor-based chaotic system and its slave system are carried out. Furthermore, the modularized circuit designs method is used in the new chaotic system circuit implementation. The Multisim simulation and the physical experiments are conducted, compared, and matched with each other which can demonstrate the existence of the attractor for the new system.


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## Body

## 1. Introduction

Memristors are the fourth kind of circuit elements except for resistors, capacitors, and conductors and are conceived by Chua in 1971 through the basic symmetric principle [1]. Furthermore, the corresponding theory was applied to memristive devices in 1976 [2]. It took a long time to develop the hardware memristor model. Until 2008 HP labs first realized the memristor of nanoscale in the form of crossbar array [3]. Since memristors have the potential applications in the wide range of fields, memristor study becomes hotter, and a huge amount of researchers have paid immense attention on memristor studies from industry and academics, respectively [4, 5]. The typical examples include nonvolatile memories of nanoscale [6], memristor-based synapse in neuromorphic systems [7, 8], logic operations through material implication [9–12], and nonlinear dynamics in chaotic system [13–17].With rapid development of memristor models, some studies combine the memristor and chaotic systems including dynamics analysis, image encryption applications and circuit implementations which have grown up quickly in recent years [18–22]. One of the typical early memristor-based chaotic systems was developed by Itoh and Chua in 2008 [19]. This paper developed some nonlinear oscillators by using memristors based on Chua’s oscillators. Petráš derived and investigated a fractional-order memristor-based Chua’s circuit in [20]. Chua and Muthuswamy also discussed circuit topology and developed the simplest memristor-based circuits [21]. These papers demonstrate that the memristor oscillators own the special nonlinear dynamics due to memristors’ extinguished characteristics. One of these significant characteristics is that the behaviors are dependent on initial states and circuit parameters. Li et al. proposed a scroll chaotic system circuit implementation by using HP memristor [22]. Ma et al. developed a four-wing hyperchaotic system by using a memristor adding over a three-dimensional chaotic system [16]. Dimitrios et al. found a new 4-D memristive chaotic system and investigated the behavior with hidden attractors of the system through numerical simulations [23]. Wang et al. proposed a flux-controlled memristor model and established a 4-D chaotic system with this model. The numerical analysis and circuit implementation simulation verification were conducted [24]. Mou et al. discussed the characteristics of dynamical behaviors of a fractional-order 4D hyperchaotic memristive system and circuit simulation verification [25]. Other memristor-based hyperchaotic systems were also investigated such as numerical analysis about a four-dimensional hyperchaotic system with memristor and conducted circuit simulation verifications [26–29]. There are also some other chaotic systems developed by memristive models and its applications [30–32].The above memristor-based chaotic system literatures focus on four or lower-dimensional chaotic systems and investigate the detailed numerical analysis and the corresponding numerical and circuit implementation simulation verifications. However, the higher-dimensional (5-D or above) memristor-based hyperchaotic systems and the corresponding physical hardware experiments are not found. Therefore, the paper analyzes a new memristor-based hyperchaotic system and develops a circuit physical implementation method by using the modularized design method. This method is used to design the circuit without dimensions for chaotic circuit designs and is easy to be implemented in the circuit by using less circuit parts [33–38].The novelty of this paper is to develop a new memristor-based 5D hyperchaotic system, design and analyze the adaptive synchronization of this new system, implement the physical experiment circuit hardware, and verify the existence of system attractors. The improved modularized design method is used to implement the circuit of the system to verify the existence of attractors.The rest of the paper is organized as follows. Section2 analyzes the fundamental characteristics of the new memristor-based chaotic system. Section 3 investigates the adaptive synchronization of the new memristor-based hyperchaotic system. Section 4 discusses the circuit implementation of the new system and verifies the existence of attractors. Conclusions are presented in Section 5.


## 2. Analysis of a New Memristor-Based Hyperchaotic System

In this section, numerical analyses are conducted for a flux-controlled memristor-based new 5D hyperchaotic system derived from Wang’s 4D hyperchaotic system.As illustrated in [16], memristor model is based on the fundamental characteristics of a flux-controlled memristor described below.(1)i=Wφvwhere i and v are the current and the voltage of the device terminal, respectively. Wφ is the incremental memductance defined as (2)Wφ=dqφdφThis demonstrates that the characteristics of a memristor are a nonlinear function reflecting the relationship among the charge and flux across and through the device.Furthermore, this paper consistently uses the smooth cubic monotone-increasing continuous nonlinearity described as follows [20–22, 33].(3)dqφ=mφ+nφ3where m,n>0.Then the memductance is shown below(4)Wφ=dqφdφ=m+3nφ2This paper develops a 5D memristor-based chaotic system which is derived from four-wing autonomous chaotic dynamics systems reported by Wang et al. [34, 37](5)x˙=ay-x+4yzy˙=-x+16y-xz+wz˙=-bz+xyw˙=-10y+0.15xzu˙=-xThis system has four state variables x, y, z, and w, and a,b∈R+.Substitute (4) into (5), a 5D memristor-based system is obtained.(6)x˙=ay-x+4yz-kxWuy˙=-x+16y-xz+wz˙=-bz+xy-xu-yww˙=-10y+0.15xz-gzuu˙=-xwhere (7)Wu=m+3nu2,and k, m, n, g are positive parameters.


### 2.1. Equilibria and Stability

The equilibrium points of System (6) can be calculated by solving the equations as shown below(8)ay-x+4yz-kxWu=0-x+16y-xz+w=0-bz+xy-xu-yw=0-10y+0.15xz-gzu=0-x=0where m=0.1, n=0.01, and g=0.3.System (6) has only one real equilibrium point with (0, 0, 0, 0, 0) and has typical characteristics with a line equilibrium in (0, 0, 0, 0,δ), given δ a real constant.First, analyze the zero equilibrium point (0, 0, 0, 0, 0). The Jacobian matrix of System (6) on the zero equilibrium point is (9)J0=-a-0.1ka000-11601000b000-1-1000000000,0,0,0,0and the corresponding polynomial is(10)fλ=λλ-bf1λand(11)f1λ=λ3+a-16+0.1kλ2+10-15a-1.6kλ+10a+kIt is obvious that 0 and -b are eigenvalues of System(6) for the (0, 0, 0, 0, 0) equilibrium point. According to Routh-Hurwitz condition, if and only if a-16+0.1k>0, 10-15a-1.6k>0, (10a+k)>0, and (a-16+0.1k)(10-15a-1.6k)-(10a+k)>0 coexist, f1(λ) has the negative real number. However, the above four inequalities are not able to be realized simultaneously. Then not all real parts of the eigenvalues are negative. Therefore, it is not a stable equilibrium point.Second, analyze the eigenvalues of Jacobian matrix of System (6) on the line equilibrium in (0, 0, 0, 0,δ).(12)J1=-a-k0.1+0.03δ2a000-116010-δ0-b000-1-100-0.3δ00000Typically, when a=14 and b=78, J1∗ is calculated as shown below(13)J1∗=-14-k0.1+0.03δ214000-116010-δ0-78000-1-100-0.3δ00000Two of the five eigenvalues of J1∗ are complex conjugates; therefore, it is difficult to determine the stability of the line equilibria.


### 2.2. Symmetry

System (6) is symmetric with respect to z axis since it is invariant when applying the coordinate transformations. (14)x,y,z,w,u↔-x,-y,z,-w,-u


### 2.3. Dissipativity

Furthermore, dissipative characteristics analysis of System (6) is shown below. The system divergence is given by(15)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w+∂u˙∂u=-a-kWu+16-b=-km+3nu2+16-a-bwhen k>0, m>0, n>0, 16-a-b<0,-k(m+3nu2)+16-a-b<0, System (6) is dissipative. The paper selects m=0.1, n=0.01 for equation (7), and k=0.02 for System (6).


### 2.4. Lyapunov Spectrum and Bifurcation Diagram

Fix parametersa=14, b=78 and vary the parameter k, and the graphs about the Lyapunov exponents versus k, bifurcation diagram, and phase portraits are shown in Figure 1.Figure 1
Lyapunov exponents versus the parameterk: (a) the full five exponents, (b) Lyapunov exponentsL1-L4 of System (6), (c) the bifurcation diagram of x vs. k for System (6) when m=0.1, n=0.01, and g=0.3. (d)-(e) The phase portraits x-y and x-z for System (6) when k=0.02, m=0.1, n=0.01, and g=0.3.
(a) (b) (c) (d) (e)Figure1 shows that the Lyapunov exponents vary with the parameter k changes. In the five Lyapunov exponents, three of them are obviously negative when k∈[0,5.9]. The top two lines in Figure 1(b) demonstrate that these two kinds of Lyapunov exponents are bigger than zero, and systems are hyperchaotic systems when k lies in this range. In this paper, k will be selected in this range.Whena=14, b=78, k=0.02, m=0.1, n=0.01, and g=0.3, the Lyapunov exponents are calculated asL1 = 1.0241,L2 = 0.0137,L3 = -0.1735,L4 = -2.3787, andL5 = -70.3244. This system is a hyperchaotic system with two positive Lyapunov exponents. Therefore, the Kaplan-Yorke dimension of System (6) can be found below(16)DKY=4+L1+L2+L3+L4L5=3.978466


## 2.1. Equilibria and Stability

The equilibrium points of System (6) can be calculated by solving the equations as shown below(8)ay-x+4yz-kxWu=0-x+16y-xz+w=0-bz+xy-xu-yw=0-10y+0.15xz-gzu=0-x=0where m=0.1, n=0.01, and g=0.3.System (6) has only one real equilibrium point with (0, 0, 0, 0, 0) and has typical characteristics with a line equilibrium in (0, 0, 0, 0,δ), given δ a real constant.First, analyze the zero equilibrium point (0, 0, 0, 0, 0). The Jacobian matrix of System (6) on the zero equilibrium point is (9)J0=-a-0.1ka000-11601000b000-1-1000000000,0,0,0,0and the corresponding polynomial is(10)fλ=λλ-bf1λand(11)f1λ=λ3+a-16+0.1kλ2+10-15a-1.6kλ+10a+kIt is obvious that 0 and -b are eigenvalues of System(6) for the (0, 0, 0, 0, 0) equilibrium point. According to Routh-Hurwitz condition, if and only if a-16+0.1k>0, 10-15a-1.6k>0, (10a+k)>0, and (a-16+0.1k)(10-15a-1.6k)-(10a+k)>0 coexist, f1(λ) has the negative real number. However, the above four inequalities are not able to be realized simultaneously. Then not all real parts of the eigenvalues are negative. Therefore, it is not a stable equilibrium point.Second, analyze the eigenvalues of Jacobian matrix of System (6) on the line equilibrium in (0, 0, 0, 0,δ).(12)J1=-a-k0.1+0.03δ2a000-116010-δ0-b000-1-100-0.3δ00000Typically, when a=14 and b=78, J1∗ is calculated as shown below(13)J1∗=-14-k0.1+0.03δ214000-116010-δ0-78000-1-100-0.3δ00000Two of the five eigenvalues of J1∗ are complex conjugates; therefore, it is difficult to determine the stability of the line equilibria.


## 2.2. Symmetry

System (6) is symmetric with respect to z axis since it is invariant when applying the coordinate transformations. (14)x,y,z,w,u↔-x,-y,z,-w,-u


## 2.3. Dissipativity

Furthermore, dissipative characteristics analysis of System (6) is shown below. The system divergence is given by(15)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w+∂u˙∂u=-a-kWu+16-b=-km+3nu2+16-a-bwhen k>0, m>0, n>0, 16-a-b<0,-k(m+3nu2)+16-a-b<0, System (6) is dissipative. The paper selects m=0.1, n=0.01 for equation (7), and k=0.02 for System (6).


## 2.4. Lyapunov Spectrum and Bifurcation Diagram

Fix parametersa=14, b=78 and vary the parameter k, and the graphs about the Lyapunov exponents versus k, bifurcation diagram, and phase portraits are shown in Figure 1.Figure 1
Lyapunov exponents versus the parameterk: (a) the full five exponents, (b) Lyapunov exponentsL1-L4 of System (6), (c) the bifurcation diagram of x vs. k for System (6) when m=0.1, n=0.01, and g=0.3. (d)-(e) The phase portraits x-y and x-z for System (6) when k=0.02, m=0.1, n=0.01, and g=0.3.
(a) (b) (c) (d) (e)Figure1 shows that the Lyapunov exponents vary with the parameter k changes. In the five Lyapunov exponents, three of them are obviously negative when k∈[0,5.9]. The top two lines in Figure 1(b) demonstrate that these two kinds of Lyapunov exponents are bigger than zero, and systems are hyperchaotic systems when k lies in this range. In this paper, k will be selected in this range.Whena=14, b=78, k=0.02, m=0.1, n=0.01, and g=0.3, the Lyapunov exponents are calculated asL1 = 1.0241,L2 = 0.0137,L3 = -0.1735,L4 = -2.3787, andL5 = -70.3244. This system is a hyperchaotic system with two positive Lyapunov exponents. Therefore, the Kaplan-Yorke dimension of System (6) can be found below(16)DKY=4+L1+L2+L3+L4L5=3.978466


## 3. Adaptive Synchronization of a New Memristor-Based Hyperchaotic System

In this section, the adaptive controller was designed for the new memristor-based hyperchaotic system which was derived by Lyapunov stability theory inspired by [39].First, consider the master System (6) with a=14, b=78, k=0.02, m=0.1, n=0.01, and g=0.3 shown as below(17)x˙=14y-x+4yz-kx0.1+0.01u2y˙=-x+16y-xz+wz˙=-78z+xy-xu-yww˙=-10y+0.15xz-0.3zuu˙=-xSecond, consider the slave system shown as below(18)x˙′=14y′-x′+4y′z′-kx′0.1+0.01u′2+u1y˙′=-x′+16y′-x′z′+w′+u2z˙′=-78z′+x′y′-x′u-y′w′+u3w˙′=-10y′+0.15x′z′-0.3z′u′+u4u˙′=-x′+u5where ui(i=1⋯5) are adaptive controller, and x′, y′, z′, w′, u′ are the new state variables.Therefore, the synchronization errors among the Systems (17) and (18) are defined as follows.(19)e1=x′-xe2=y′-ye3=z′-ze4=w′-we5=u′-uThen the dynamics of the error can be calculated by substituting the master and slave systems as shown below(20)e˙1=ae2-e1-0.1ke1+4y′z′-yz-0.03kx′u′2-xu2+u1e˙2=-e1+16e2+e4-x′z′+xz+u2e˙3=-be3+x′y′-x′u′-y′w′-xy+xu+yw+u3e˙4=-10e2+0.15x′z′-0.15xz-0.3z′u′+0.3zu+u4e˙5=-e1+u5where the adaptive controller ui are defined by(21)u1=-a^te2-e1+0.1ke1-4y′z′-yz+0.03kx′u′2-xu2-k1e1u2=e1-16e2-e4+x′z′-xz-k2e2u3=b^te3-x′y′+x′u′+y′w′+xy-xu-yw-k3e3u4=10e2-0.15x′z′+0.15xz+0.3z′u′-0.3zu-k4e4u5=e1-k5e5where ki(i=1,…,5) are the positive gains and a^(t), b^(t) are the estimations of the corresponding parameters a and b, respectively.Correspondingly, the dynamics of the error is changed into(22)e˙1=a-a^te2-e1-k1e1e˙2=-k2e2e˙3=b-b^te3-k3e3e˙4=-k4e4e˙5=-k5e5Define the parameters for estimating errors as(23)eat=a-a^tebt=b-b^tSubstitute (23) into the dynamics (22), the simplified error dynamics is described as(24)e˙1=eate2-e1-k1e1e˙2=-k2e2e˙3=-ebte3-k3e3e˙4=-k4e4e˙5=-k5e5Correspondingly, (25)e˙at=-a^te˙bt=-b^tDepending on the master and slave systems definitions and error dynamics transformations mentioned above, consider a Lyapunov function defined by (26)V=12e12+e22+e32+e42+e52+ea2+eb2Differentiate V along the trajectories for equations (24)-(25), V˙ can be obtained as(27)V˙=e1e˙1+e2e˙2+e3e˙3+e4e˙4+e5e˙5+eae˙a+ebe˙b=-k1e12-k2e22-k3e32-k4e42-k5e52+ea-e12-e1e2-a^˙t-ebte32+b^˙tThen, the parameter update law is (28)a^˙t=-e12-e1e2b^˙t=-e32Theorem 1.
The master and slave Systems (17)-(18) for a new memristor-based hyperchaotic system along with unknown parameters and positive gains ki(i=1⋯5) synchronize exponentially by using the adaptive controller (21) and the parameter update law (28).Proof.
Substitute the parameter update law (28) into the adaptive controller (21), then(29)V˙=-k1e12-k2e22-k3e32-k4e42-k5e52≤-k′e2≤0where k′=min⁡k1,k2,k3,k4,k5∣ki∈R+,i=1⋯5.It is obvious thate(t)→0 exponentially as time goes to infinite for all initial conditions of e(0). According to Lyapunov stability theory, two systems are synchronized.Figure2 is the simulation verification of synchronization for the master and the slave memristor-based hyperchaotic Systems (21) and (22) when selecting k′=ki=4(i=1⋯5), and the systems choose the initial conditions for both systems, respectively, such that x(0)=4, y(0)=1.2, z(0)=0.5, w(0)=-3.6, u(0)=6 and x′0=-5, y′0=2, z′0=1, w′0=-0.8, u′0=5. These figures demonstrate the synchronization of the master and slave Systems (21) and (22).Figure 2
Synchronization of states for Systems (21) and (22).
(a)
Synchronization of statesx and x′ for Systems (21) and (22) (b)
Synchronization of statesy and y′ for Systems (21) and (22) (c)
Synchronization of statesz and z′ for Systems (21) and (22) (d)
Synchronization of statesw and w′ for Systems (21) and (22) (e)
Synchronization of statesu and u′ for Systems (21) and (22)


## 4. A New Memristor-Based Hyperchaotic System Circuit Implementation

In this section, modularized design methods will be implemented in the new memristor-based hyperchaotic systems mentioned above. Multisim software is used to generate the circuit simulation results, and the corresponding physical circuit experiments are conducted to verify the hyperchaotic attractor existences of this hyperchaotic system.(A1) Memristors Circuit Implementation. Figure 3 shows the circuit configuration of memristor inspired by [16].Figure 3
W ( u ) function-memristor channel.In this memristor circuit, a factor 0.1/V multiplier AD633JN is used. Therefore,m=Rf/Rm=0.1, n=100Rf/Rn0.01/V=0.01, and (30)W=RfRm+3·100RfRn0.01V·u2where Rf is the feedback resistor of the amplifier.Parameterk represents the strength of a memristor.(31)k=RfmRm=Rf300nRnIn Figure 4, select the following resistors and the capacitor, Rf=2kΩ, Rm=1MΩ, Rn=66kΩ, R15=200kΩ, Rc=1kΩ, C4=10nF.Figure 4
The relationship betweenx and W(x) for the circuit as shown in Figure 3.Figure4 is the Multisim simulation for the flux-controlled memristor with x input with sinusoid signal. This demonstrates the typical closed loop characteristics of the memristor developed by equation (7) and part of equation (6).(A2) Memristor-Based Chaotic System Circuit Analysis. The implementation of memristor-based chaotic system uses the modularized design method applied on System (17). After the time-scale transformation method is applied on System (17), System (17) becomes System (32).(32)dxdt=-1τ0R1C1x-1τ0R2C1-y-110τ0R3C1-yz-110τ0R4C1kxWudydt=-1τ0R5C2x-1τ0R6C2-y-110τ0R7C2xz-1τ0R8C2-wdzdt=-1τ0R9C3z-110τ0R10C3x-y-110τ0R11C3xu-110τ0R12C3ywdwdt=-1τ0R13C4y-110τ0R14C4x-z-110τ0R15C4zududt=-1τ0R16C5xwhere τ0=100.Therefore,(33)dxdt=-1400x-1400-y-400-yz-100kxWudydt=-100x-1600-y-100xz-100-wdzdt=-7800z-100x-y-100xu-100ywdwdt=-1000y-15x-z-30zududt=-100xFurthermore, the paper employs the unified compression coefficient,s=1/20, for each variable in order not to exceed the range of the oscilloscope used to the circuit. Then(34)sx→xsy→ysz→zsw→wSubstituting (34) into (33), we can obtain (35)dxdt=-1400x-1400-y-8000-yz-100kxWudydt=-100x-1600-y-2000xz-100-wdzdt=-7800z-100x-y-2000xu-2000ywdwdt=-1000y-300x-z-600zududt=-100xComparing equations (35) to (32), parametersR andCin the circuit can be calculated when using the factor of 0.1 multiplier.R1=R2=75kΩ,R3= 1kΩ,R4=1MΩ,R5=1MΩ,R6=68kΩ,R7=5.1kΩ,R8=1MΩ,R9=13kΩ,R10=5.1kΩ,R11=R12=1MΩ,R13= 100kΩ,R14= 33.3kΩ,R15= 330kΩ,R16= 220kΩ.In the modularized circuit design for System (17), there are five channels for each variable. The circuit implementation uses analog amplifiers LF347N and AD633JN multipliers to implement the addition and the integral operations. Some additionalR andCcomponents are also selected in the circuit implementation.R17 =R18 =R19 =R20=10kΩ,R21 =R22 =R23 =R24 = 20kΩ,C1 =C2 =C3 =C4 =C5 = 10nF,Rf=2.2kΩ,Rc=1kΩ,Rm=1MΩ,Rn=68kΩ.The circuit channels for first four variables are shown in Figure5, and the fifth one is shown in Figure 3.Figure 5
Modularized circuit channels for circuit implementation of the new memristor-based 5D hyperchaotic system.
(a)
x channel (b)
y channel (c)
z channel (d)
w channel(A3) Circuit Implementation for the New 5D Chaotic Systems. According to the circuit design, the Multisim simulation implementation for the memristor-based hyperchaotic System (17) is demonstrated in Figure 6. That shows the attractor phases graphs of different planes for System (17).Figure 6
Multisim implementation of the chaotic System (17): (a)x-y plane with scales 5V/div and 2V/div, (b)x-z plane with scales 2V/div and 1V/div, (c)x-w plane with scales 2V/div and 1V/div, (d)z-w plane with scales 2V/div and 1V/div, (e)y-w plane with scales 2V/div and 1V/div, and (f)y-uplane with scales 2V/div and 2V/div.
(a) (b) (c) (d) (e) (f)Attractor phases of System (17) as shown in Figure 6 own similar hyperchaotic characteristics with those of System (5) which are elaborated in [37]. It is known that practical analog devices are not ideal. Therefore, in reality, the voltages added on the amplifiers are not exceeding ±13.5V in order to guarantee the system variable outputs do not exceed the amplifier linear region. This paper employs ±12.5V across the amplifier in the physical circuits hardware, and the corresponding attractor phases graphs are shown in Figure 7.Figure 7
Physical circuit implementation hardware and digital oscilloscope diagrams of the chaotic System (17): (a)x-y plane with scales 2V/div and 1V/div, (b)x-z plane with scales 2V/div and 1V/div, (c)x-w plane with scales 2V/div and 1V/div, (d)z-w plane with scales 1V/div and 1V/div, (e)y-w plane with scales 1V/div and 1V/div, and (f)y-uplane with scales 1V/div and 2V/div.
(a) (b) (c) (d) (e) (f)Comparing Figure6 to Figure 7, it is shown that the physical experiment attractor phase results match those conducted by Multisim and demonstrate the existence of the memristor-based hyperchaotic attractor for System (17). Furthermore, the memristor-based Wang hyperchaotic system has similar characteristics with those of the original Wang hyperchaotic system which is verified by the Multisim simulation and physical experimental results. The improved modularized method with compression coefficients is flexible, reliable, and straightforward to realize the physical implementation. The memristor-based hyperchaotic system has potential and bright application physically. For example, it can realize the physical communication encryptions more reliably and easily. Future works will focus on the physical applications for communication encryptions by using the memristor-based hyperchaotic system.


## 5. Conclusions

The paper developed a new 5D memristor-based chaotic system with a flux-controlled memristor. The dynamics analysis of the system showed that the new system is a hyperchaotic system. The adaptive controller and update laws for the synchronization of the new system were designed and conducted. Furthermore, the modularized design method for the physical circuit experiment implementation is applied to realize the circuit by Multisim and physical experiments hardware. After the implementation of these circuits, then the comparisons between Multisim simulation and results from the physical experiments showed that these two kinds of results matched with each other and verified the existence of the attractors from the phase plane graphs. Furthermore, this new higher-dimensional chaotic system with memristor owns more complex dynamics and can be applied in a wide range of applications such as encryptions. This is also the future work for this work.



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