# Dynamic Analysis, Circuit Design, and Synchronization of a Novel 6D Memristive Four-Wing Hyperchaotic System with Multiple Coexisting Attractors

Authors: Fei Yu; Li Liu; Hui Shen; Zinan Zhang; Yuanyuan Huang; Changqiong Shi; Shuo Cai; Xianming Wu; Sichun Du; Qiuzhen Wan
Journal: Complexity
 (2020)

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## Abstract

In this work, a novel 6D four-wing hyperchaotic system with a line equilibrium based on a flux-controlled memristor model is proposed. The novel system is inspired from an existing 5D four-wing hyperchaotic system introduced by Zarei (2015). Fundamental properties of the novel system are discussed, and its complex behaviors are characterized using phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. When a suitable set of parameters are chosen, the system exhibits a rich repertoire of dynamic behaviors including double-period bifurcation of the quasiperiod, a single two-wing, and four-wing chaotic attractors. Further analysis of the novel system shows that the multiple coexisting attractors can be observed with different system parameter values and initial values. Moreover, the feasibility of the proposed mathematical model is also presented by using Multisim simulations based on an electronic analog of the model. Finally, the active control method is used to design the appropriate controller to realize the synchronization between the proposed 6D memristive hyperchaotic system and the 6D hyperchaotic Yang system with different structures. The Routh–Hurwitz criterion is used to prove the rationality of the controller, and the feasibility and effectiveness of the proposed synchronization method are proved by numerical simulations.


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## Body

## 1. Introduction

Since the 1960s, nonlinear science has developed rapidly in various branches of disciplines. The in-depth study of nonlinear science not only has important theoretical value to the academic community, but also has a broad prospect for the practical application in life [1]. Chaos is one of the most important subjects in nonlinear motion, which creates a new situation of nonlinear science. Since the discovery of chaotic motion, chaotic dynamics has made rapid progress, and scientists from all over the world have made in-depth analysis and research on the characteristics of chaos [2–7]. Chaotic motion is a random behavior occurring in a defined nonlinear system. It is highly sensitive to initial conditions, has complex dynamic properties, and is difficult to predict. At present, it is widely used in complex networks [8–11], electronic circuits [12–15], image processing [16–20], random number generator [21–23], secure communication [24, 25], and other engineering fields.For the application of chaos in engineering, it is sometimes a key problem to generate a chaotic attractor with a complex topological structure. Most research in this field has been focused on the multiwing attractors [26–28], multiscroll attractors [29–32], and chaotic systems in the fractional-order form [33–35]. More and more articles are written on this topic every day, and numerous articles are devoted to explain the new high-dimensional chaotic systems and more complicated topological structure.Compared with chaotic systems, hyperchaotic systems have more complex dynamic behaviors, which have two or more positive Lyapunov indices, more complex topological structures, and more unpredictable dynamic behaviors and are more difficult to crack. The most common method to construct hyperchaotic systems is to introduce new variables to the proposed chaotic systems to increase the dimensions of the differential equations and increase the nonlinear terms. Since the discovery of a first 4D hyperchaotic system by Rossler in 1979 [36], many 4D hyperchaotic systems have been found in the literature such as hyperchaotic Lorenz system [37], hyperchaotic Chen system [38], hyperchaotic Lü system [39], hyperchaotic Yu system [40], hyperchaotic Wang system [41], and hyperchaotic Vaidyanathan system [42]. Recently, people have developed a strong interest in searching for 5D and 6D hyperchaotic systems with more complex dynamic behavior and such 5D and 6D hyperchaotic systems have been found in the literature such as hyperchaotic Vaidyanathan system [43], hyperchaotic Kemih system [44], hyperchaotic Lorenz system [45], and hyperchaotic Yang system [46]. Hyperchaotic systems can also produce multiscroll or multiwing attractors, which is a very important phenomenon. In recent years, some four-wing hyperchaotic attractors have appeared [47, 48]. These attractors generally have five equilibrium points, and each wing hovers near a nonzero equilibrium point. The three or five equilibrium points of the chaotic system are very important, especially in the multiscroll or multiwing chaotic system, but the multiscroll or multiwing hyperchaotic attractor with a linear equilibrium point is exciting.Memristor is a nonlinear passive element with nonlinearity and nonvolatility. In recent years, the research work has made gratifying progress, and the application of various memristors has become a research hotspot [49–51]. In 2008, scientists at HP LABS successfully built the first physically realized memristor [52], confirming the prediction of professor Chua in 1971 [53]. Since then, memristors have received extensive attention and research. Due to its small size and low power consumption, a memristor is an ideal choice for nonlinear circuits in chaos [54]. The common methods to produce hyperchaos are the linear feedback method and the nonlinear feedback method. Among them, the nonlinear feedback method is better than the linear feedback method. However, the product term of the nonlinear function makes the realization circuit more complex. If the memristor is used as the nonlinear feedback, it will greatly reduce the difficulty of circuit realization. At the same time, the memory ability of a memristor to flow through current is not possessed by conventional chaotic circuit elements [55]. Therefore, it is of practical significance to study the application of a memristor in a hyperchaotic system, and various hyperchaotic systems based on memristors have been paid close attention by researchers [56–59].In order to construct memristive hyperchaotic systems with more complex dynamics, some kind of 5D and 6D memristive hyperchaotic systems have been proposed recently [60–62]. In [60], a novel 5D hyperchaotic four-wing memristive system (HFWMS) was proposed by introducing a flux-controlled memristor with quadratic nonlinearity into a 4D hyperchaotic system as a feedback term. The HFWMS with multiline equilibrium and three positive Lyapunov exponents presented very complex dynamic characteristics, such as the existence of chaos, hyperchaos, limit cycles, and periods. In [62], a 6D autonomous system was presented by introducing a flux-controlled memristor model into an existing 5D hyperchaotic autonomous system, which exhibited hyperchaotic under a line or a plane of equilibria. Some other attractive dynamics were also observed, like hidden extreme multistability, transient chaos, bursting, and offset boosting phenomenon. It can be seen that such super-high-dimensional attractors cannot be ignored. Because of their complexity, the generated signals are usually suitable for secure communication and random number generation, so the super-high-dimensional attractors will be an added value to their randomness.Coexistence of multiple attractors is a kind of singular physical phenomenon often encountered in a nonlinear dynamic system. Under the condition of constant system parameters, when the initial state is changed, the trajectory of the system may asymptotically approach different stable states such as trend point, chaos, period, and quasiperiod [15, 23, 46]. In some special coupling systems and novel memristive chaotic systems, the coexistence of infinite number of attractors can also be observed [62]. Common multiple coexisting attractors generally have symmetry, and there is symmetric coexistence of left and right or upper and lower attractors. Recently, it has been found that the coexistence of asymmetric multiattractors also exists in some special systems, which is a new nonlinear phenomenon [61, 62]. Multiple coexisting attractors provide a great degree of freedom for the engineering application of nonlinear dynamic systems and also present a new challenge to the multistability state switching control technology. Therefore, the study of multiple coexisting attractors and their synchronization has important theoretical physical significance and engineering application value.With the rapid development of network communication technology, the confidentiality of information and the security of the system is not considered complete, resulting in increasingly serious information security problems. Information security technology mainly includes monitoring, scanning, detection, encryption, authentication, and attack prevention [63–72]. Due to the characteristics of chaotic systems such as aperiodic, continuous wideband, and noise-like, the use of chaotic synchronization has more stringent communication confidentiality, so it has received great attention in the field of information security. Pecora and Carroll [73] first proposed the concept of chaotic synchronization in 1990 and observed the phenomenon of chaotic synchronization on electronic circuits. This pioneering work greatly promoted the study of chaotic synchronization theory. Since then, complete synchronization [74], antisynchronization [40], generalized synchronization [75], projection synchronization [76, 77], lag synchronization [78], function projection synchronization [79], and shape synchronization [80] methods have been widely studied in the literature.In this paper, a novel 6D memristive hyperchaotic system is proposed based on a flux-controlled memristor model and the 5D hyperchaotic system introduced in [48]. Most importantly, the novel system generates the striking phenomenon of multiple coexisting chaotic attractors and exhibits hyperchaos with a line equilibrium. Under certain parameters and initial conditions, the system exits double-period bifurcation of the quasiperiod, which can produce four-wing hyperchaotic and chaotic attractors. A notable feature of the new system is the ability to generate two-wing and four-wing smooth chaotic attractors with special appearance. Then, an electronic circuit realization of the novel 6D memristive four-wing hyperchaotic system is presented to confirm the feasibility of the theoretical model. Finally, an adaptive active controller is designed to realize the global hyperchaos synchronization of the novel 6D memristive four-wing hyperchaotic systems and the 6D Yang hyperchaotic system with different structures.The rest of this work is structured as follows: InSection 2, the mathematical model of the novel 6D memristive hyperchaotic system that can generate two-wing and four-wing attractors is introduced. Numerical findings of the novel 6D memristive hyperchaotic system are carried out in Section 3 by using classical nonlinear diagnostic tools. The multiple coexisting attractors of the system are investigated, and the spectral entropy complexity is also reported. Some Multisim simulations based on a suitable designed electronic analog circuit diagram of the model are carried out to show its feasibility in Section 4. In Section 5, the novel chaotic system’s active control synchronization is derived. Finally, Section 6 draws the concluding remarks of this work.


## 2. A Novel 6D Memristive Four-Wing Hyperchaotic System

Recently, Zarei [48] proposed a 5D hyperchaotic system, whose differential equation can be described as(1)x˙=−ax+yz,y˙=−by+fu,z˙=−cz+xy+gw,w˙=dw−hz,u˙=eu−x2y,where x,y,z,w,andu are the state variables of the system and a,b,c,d,e,f,g,andh are the system parameters. The system has many interesting complex dynamical behaviors such as periodic orbit, chaos, and hyperchaos with only one equilibrium point. When proper system parameters and initial values are selected, the system can exhibit four-wing hyperchaotic attractors. The system has been well studied in [48], which shows the coexistence attractor and hyperchaotic attractor of two positive Lyapunov exponents. However, memristor chaos is not part of this feature. Our goal is to construct a high-dimensional system with coexistence attractors and memristor, thus forming a system of ordinary differential equations of memristive four-wing high-dimensional hyperchaos.Memristor is a passive two-terminal device that describes the relationship between magnetic fluxφ and charge q. The memristor used in this work is a flux-controlled memristor, which is described by the nonlinear constitutive relation between the terminal voltage u and the terminal current i of the device, i.e.,(2)i=Wφu,φ˙=u,where Wφ is a memductance function which is called the incremental memductance, defined as Wφ≡dqφ/φ.In this paper, theφ−q characteristic curve of the memristor is given by a smooth continuous cubic monotone-increasing nonlinearity, i.e., qφ=m+nφ3, where m,n>0. Thus, the memductance in this paper is given by(3)Wφ=m+3nφ2.By introducing the lux-controlled memristor model (3) into the second equation of system (1), a novel 6D memristive autonomous hyperchaotic system is constructed(4)x˙=−ax+yz,y˙=−by+fm+3nφ2u,z˙=−cz+xy+gw,w˙=dw−hz,u˙=eu−x2y,φ˙=u,where x,y,z,w,u,andφ are the state variables; a,b,c,d,e,f,g,h,m,andn are the system parameters. When a=10,b=60,c=20,d=15,e=40,f=1,g=50,h=10,m=1,3n=0.02, and the initial condition is set to 1,1,1,1,1,1, we use the Runge–Kutta algorithm (RK45) to solve the differential equation. Figure 1 shows the phase portraits of system (4) obtained through MATLAB simulation. It can be seen from the figure that the proposed system presents four-wing chaos in different phase planes.Figure 1
The four-wing chaotic attractor of system (4) in the (a) y−u−x plane, (b) y−z plane, (c) x−z plane, and (d) time-domain waveform of x−u.
(a)(b)(c)(d)In general, symmetry is widespread in chaotic systems, and system (4) is invariant under the coordinate transformation x,y,z,w,u,φ⟶−x,−y,z,w,−u,−φ and has the same symmetry as the original 5D system (1).Let the six equations at the right end of system (4) be zero, and the equilibrium point of system (4) can be obtained by solving the following equations:(5)−ax+yz=0,−by+fm+3nφ2u=0,−cz+xy+gw=0,dw−hz=0,eu−x2y=0,u=0.According to equation (5), system (4) has a line equilibrium point O=x,y,z,w,u,φx=y=z=w=0,u=0,φ=l, which means that every point on the φ-axis is the system equilibrium point, where l is an arbitrary real constant. The Jacobian matrix at the line equilibrium point O of system (4) is(6)Jo=−azy0000−b00fm+3nφ26fnφuyx−cg0000−hd00−2xy−x200e0000010.According to (6), the characteristic equation can be obtained:(7)λλ−eλ+aλ+bλ−m1λ−m2=0,where(8)m1=d−c+c−d2−4gh−cd2,m2=d−c−c−d2−4gh−cd2.According to the characteristic equation and system parameters,λ1=0,λ2=40,λ3=−10,λ4=−60,λ5=−2.5+13.9194i,andλ6=−2.5−13.9194i can be obtained. Therefore, there are one positive eigenvalue, one zero eigenvalue, and two negative eigenvalues, and the line equilibrium of system (4) is unstable saddle points.The divergence of system (4) is given by(9)∇V=dx˙dx+dy˙dy+dz˙dz+dw˙dw+du˙du+dφ˙dφ=−a−b−c+d+e,since −a+b−c−e=−35 satisfies ∇V<0, system (4) is dissipative and converges exponentially.


## 3. Dynamic Analysis of the Novel 6D Memristive Chaotic System

In this section, with the help of a bifurcation diagram, Lyapunov exponent spectrum, and phase portraits, we will use the fourth-order Runge–Kutta algorithm to numerically study the complex dynamic behavior of system (4) by MATLAB.


### 3.1. Fix Other Parameters and Change Parametera

Given parametersb=60,c=20,d=15,e=40,f=1,g=50,h=10,m=1,and3n=0.02 and initial conditions 0=1,y0=1,z0=1,w0=1,u0=1,andφ0=1, let parameter a be the bifurcation parameter of system (4), where Figure 2(a) shows the bifurcation diagram when system parameter a changes from 0 to 12, and Figure 2(b) shows the corresponding Lyapunov exponent spectrum. It can be seen from Figure 2 that the system is chaotic in 0,4.6 and hyperchaotic in 4.6,12. When a=12, the value of the Lyapunov exponent is 12.56, which is the maximum value of the simulation interval and larger than the maximum Lyapunov exponent of system (1) (LEmax=9.979). Suffice it to say, the introduction of a memristor can make the system more complex. When a=10, we use the famous wolf method to calculate the Lyapunov exponents. The LEs are LE1=10.16,LE2=2.187,LE3=0.0136,LE4=−0.5759,LE5=−16.08,andLE6=−18.86. There are two positive Lyapunov exponents, so system (4) is hyperchaotic. Based on the Lyapunov exponents, we also get the Kaplan–Yorke dimension that describes the complexity of the attractor. It can be computed by(10)DKY=D+∑i=1DLEiLED,where D is a constant satisfying ∑i=1DLEi≥0 and ∑i=1D+1LEi<0. According to equation (10), the Kaplan–Yorke dimension of system (4) is 4.7723, so the attractors generated by the new system are strange attractors.Figure 2
Lyapunov exponent spectrum and bifurcation diagram for parametera∈0,12: (a) bifurcation diagram; (b) Lyapunov exponent spectrum.
(a)(b)


### 3.2. Fix Other Parameters and Change Parameterd

Given parametersa=10,b=60,c=20,e=40,f=1,g=50,h=10,m=1,and3n=0.02 and initial conditions 0=1,y0=1,z0=1,w0=1,u0=1,andφ0=1, when parameter d∈−10,20, Figure 3(a) shows the bifurcation diagram changing with parameter d, and Figure 3(b) shows the corresponding Lyapunov exponent spectrum. It can be seen from Figure 3 that the system has doubly periodic bifurcation, chaos, and hyperchaos phenomena. The double-period bifurcation simulated in this paper is different from the simulation results of most papers, which are double-period bifurcation of the period, while in this paper, it is the double-period bifurcation of the quasiperiod. Table 1 gives a summary of dynamic characteristics of parameter d. The following analysis shows the dynamic behavior with respect to parameter d:(i)
Whend=−2, the maximum Lyapunov exponent of system (4) is zero (LE1,2=0,LE3,4,5,6<0), and the system is in a quasiperiodic 1 state. Figure 4(a) shows the corresponding phase portraits;(ii)
Whend=−1, the maximum Lyapunov exponent of system (4) is zero (LE1,2=0,LE3,4,5,6<0), and the system is in a quasiperiodic 2 state. Figure 4(b) shows the corresponding phase portraits;(iii)
Whend=0, system (4) has a positive Lyapunov exponent (LE1>0,LE2=0,LE3,4,5,6<0), and the system behaves as a two-wing chaotic attractor state. The corresponding phase portrait is shown in Figure 4(c);(iv)
Whend=16, system (4) has two positive Lyapunov exponents (LE1,2>0,LE3=0,andLE4,5,6<0), and the system is in a four-wing hyperchaos state. The corresponding four-wing phase portrait is shown in Figure 4(d).Figure 3
Lyapunov exponent spectrum and bifurcation diagram for parameterd∈−10,20: (a) bifurcation diagram; (b) Lyapunov exponent spectrum.
(a)(b)Table 1
Dynamical behavior and Lyapunov exponents under different parameter range ofd.
dLE1,LE2,LE3,LE4,LE5,LE6DynamicFigure−10,−1.20,0,−,−,−,−Quasiperiodic 1Figure4(a)−1.2,00,0,−,−,−,−Quasiperiodic 2Figure4(b)0,5+,0,−,−,−,−ChaoticFigure4(c)5,20+,+,0,−,−,−HyperchaoticFigure4(d)Figure 4
The phase portraits: (a) quasiperiodic 1, (b) quasiperiodic 2, (c) two-wing chaotic attractor, and (d) four-wing hyperchaotic attractor.
(a)(b)(c)(d)


### 3.3. Multiple Coexisting Attractors

In this section, we will study the multiple coexisting attractors of the proposed 6D memristive hyperchaotic system. Fixed system parameters area=10,b=60,c=20,e=40,f=1,g=50,h=10,m=1,and3n=0.02. When d=−3 and d=−0.5, two different initial conditions 1,1,1,1,1,1 and 1,1,−1,1,1,1 are taken to observe the phenomenon of coexistence quasiperiodic 1 and coexistence quasiperiodic 2 as shown in Figures 5(a) and 5(b). When d=0, two different initial conditions 1,1,1,1,1,1 and −1,1,1,1,1,1 are taken to observe the coexistence of two-wing chaotic attractors presented in Figure 5(c). Choosing d=15 and taking two different initial conditions 1,1,0.001,1,1,1 and 1,1,−0.001,1,1,1, the coexistence of four-wing hyperchaotic attractors is observed in Figure 5(d).When d=−7 is selected, the initial conditions 1,1,1,1,1,1, −1,1,1,1,1,1, 20,1,1,1,1,1, and −20,1,1,1,1,1 are selected, as shown in Figure 5(e); there are four quasiperiodic attractors coexisting, and the four attractors are symmetric.Figure 5
Various coexisting attractors with different values of parameterd in the x−z plane: (a) d=−3, (b) d=−0.5, (c) d=0, (d) d=15, and (e) d=−7.
(a)(b)(c)(d)(e)When the system parameters are selected asa=1,b=8,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02, the phase portraits of system (4) under different initial conditions are shown in Figure 6. Figure 6(a) shows the coexistence of four one-wing period-1 attractors, Figure 6(b) shows the coexistence of four one-wing multiperiod attractors, Figure 6(c) shows the coexistence of two-wing multiperiod attractors, and Figure 6(d) shows the coexistence of four two-wing multiperiod attractors. When the system parameters are selected as a=1,b=5,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02, the phase portraits of system (4) under different initial conditions are shown in Figure 7. In Figure 7(a), two-wing period-1 attractors coexist; in Figure 7(b), two-wing period-1 attractors coexist; in Figure 7(c), four-wing period-1 attractors coexist; in Figure 7(d), two-wing multiperiod attractors coexist, among which cyan, red, yellow, and earthy yellow are one group; black, green, blue, and magenta are the other. Figure 8 shows the phase portraits of different attractors when the initial conditions are 1,1,1,1,1,1 and 1,1,−1,1,1,1, but the parameter values in Figures 8(a) and 8(b) are different. The parameter values in Figure 8(a) are fixed to a=2,b=6,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02. It can be seen from the figure that the system has the coexistence of two-wing chaotic attractors. The parameter values in Figure 8(b) are fixed to a=2,b=6,c=2,d=−2,e=2,f=2,g=−1,h=−1,m=1,and3n=0.02. It can be seen from the Figure that the limit cycle presented by the system is completely symmetric. In conclusion, the attractors generated by the new system are symmetric with respect to different initial conditions.Figure 6
Coexisting attractors in the memristive hyperchaotic system: projections of different attractors on they−z plane for different initial conditions. (a) Symmetric period-1, 1,1,1,1,1,1 (red), −1,−1,−1,1,1,1 (blue), −1,−1,1,1,−1,−1 (green), and 1,1,−1,−1,−1,1 (yellow). (b) Symmetric one-wing multiperiod, −1,1,1,−1,1,1 (red), 1,1,−1,1,1,1 (blue), −1,−1,−1,1,−1,−1 (green), and 1,−1,1,−1,−1,−1 (yellow). (c) Symmetric two-wing multiperiod, −1,1,1,1,1,1 (red) and 1,1,−1,−1,1,1 (blue). (d) Symmetric two-wing multiperiod, −1,1,−1,1,−1,−1 (red), −1,−1,1,−1,1,1 (blue), 1,−1,−1,1,1,1 (green), and 1,1,1,−1,−1,−1 (yellow).
(a)(b)(c)(d)Figure 7
Coexisting attractors in the memristive hyperchaotic system: projections of different attractors on they−z plane for different initial conditions. (a) Symmetric two-wing period-1, 1,1,1,1,1,1 (blue) and 1,−1,−1,1,−1,1 (red). (b) Symmetric two-wing multiperiod, −1,1,1,1,1,−1 (red) and −1,−1,−1,−1,−1,1 (blue). (c) Symmetric one-wing period-1, −1,−1,−1,−1,−1,−1 (red), −1,1,1,1,1,1 (blue), 1,1,1,−1,−1,1 (green), and −1,1,−1,−1,−1,−1 (yellow). (d) Symmetric one-wing multiperiod, −1,−1,1,−1,1,1 (red), 1,−1,1,−1,−1,−1 (blue), 1,1,−1,1,1,1 (green), 1,−1,−1,1,1,1 (yellow), −1,−1,−1,1,−1,−1 (black), −1,1,−1,1,−1,−1 (khaki), −1,1,1,−1,1,1 (magenta), and 1,1,1,−1,−1,−1 (cyan).
(a)(b)(c)(d)Figure 8
Various coexisting attractors in they−z plane under initial conditions 1,1,±1,1,1,1: (a) coexistence of two-wing chaotic attractors and (b) coexistence of limit cycles.
(a)(b)


### 3.4. Complexity Analysis of Spectral Entropy

Spectral entropy (SE) algorithm is based on the Fourier transform to calculate the relative power spectrum and the Shannon entropy to calculate the SE complexity of the sequence, which reflects the disorder of time series in the frequency domain [81]. If the spectrum of the sequence is more complex, the SE of the chaotic system will be larger, making the system more complex, otherwise the system complexity is low [82]. Generally, the SE algorithm can be described as follows: given a chaotic random sequence xn,n=0,1,2,…,N−1 of length N, xn=xn−x¯ is adopted to remove the dc part, where x¯ is the mean value of the given sequence, and discrete Fourier transform is performed on sequence xn:(11)Xk=∑n=0N−1xne−j2πnk/N,where k=0,1,2,…,N−1. Taking half the total power of the calculation sequence for Xk:(12)pt=1N∑k=0N/2−1Xk2.According to the total power of the sequence, the relative power spectrum probability of the sequence is obtained:(13)pk=Xk2∑k=0N/2−1Xk2.The normalized SE is(14)SE=selnN/2,where se=−∑k=0N/2−1PklnPk. Using pk and the Shannon entropy, the spectral entropy of the system is obtained.The complexity of system (4) is analyzed by the SE algorithm. The control parameters a and d of the chaotic system are divided into 101×101 parts, where a∈0,12 and d∈−10,20, and then the SE of each point a,d in the parameter space is calculated. Figure 7 shows the SE diagram of system (4) based on the previous algorithm. It can be seen from the figure that Figures 9(a) and 9(b) well correspond to the largest Lyapunov exponents in Figures 2 and 3. The results show that with the increase of parameters a and d, the higher the complexity of the chaotic system is, the higher the complexity of the system is mainly concentrated in a∈4.6,12 and d∈0,20. Figure 9(c) shows the SE complexity in control parameters a and d planes. It can be seen from the figure that the system has high complexity in a large range, which means chaos or hyperchaos in these ranges.Figure 9
Spectral entropy (SE) complexity of system (4): (a) SE complexity versus a (d=15); (b) SE complexity versus d (a=10); (c) SE complexity in the a−d plane.
(a)(b)(c)


## 3.1. Fix Other Parameters and Change Parametera

Given parametersb=60,c=20,d=15,e=40,f=1,g=50,h=10,m=1,and3n=0.02 and initial conditions 0=1,y0=1,z0=1,w0=1,u0=1,andφ0=1, let parameter a be the bifurcation parameter of system (4), where Figure 2(a) shows the bifurcation diagram when system parameter a changes from 0 to 12, and Figure 2(b) shows the corresponding Lyapunov exponent spectrum. It can be seen from Figure 2 that the system is chaotic in 0,4.6 and hyperchaotic in 4.6,12. When a=12, the value of the Lyapunov exponent is 12.56, which is the maximum value of the simulation interval and larger than the maximum Lyapunov exponent of system (1) (LEmax=9.979). Suffice it to say, the introduction of a memristor can make the system more complex. When a=10, we use the famous wolf method to calculate the Lyapunov exponents. The LEs are LE1=10.16,LE2=2.187,LE3=0.0136,LE4=−0.5759,LE5=−16.08,andLE6=−18.86. There are two positive Lyapunov exponents, so system (4) is hyperchaotic. Based on the Lyapunov exponents, we also get the Kaplan–Yorke dimension that describes the complexity of the attractor. It can be computed by(10)DKY=D+∑i=1DLEiLED,where D is a constant satisfying ∑i=1DLEi≥0 and ∑i=1D+1LEi<0. According to equation (10), the Kaplan–Yorke dimension of system (4) is 4.7723, so the attractors generated by the new system are strange attractors.Figure 2
Lyapunov exponent spectrum and bifurcation diagram for parametera∈0,12: (a) bifurcation diagram; (b) Lyapunov exponent spectrum.
(a)(b)


## 3.2. Fix Other Parameters and Change Parameterd

Given parametersa=10,b=60,c=20,e=40,f=1,g=50,h=10,m=1,and3n=0.02 and initial conditions 0=1,y0=1,z0=1,w0=1,u0=1,andφ0=1, when parameter d∈−10,20, Figure 3(a) shows the bifurcation diagram changing with parameter d, and Figure 3(b) shows the corresponding Lyapunov exponent spectrum. It can be seen from Figure 3 that the system has doubly periodic bifurcation, chaos, and hyperchaos phenomena. The double-period bifurcation simulated in this paper is different from the simulation results of most papers, which are double-period bifurcation of the period, while in this paper, it is the double-period bifurcation of the quasiperiod. Table 1 gives a summary of dynamic characteristics of parameter d. The following analysis shows the dynamic behavior with respect to parameter d:(i)
Whend=−2, the maximum Lyapunov exponent of system (4) is zero (LE1,2=0,LE3,4,5,6<0), and the system is in a quasiperiodic 1 state. Figure 4(a) shows the corresponding phase portraits;(ii)
Whend=−1, the maximum Lyapunov exponent of system (4) is zero (LE1,2=0,LE3,4,5,6<0), and the system is in a quasiperiodic 2 state. Figure 4(b) shows the corresponding phase portraits;(iii)
Whend=0, system (4) has a positive Lyapunov exponent (LE1>0,LE2=0,LE3,4,5,6<0), and the system behaves as a two-wing chaotic attractor state. The corresponding phase portrait is shown in Figure 4(c);(iv)
Whend=16, system (4) has two positive Lyapunov exponents (LE1,2>0,LE3=0,andLE4,5,6<0), and the system is in a four-wing hyperchaos state. The corresponding four-wing phase portrait is shown in Figure 4(d).Figure 3
Lyapunov exponent spectrum and bifurcation diagram for parameterd∈−10,20: (a) bifurcation diagram; (b) Lyapunov exponent spectrum.
(a)(b)Table 1
Dynamical behavior and Lyapunov exponents under different parameter range ofd.
dLE1,LE2,LE3,LE4,LE5,LE6DynamicFigure−10,−1.20,0,−,−,−,−Quasiperiodic 1Figure4(a)−1.2,00,0,−,−,−,−Quasiperiodic 2Figure4(b)0,5+,0,−,−,−,−ChaoticFigure4(c)5,20+,+,0,−,−,−HyperchaoticFigure4(d)Figure 4
The phase portraits: (a) quasiperiodic 1, (b) quasiperiodic 2, (c) two-wing chaotic attractor, and (d) four-wing hyperchaotic attractor.
(a)(b)(c)(d)


## 3.3. Multiple Coexisting Attractors

In this section, we will study the multiple coexisting attractors of the proposed 6D memristive hyperchaotic system. Fixed system parameters area=10,b=60,c=20,e=40,f=1,g=50,h=10,m=1,and3n=0.02. When d=−3 and d=−0.5, two different initial conditions 1,1,1,1,1,1 and 1,1,−1,1,1,1 are taken to observe the phenomenon of coexistence quasiperiodic 1 and coexistence quasiperiodic 2 as shown in Figures 5(a) and 5(b). When d=0, two different initial conditions 1,1,1,1,1,1 and −1,1,1,1,1,1 are taken to observe the coexistence of two-wing chaotic attractors presented in Figure 5(c). Choosing d=15 and taking two different initial conditions 1,1,0.001,1,1,1 and 1,1,−0.001,1,1,1, the coexistence of four-wing hyperchaotic attractors is observed in Figure 5(d).When d=−7 is selected, the initial conditions 1,1,1,1,1,1, −1,1,1,1,1,1, 20,1,1,1,1,1, and −20,1,1,1,1,1 are selected, as shown in Figure 5(e); there are four quasiperiodic attractors coexisting, and the four attractors are symmetric.Figure 5
Various coexisting attractors with different values of parameterd in the x−z plane: (a) d=−3, (b) d=−0.5, (c) d=0, (d) d=15, and (e) d=−7.
(a)(b)(c)(d)(e)When the system parameters are selected asa=1,b=8,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02, the phase portraits of system (4) under different initial conditions are shown in Figure 6. Figure 6(a) shows the coexistence of four one-wing period-1 attractors, Figure 6(b) shows the coexistence of four one-wing multiperiod attractors, Figure 6(c) shows the coexistence of two-wing multiperiod attractors, and Figure 6(d) shows the coexistence of four two-wing multiperiod attractors. When the system parameters are selected as a=1,b=5,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02, the phase portraits of system (4) under different initial conditions are shown in Figure 7. In Figure 7(a), two-wing period-1 attractors coexist; in Figure 7(b), two-wing period-1 attractors coexist; in Figure 7(c), four-wing period-1 attractors coexist; in Figure 7(d), two-wing multiperiod attractors coexist, among which cyan, red, yellow, and earthy yellow are one group; black, green, blue, and magenta are the other. Figure 8 shows the phase portraits of different attractors when the initial conditions are 1,1,1,1,1,1 and 1,1,−1,1,1,1, but the parameter values in Figures 8(a) and 8(b) are different. The parameter values in Figure 8(a) are fixed to a=2,b=6,c=1,d=−20,e=1,f=2,g=1,h=−1,m=1,and3n=0.02. It can be seen from the figure that the system has the coexistence of two-wing chaotic attractors. The parameter values in Figure 8(b) are fixed to a=2,b=6,c=2,d=−2,e=2,f=2,g=−1,h=−1,m=1,and3n=0.02. It can be seen from the Figure that the limit cycle presented by the system is completely symmetric. In conclusion, the attractors generated by the new system are symmetric with respect to different initial conditions.Figure 6
Coexisting attractors in the memristive hyperchaotic system: projections of different attractors on they−z plane for different initial conditions. (a) Symmetric period-1, 1,1,1,1,1,1 (red), −1,−1,−1,1,1,1 (blue), −1,−1,1,1,−1,−1 (green), and 1,1,−1,−1,−1,1 (yellow). (b) Symmetric one-wing multiperiod, −1,1,1,−1,1,1 (red), 1,1,−1,1,1,1 (blue), −1,−1,−1,1,−1,−1 (green), and 1,−1,1,−1,−1,−1 (yellow). (c) Symmetric two-wing multiperiod, −1,1,1,1,1,1 (red) and 1,1,−1,−1,1,1 (blue). (d) Symmetric two-wing multiperiod, −1,1,−1,1,−1,−1 (red), −1,−1,1,−1,1,1 (blue), 1,−1,−1,1,1,1 (green), and 1,1,1,−1,−1,−1 (yellow).
(a)(b)(c)(d)Figure 7
Coexisting attractors in the memristive hyperchaotic system: projections of different attractors on they−z plane for different initial conditions. (a) Symmetric two-wing period-1, 1,1,1,1,1,1 (blue) and 1,−1,−1,1,−1,1 (red). (b) Symmetric two-wing multiperiod, −1,1,1,1,1,−1 (red) and −1,−1,−1,−1,−1,1 (blue). (c) Symmetric one-wing period-1, −1,−1,−1,−1,−1,−1 (red), −1,1,1,1,1,1 (blue), 1,1,1,−1,−1,1 (green), and −1,1,−1,−1,−1,−1 (yellow). (d) Symmetric one-wing multiperiod, −1,−1,1,−1,1,1 (red), 1,−1,1,−1,−1,−1 (blue), 1,1,−1,1,1,1 (green), 1,−1,−1,1,1,1 (yellow), −1,−1,−1,1,−1,−1 (black), −1,1,−1,1,−1,−1 (khaki), −1,1,1,−1,1,1 (magenta), and 1,1,1,−1,−1,−1 (cyan).
(a)(b)(c)(d)Figure 8
Various coexisting attractors in they−z plane under initial conditions 1,1,±1,1,1,1: (a) coexistence of two-wing chaotic attractors and (b) coexistence of limit cycles.
(a)(b)


## 3.4. Complexity Analysis of Spectral Entropy

Spectral entropy (SE) algorithm is based on the Fourier transform to calculate the relative power spectrum and the Shannon entropy to calculate the SE complexity of the sequence, which reflects the disorder of time series in the frequency domain [81]. If the spectrum of the sequence is more complex, the SE of the chaotic system will be larger, making the system more complex, otherwise the system complexity is low [82]. Generally, the SE algorithm can be described as follows: given a chaotic random sequence xn,n=0,1,2,…,N−1 of length N, xn=xn−x¯ is adopted to remove the dc part, where x¯ is the mean value of the given sequence, and discrete Fourier transform is performed on sequence xn:(11)Xk=∑n=0N−1xne−j2πnk/N,where k=0,1,2,…,N−1. Taking half the total power of the calculation sequence for Xk:(12)pt=1N∑k=0N/2−1Xk2.According to the total power of the sequence, the relative power spectrum probability of the sequence is obtained:(13)pk=Xk2∑k=0N/2−1Xk2.The normalized SE is(14)SE=selnN/2,where se=−∑k=0N/2−1PklnPk. Using pk and the Shannon entropy, the spectral entropy of the system is obtained.The complexity of system (4) is analyzed by the SE algorithm. The control parameters a and d of the chaotic system are divided into 101×101 parts, where a∈0,12 and d∈−10,20, and then the SE of each point a,d in the parameter space is calculated. Figure 7 shows the SE diagram of system (4) based on the previous algorithm. It can be seen from the figure that Figures 9(a) and 9(b) well correspond to the largest Lyapunov exponents in Figures 2 and 3. The results show that with the increase of parameters a and d, the higher the complexity of the chaotic system is, the higher the complexity of the system is mainly concentrated in a∈4.6,12 and d∈0,20. Figure 9(c) shows the SE complexity in control parameters a and d planes. It can be seen from the figure that the system has high complexity in a large range, which means chaos or hyperchaos in these ranges.Figure 9
Spectral entropy (SE) complexity of system (4): (a) SE complexity versus a (d=15); (b) SE complexity versus d (a=10); (c) SE complexity in the a−d plane.
(a)(b)(c)


## 4. Circuit Design

In recent years, the implementation of a chaotic system by hardware mainly includes analog discrete component circuit, CMOS integrated circuit, and continuous chaotic signal by modern digital signal processing technology, such as FPGA. CMOS technology is used to realize the chaotic oscillator circuit, which has the characteristics of low power consumption and small area [12–14, 49], but the design needs a long period, high cost, and difficult tuning [83–85]. Because of its large capacity and high reliability, FPGA is widely used in modern digital signal processing. However, FPGA needs a discrete continuous system, writing the underlying hardware code and requiring the computational intensive reading [15, 21, 60]. It is the most common method to generate a chaotic signal by using discrete components to design an analog circuit with simple structure, low cost, and easy operation [26–28, 30–32, 57–59, 61]. To further verify the dynamic characteristics of system (4), the system circuit was designed using discrete components: resistors, capacitors, operational amplifiers, and multipliers. In the circuit design, LF347 is used as the operational amplifier, the multiplier is AD633JN, and the multiplication factor is 0.1/V. The operating voltage of operational amplifier is ±E=±15V, and the saturation voltage measured by the operational amplifier and the multiplier is ±Vsat≈±13.5V. The relevant circuit equations are as follows:(15)v˙x=−1R1Cxvx+110⋅R2Cxvyvz,v˙y=−1R3Cyvy+1R4CyRvuR13+R100R14vφ2vu,v˙z=110⋅R5Czvxvy−1R6Czvz+1R7Czvw,v˙w=1R8Cwvw−1R9Cwvz,v˙u=1R10Cuvu−1100⋅R11Cuvx2vy,v˙φ=1R12Cφvu,where R1=R/a, R3=R/b, R4=R/f, R6=R/c, R7=R/g, R8=R/d, R9=R/h, R10=R/e, R13=R/fm, and R14=R/100⋅3fn. The hardware experiment simulation circuit of system (4) is shown in Figure 10. According to the parameter values in the four cases given in Table 2, the resistance values of the parameters in the equation are calculated when Cx=Cy=Cz=Cw=Cu=Cφ=10nF, R=100kΩ, R2=R5=10kΩ, R11=1kΩ, and R12=100kΩ. Figure 11 shows a group of phase portraits obtained by the Multisim simulator, which is basically consistent with the MATLAB numerical simulation results in the previous dynamic analysis and verifies the correctness of the chaotic circuit.Figure 10
The circuit diagram of system (4).Table 2
Resistance values corresponding to different parameter values and simulation Figure11.
CaseParametersResistorFigureCase 1a=1,b=8,c=1,d=−20,e=1, f=2,g=1,h=−1,m=1,3n=0.02R1=100kΩ,R3=12.5kΩ,R4=50kΩ,R6=100kΩ, R7=100kΩ,R8=5kΩ,R9=100kΩ,R10=100kΩFigure11(a)Case 2a=1,b=5,c=1,d=−20,e=1, f=2,g=1,h=−1,m=1,3n=0.02R1=100kΩ,R3=20kΩ,R4=50kΩ,R6=100kΩ, R7=100kΩ,R8=5kΩ,R9=100kΩ,R10=100kΩFigures11(b) and 11(c)Case 3a=2,b=6,c=1,d=−20,e=1, f=2,g=1,h=−1,m=1,3n=0.02R1=50kΩ,R3=16.5kΩ,R4=50kΩ,R6=100kΩ, R7=100kΩ,R8=5kΩ,R9=100kΩ,R10=100kΩFigures11(d) and 11(e)Case 4a=2,b=6,c=2,d=−2,e=2, f=2,g=−1,h=−1,m=1,3n=0.02R1=50kΩ,R3=16.5kΩ,R4=50kΩ,R6=50kΩ, R7=100kΩ,R8=50kΩ,R9=100kΩ,R10=50kΩFigure11(f)Figure 11
The circuit simulation diagram of system (4): (a) one-wing multiperiod, (b) period-1, (c) two-wing multiperiod, (d) two-wing, (e) four-wing, and (f) limit cycle.


## 5. Active Control Synchronization of the Novel 6D Memristive Hyperchaotic System

At present, many synchronization methods are based on the synchronization between two identical systems, but between practical engineering applications, many systems are of different structures, so it is very important to realize the synchronization between two systems with different structures. The system mainly consists of two parts: one is the main system and the other is the slave system. This section mainly uses the method of active control to realize the synchronization of system (4). Set the main system as(16)x˙1=a1x1+y1z1,y˙1=−b1y1+fm+3nφ12l1,z˙1=−c1z1+x1y1+g1w1,w˙1=d1w1−h1z1,l˙1=el1−x12y1,φ˙1=l1.The slave system is different from the main system in structure. The 6D hyperchaotic system designed by Yang et al. [46] is used as the slave system:(17)x˙2=a2y2−x2+w2+u1,y˙2=c2x2−y2−x2z2+l2+u2,z˙2=−b2z2−x2y2+u3,w˙2=d2w2−x2z2+u4,l˙2=−ky2+u5,φ˙2=h2φ2+u6,where u=u1,u2,u3,u4,u5,u6T is the active controller of the synchronous system, which can make the main system and the slave system tend to be synchronous under different parameters and initial conditions. The error variable is made as shown in the following equation:(18)e1=x2−x1,e2=y2−y1,e3=z2−z1,e4=w2−w1,e5=l2−l1,e6=φ2−φ1.Therefore, from the error variable, the main system (16), and the slave system (17), the error system equation can be obtained:(19)e˙1=a2e2−a2+a1e1+e4+a2y1−a2x1+a1x2−y1z1+w1+u1,e˙2=c2e1+c2x1−1+b1e2−y1−x2z2+1+fme5+b1y2−fml2−3nfφ12l1+l1+u2,e˙3=−b2−c1e3−b2z1+x2y2+c1z2−x1y1+g1e4−g1w2+u3,e˙4=d2+d1e4+d2w1−h1e3−x2z2−d1w2+h1z2+u4,e˙5=−ke2+ee5−ky1−el2+x12y1+u5,e˙6=h2e6+g2e2+e5+h2φ1+g2y1+u6.By simplifying the linear term of equation (19), the active control function is obtained:(20)u1=−a2y1+a2x1−a1x2+y1z1−w1+v1,u2=−c2x1+y1+x2z2−b1y2+fml2+3nfφ12l1−l1+v2,u3=b2z1−x2y2−c1z2+x1y1+g1w2+v3,u4=d2w1+x2z2+d1w2−h1z2+v4,u5=ky1+el2−x12y1+v5,u6=−h2φ1−g2y1+v6,where v=v1,v2,v3,v4,v5,v6T is the control input, and the linear error system without an active controller can be obtained by taking (20) into (19):(21)e˙1=a2e2−a2+a1e1+e4+v1,e˙2=c2e1+1+fme5−1+b1e2+v2,e˙3=g1e4+−b2−c1e3+v3,e˙4=d2+d1e4−h1e3+v4,e˙5=−ke2+ee5+v5,e˙6=h2e6+g2e2+e5+v6.To synchronize the system, we need to(22)limx⟶∞ei=0,i=1,2,3,4,5,6.The above formula shows that if system (21) tends to be stable with time and under the control input v=v1,v2,v3,v4,v5,v6T, then the error variable e=e1,e2,e3,e4,e5,e6T tends to zero and then the main system (16) and the slave system (17) are synchronized. To achieve this goal, we define a matrix A to express the relationship between the error system and the control input, which can be expressed as(23)v=A⋅e.According to the criteria of Routh–Hurwitz, if equation (19) is stable, all eigenvalues of a matrix must be negative. Therefore, equation (19) can be expressed as(24)v1v2v3v4v5v6=a1+a2−1−a20100−c2b100−fm−1000b2+c1−100000h1−d1−d2−1000k00−e−100−g200−1−h2−1e1e2e3e4e5e6.Then, the eigenvalue of the error system (21) is −1, −1, −1, −1, −1, and −1, so equation (24) can be reduced to(25)v1=a1+a2−1x2−x1+w2−w1−a2y2−y1,v2=−c2x2−x1+b1y2−y1+−fm−1l2−l1,v3=b2+c1−1z2−z1,v4=h1z2−z1+−d1−d2−1w2−w1,v5=ky2−y1+−e−1l2−l1,v6=−g2y2−y1−l2−l1+−h2−1φ2−φ1.The main slave system is simulated by MATLAB to verify whether the proposed system can achieve synchronization. According to the system equation, the parameters of the main system (16) are given as a1=10,b1=60,c1=20,d1=15,e=40,f=1,g1=50,h1=10,m=1,and3n=0.02, the parameters of the slave system (17) are set as a2=10,b2=8/3,c2=28,d2=2,g2=1,k=8.4,andh2=1, and the initial conditions of the main slave system are set as 1,1,1,1,1,1 and 0.1,0.1,0.1,0.1,0.1,0.1, respectively. Figure 12 shows a simulation diagram of the system error. It can be seen from Figure 12 that when t>2, two different structure hyperchaotic systems realize global synchronization. From Figure 13, it can also be seen from the six phase planes that the two systems realize synchronization.Figure 12
The trajectories of the synchronization errorse1,e2,e3,e4,e5, and e6.Figure 13
Synchronous phase diagram of two different structure systems in the corresponding plane. (a)x1−x2, (b) y1−y2, (c) z1−z2, (d) w1−w2, (e) l1−l2, and (f) φ1−φ2.
(a)(b)(c)(d)(e)(f)


## 6. Conclusion

This work presents a novel 6D memristive four-wing hyperchaotic system. Dynamical analysis and numerical simulation of the novel chaotic system were first carried out. Further analysis of the novel system shows that the multiple coexisting attractors can be observed with different system parameter values and initial values. Then, circuitry of the novel chaotic system was designed. The numerical and electronic circuit simulation results were found to be in good accordance. Besides, synchronization between the proposed 6D memristive hyperchaotic system and the 6D hyperchaotic Yang system with different structures was realized by an active control approach for secure communication applications, and the accuracy and validity of the results were verified by theoretical analysis and numerical simulations.



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