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

School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha 410114, China School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China College of Information Science and Engineering, Hunan Normal University, Changsha 410081, China


Introduction
Since the 1960s, nonlinear science has developed rapidly in various branches of disciplines. e 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][3][4][5][6][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][9][10][11], electronic circuits [12][13][14][15], image processing [16][17][18][19][20], random number generator [21][22][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][27][28], multiscroll attractors [29][30][31][32], and chaotic systems in the fractional-order form [33][34][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. e 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 fourwing hyperchaotic attractors have appeared [47,48]. ese attractors generally have five equilibrium points, and each wing hovers near a nonzero equilibrium point. e 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][50][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]. e 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]. erefore, 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][57][58][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][61][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. e 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 superhigh-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.
erefore, 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][64][65][66][67][68][69][70][71][72]. Due to the characteristics of chaotic systems such as aperiodic, continuous wideband, and noiselike, 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.

Complexity
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 doubleperiod 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. en, 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. e rest of this work is structured as follows: In Section 2, the mathematical model of the novel 6D memristive hyperchaotic system that can generate two-wing and fourwing attractors is introduced. Numerical findings of the novel 6D memristive hyperchaotic system are carried out in Section 3 by using classical nonlinear diagnostic tools. e 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.

A Novel 6D Memristive Four-Wing Hyperchaotic System
Recently, Zarei [48] proposed a 5D hyperchaotic system, whose differential equation can be described as where x, y, z, w, and u are the state variables of the system and a, b, c, d, e, f, g, and h are the system parameters. e 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. e 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 flux φ and charge q.
e 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., where W(φ) is a memductance function which is called the incremental memductance, defined as W(φ) ≡ dq(φ)/φ. In this paper, the φ − q characteristic curve of the memristor is given by a smooth continuous cubic monotone-increasing nonlinearity, i.e., q(φ) � m + nφ 3 , where m, n > 0. us, the memductance in this paper is given by By introducing the lux-controlled memristor model (3) into the second equation of system (1), a novel 6D memristive autonomous hyperchaotic system is constructed where x, y, z, w, u, and φ are the state variables; a, b, c, d, e, f, g, h, m, and n are the system parameters.
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.
In general, symmetry is widespread in chaotic systems, and system (4) is invariant under the coordinate transfor- 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: According to equation (5), system (4) has a line equi- which means that every point on the φ-axis is the system equilibrium point, where l is an arbitrary real Complexity 3 constant. e Jacobian matrix at the line equilibrium point O of system (4) is According to (6), the characteristic equation can be obtained: where According to the characteristic equation and system parameters, λ 1 � 0, λ 2 � 40, λ 3 � − 10, λ 4 � − 60, λ 5 � − 2.5 + 13.9194i, and λ 6 � − 2.5 − 13.9194i can be obtained. erefore, there are one positive eigenvalue, one zero eigenvalue, and two negative eigenvalues, and the line equilibrium of system (4) is unstable saddle points. e divergence of system (4) is given by since − a + b − c − e � − 35 satisfies ∇V < 0, system (4) is dissipative and converges exponentially.

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.

Fix Other Parameters and Change Parameter a.
Given 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) (LE max � 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   (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 where D is a constant satisfying 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.
e 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. e following analysis shows the dynamic behavior with respect to parameter d: (i) When d � − 2, the maximum Lyapunov exponent of system (4) is zero (LE 1,2 � 0, LE 3,4,5,6 < 0), and the system is in a quasiperiodic 1 state. Figure 4(a) shows the corresponding phase portraits; (ii) When d � − 1, the maximum Lyapunov exponent of system (4) is zero (LE 1,2 � 0, LE 3,4,5,6 < 0), and the system is in a quasiperiodic 2 state. Figure 4(b) shows the corresponding phase portraits; (iii) When d � 0, system (4) has a positive Lyapunov exponent (LE 1 > 0, LE 2 � 0, LE 3,4,5,6 < 0), and the system behaves as a two-wing chaotic attractor state. e corresponding phase portrait is shown in Figure 4(c); (iv) When d � 16, system (4) has two positive Lyapunov exponents (LE 1,2 > 0, LE 3 � 0, and LE 4,5,6 < 0), and the system is in a four-wing hyperchaos state. e corresponding four-wing phase portrait is shown in Figure 4(d).

Multiple Coexisting Attractors.
In this section, we will study the multiple coexisting attractors of the proposed 6D memristive hyperchaotic system.  Figure 5(e); there are four quasiperiodic attractors coexisting, and the four attractors are symmetric.

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 x(n), n � 0, 1, 2, . . . , N − 1 { } of length N, x(n) � x(n) − 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 x(n):  Complexity 7 where k � 0, 1, 2, . . . , N − 1. Taking half the total power of the calculation sequence for X(k): According to the total power of the sequence, the relative power spectrum probability of the sequence is obtained: e normalized SE is where se � − N/2− 1 k�0 P k ln P k . Using p k and the Shannon entropy, the spectral entropy of the system is obtained. e complexity of system (4) is analyzed by the SE algorithm. e control parameters a and d of the chaotic system   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. e 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.

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][13][14]49], but the design needs a long period, high cost, and difficult tuning [83][84][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. e operating voltage of operational amplifier is ±E � ±15 V, and the saturation voltage measured by the operational amplifier and the multiplier is ±|Vsat| ≈ ±13.5 V. e relevant circuit equations are as follows: where 13 � R/fm, and R 14 � R/(100 · 3fn). e 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 C x � C y � C z � C w � C u � C φ � 10 nF, R � 100 kΩ, R 2 � R 5 � 10 kΩ, R 11 � 1 kΩ, and R 12 � 100 kΩ. 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.

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. e system mainly consists of two parts: one is the main system and the other is the slave system.
is section mainly uses the method of active control to realize the synchronization of system (4). Set the main system as e slave system is different from the main system in structure. e 6D hyperchaotic system designed by Yang et al. [46] is used as the slave system: where u � [u 1 , u 2 , u 3 , u 4 , u 5 , u 6 ] T 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. e error variable is made as shown in the following equation: Figure 10: e circuit diagram of system (4). 10 Complexity erefore, from the error variable, the main system (16), and the slave system (17), the error system equation can be obtained:  Figure 11.
By simplifying the linear term of equation (19), the active control function is obtained: where 6 ] T is the control input, and the linear error system without an active controller can be obtained by taking (20) into (19): _ e 6 � h 2 e 6 + g 2 e 2 + e 5 + v 6 .
e above formula shows that if system (21) tends to be stable with time and under the control 6 ] T , then the error variable e � [e 1 , e 2 , e 3 , e 4 , e 5 , e 6 ] T 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 According to the criteria of Routh-Hurwitz, if equation (19) is stable, all eigenvalues of a matrix must be negative. erefore, equation (19) can be expressed as en, the eigenvalue of the error system (21) is − 1, − 1, − 1, − 1, − 1, and − 1, so equation (24) can be reduced to 12 Complexity  e 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 a 1 � 10, b 1 � 60, c 1 � 20, d 1 � 15, e � 40, f � 1, g 1 � 50, h 1 � 10, m � 1, and 3n � 0.02, the parameters of the slave system (17) are set as a 2 � 10, b 2 � 8/3, c 2 � 28, d 2 � 2, g 2 � 1, k � 8.4, and h 2 � 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.

Conclusion
is 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. en, circuitry of the novel chaotic system was designed. e 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.

Data Availability
All data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.