# Analysis and FPGA Realization of a Novel 5D Hyperchaotic Four-Wing Memristive System, Active Control Synchronization, and Secure Communication Application

Authors: Fei Yu; Li Liu; Binyong He; Yuanyuan Huang; Changqiong Shi; Shuo Cai; Yun Song; Sichun Du; Qiuzhen Wan
Journal: Complexity
 (2019)

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## Abstract

By introducing a flux-controlled memristor with quadratic nonlinearity into a 4D hyperchaotic system as a feedback term, a novel 5D hyperchaotic four-wing memristive system (HFWMS) is derived in this paper. 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. The dynamic characteristics of the HFWMS are analyzed by using equilibria, phase portraits, poincare map, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. Of particular interest is that this novel system can generate two-wing hyperchaotic attractor under appropriate parameters and initial conditions. Moreover, the FPGA realization of the novel 5D HFWMS is reported, which prove that the system has complex dynamic behavior. Finally, synchronization of the 5D hyperchaotic system with different structures by active control and a secure signal masking application of the HFWMS are implemented based on numerical simulations and FPGA. This research demonstrates that the hardware-based design of the 5D HFWMS can be applied to various chaos-based embedded system applications including random number generation, cryptography, and secure communication.


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## Body

## 1. Introduction

Nonlinear science is a new interdisciplinary subject to study the universality of nonlinear phenomena, which runs through almost every subject of meteorology [1, 2], mathematics [3–6], fluid mechanics [7, 8], complex network [9–12], electronics [13–15], and social science [16, 17]. Chaos is one of the most important achievements of nonlinear science. Its random-like and sensitive initial values make chaos have good potential applications in the fields of random number generation [18–20], cryptosystem [21, 22], image encryption [23–25] and secure communication [26–29]. In recent years, many new multiwing (or multiscroll) chaotic systems have been discovered and proposed [30–35]. The continuous introduction of various complex chaotic attractor models not only provides research basis for the development of chaotic system theory, but also provides rich subjects for the practical application of chaotic theory.Since Rossler proposed the first hyperchaotic system with two positive Lyapunov exponents [36], a large number of researchers have begun to devote themselves to the study of hyperchaotic systems [37–42]. Hyperchaotic systems are more sensitive, pseudorandom, and have larger key space, which makes them more suitable for applications such as secure communication and image encryption than chaotic systems. In order to construct more complex chaotic attractors, a large number of literatures have recently reported the multiwing hyperchaotic systems [43–46]. In [43], in order to overcome the inherent difficulties of iteratively adjusting multiparameters in traditional multiparameter control, a unified step function for single-parameter control is proposed to construct a nonequilibrium multiwing hyperchaotic system. In [45], by introducing mirror symmetric transformation into the hyperchaotic system, various hyperchaotic attractors with mirror symmetric grids are obtained. In [46], a five-dimensional (5D) autonomous hyperchaotic attractor with four wing is introduced, which has only eight parameters to be controlled and only one equilibrium point.Memristor is an electronic device that describes the relationship between charge and magnetic flux, which was proposed in 1971 by Chua [47] and for the first time realized by HP Labs in 2008 [48]. Because of its strong nonlinear characteristics, memristors have potential applications in many engineering fields, and their research has attracted more and more attention [49, 50]. Recently, many nonlinear memristive oscillators have been proposed [51–53], creating a memristive hyperchaotic system with a multiwing attractor having a practical significance, which has become a research hotspot [54–57]. In [55], by introducing a flux-controlled memristor into a multiwing system, a multiwing hyperchaotic attractor is observed in a no-equilibrium memristive system. In [56], a hyperchaotic system is constructed by adding only a smooth flux-controlled memristor to the 3D pseudo four-wing chaotic system, which can generate a real four-wing hyperchaotic attractor with a line of equilibrium. In [57], flux-controlled memristors are used to replace the resistors in the circuit of the modified Lü system, and this new memristive system can exhibit the hyperchaotic multiwing attractor with two relatively large positive Lyapunov exponents. However, a review of literature revealed there are no research studies that examined the four-wing behavior and three-positive Lyapunov exponents in memristive hyperchaotic systems with dimensions greater than four. This kind of high-dimensional hyperchaotic systems cannot be ignored. Because of their complexity, signal generation is usually used for random number generation and secure communication just to name a few.In recent years, the main methods to realize chaotic or hyperchaotic attractors are analog circuits, such as breadboards based on discrete components [55–57] and integrated circuits (ICs) based on CMOS technology [14, 33, 34, 37]. With the change of time and temperature in analog circuit, the device will have temperature drift and poor control accuracy. Therefore, it is difficult to realize the chaotic system with high precision by analog circuit, and the breadboard is not easy to carry and digitally store. The design of high-dimensional chaotic systems using CMOS technology generally requires multipliers, which are difficult to design. At the same time, ICs have the shortcomings of long development cycle and high cost [58–60]. Therefore, researchers began to focus on digital circuits with low cost, short design cycle, fast speed, low power consumption, and high accuracy, such as digital signal processor (DSP) [61, 62] and field programmable gate array (FPGA) [63–66]. It takes a long time for DSPs to generate chaotic signals at high frequencies and DSP chips to perform operations in order to calculate the value of output signals. On the other hand, FPGA chips have a relatively flexible architecture to achieve parallel operation, and the design and test cycles of the chips is particularly low [67]. In order to increase and expand engineering applications based on chaos, chaotic systems are diversified and need flexible architecture support. With the digitalization and reconfigurability of the FPGA, chaotic systems and their applications can be more flexible. Thus, different forms of signals can be easily generated with the change of parameters of chaotic systems. In addition, the related memristive chaotic system can also be realized alternately by various memristor functions. At present, there are several studies related to designs of chaotic systems based on FPGA. Tuna et al. [63] implemented the Liu-Chen chaotic system on the Xilinx virtex-6 FPGA chip using the 32-bit IQ-Math fixed-point number Heun algorithm. Ahmadi et al. [64] designed a 5D chaotic system on the Xilinx Kintex-7 KC-705 kit FPGA chip using the IEEE 754 32-bit fixed-point number Euler’s method. Xu et al. [65] designed a 3D memristive chaotic system on the Xilinx Spartan-6 FPGA chip using the Euler’s method of IEEE 754 32-bit floating-point number standard. As far as we know, few literatures have reported the realization of the 5D memristive hyperchaotic system based on FPGA.Synchronization of chaotic systems has attracted much attention in recent years due to their applications in chemical reactors, secure communication, and the development of secure cryptosystems [68, 69]. Aiming at chaotic synchronization, several methods such as linear feedback control [70], sliding mode control [71], adaptive control [72–74], backstepping nonlinear control [75], shape control [76, 77], and active control [78–80] have been proposed for synchronization of chaotic systems. Compared with other synchronization methods, the active control method is simple, efficient, and flexible which has been successfully applied to the synchronization of chaotic systems. In [79], the synchronization and antisynchronization of the fractional-order chaotic financial system with market confidence are studied by using the active control method. The results show that the speed of synchronization (antisynchronization) increases with the increase of the order. In [80], the synchronization of chaotic systems with different orders under the influence of unknown model uncertainties and external disturbances is studied by using robust generalized active control approach. With the rapid development of computer technology, more and more attention has been paid to information security [81–91]. Secure communication based on chaotic synchronization is an important branch of information security research, which has been widely studied by many scholars [92–96]. Because of the pseudorandomness, unpredictability, and initial sensitivity of memristive chaotic systems, the encrypted information can be hidden in chaotic signals which are highly similar to the noise. In [97], based on the synchronization of the memristive chaotic system, the encrypting and decrypting of information signals are carried out to realize the secure communication with the help of LabVIEW. However, whether chaotic synchronization is achieved by the active control method [78–80] or secure communication based on chaotic synchronization [92–96], numerical simulations is used to achieve these designed methods. In some chaotic information systems, such as chaotic-based CDMA communications and many other chaotic digital information systems, digital implementation may be required [98–100].Motivated by the above discussions, based on a flux-controlled memristor model and the 4D hyperchaotic system introduced in [39], a 5D hyperchaotic system is proposed. Most importantly, the new system generates four-wing and two-wing hyperchaotic attractor phenomenon with three and two positive Lyapunov exponents, respectively and exhibits hyperchaos with multiline equilibrium. Complete dynamic properties of this new system are studied. Also, with the help of FPGA implementation, this 5D hyperchaotic four-wing memristive system (HFWMS) is realized. Finally, active control synchronization of the 5D hyperchaotic system with different structures and a secure signal masking application of the 5D HFWMS are implemented based on numerical simulations and FPGA.This paper is organized as follows. In Section2, the novel 5D HFWMS with multiline equilibrium is introduced and its dynamic properties are discussed. Section 3 is devoted to design, test, and analysis results of FPGA-based HFWMS. In addition, the active control synchronization and chaotic secure communication design of the 5D HFWMS are achieved, and the FPGA experimental results are presented. Section 4 concludes this paper with a summary of the main results.


## 2. Novel 5D HFWMS and Its Dynamic Properties

### 2.1. The 5D HFWMS

Recently, Volos et al. [39] have announced a novel 4D four-wing hyperchaotic system, which is described by(1)x˙=ax+y+yz−cw,y˙=yz−xz,z˙=−z−mxy+b,w˙=x,where a,b,c, and m are all constants and x,y,z,andw are the state variables. System (1) has two positive Lyapunov exponents, showing the four-wing hyperchaotic attractor. Unlike most existing hyperchaotic systems, this hyperchaotic system has a saddle-focus equilibrium and the second equation of the system has no linear term.In this paper, by introducing a flux-controlled memristor to the first equation of system (1), a novel 5D HFWMS is derived by(2)x˙=ax+dWφy+yz−cw,y˙=yz−xz,z˙=−z−mxy+b,w˙=x,φ˙=y,where d is a positive parameter, expressed as memristor strength and Wφ is a memductance function, defined as Wφ=dqφ/dφ. Here, the φ−q characteristic curve of the memristor is given by a smooth continuous cubic monotone increasing nonlinearity [46–49], and then the memductance is given by(3)Wφ=dqφdφ=deφ+nφ3dφ=e+3nφ2,where e and n are two positive constants. This flux-controlled memristor is easier to analyze and implement. At present, many researchers use the special nonlinear dynamic characteristics of this memristor to construct complex chaotic oscillators [54–57].


### 2.2. Equilibria and Stability

The equilibria of system (2) are obtained by setting its right-hand side to zero, that is,(4)ax+de+3nφ2y+yz−cw=0,yz−xz=0,−z−mxy+b=0,x=0,y=0.Through equation (4), we can easily observe that the equilibria of system (2) is a multiline equilibrium point O=x,y,z,wx=y=w=0,z=b,φ=η, where b is an integer and η is an arbitrary real constant.The Jacobian matrix of system (2) at the multiline equilibrium point is(5)JO=ade+3nφ2+zy−c6nφdy−zzy−x00−my−mx−1001000001000.According to the Jacobian matrix (5), we can obtain the characteristic equation of system (2) as follows:(6)λλ+1λ3−a+bλ2+b2+dbe+3dbnφ2+c+abλ−bc=0.Equation (6) can be rewritten to as follows:(7)λλ+1λ3+m1λ2+m2λ+m3=0,where(8)m1=−a+b,m2=b2+dbe+3dbnφ2+c+ab,m3=−bc.From the eigenvalue equation (7), it can be seen that Jacobian matrix (5) has one zero eigenvalue, one negative eigenvalue, and three nonzero eigenvalues. To judge whether system (2) is stable, it is necessary to discriminate the three nonzero eigenvalues. According to the Routh–Hurwitz stability criterion, the need equation (8) satisfies(9)m1>0,m3>0,m1m2−m3>0.If all three conditions in equation (9) are satisfied, the multiline equilibrium point O is stable, otherwise it is unstable. The unstable equilibrium of the system will lead to chaotic behavior. When a=1,b=1,c=0.7,m=1,d=0.2,e=0.1,n=0.01, m1, and m2 are both less than zero, so we can judge that system (2) is unstable.


### 2.3. Symmetry and Dissipativity

The proposed 5D HFWMS has the same symmetry as system (1), and both of them are invariant with respect to z-axis symmetry under coordinate transformation x,y,z,w,φ⟶−x,−y,z,−w,−φ.Furthermore, by calculating(10)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w+∂φ˙∂φ=a+z−1,when a+z<1, system (2) is dissipative and converges exponentially.


### 2.4. Analysis of the 5D HFWMS

Here, the dynamic behavior of the 5D HFWMS is numerically investigated by the use of several tools such as the phase portraits, poincare map, Lyapunov exponentials, and bifurcation diagram.


#### 2.4.1. Four-Wing Hyperchaotic Attractor

When the system parameters are selecteda=1,b=1,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 1,−1,1,1,1, the phase portraits of system (2) shown in Figure 1 are obtained by Matlab simulation, which is a typical four-wing chaotic attractor, and the time variations of state equations x,y,z,andw are provided in Figure 2.Figure 1
The four-wing hyperchaotic attractor of system (2): (a) in the x−y plane, (b) in the x−z plane, (c) in the y−z plane, and (d) in the x−y−z space.
(a)
(b)
(c)
(d)Figure 2
The time variation of state equationsx,y,z,andw of system (2): (a) t−x, (b) t−y, (c) t−z, and (d) t−w.
(a)
(b)
(c)
(d)The Lyapunov exponent is used to measure the perturbation caused by initial conditions. If there are slight differences in the system, two adjacent trajectories in the phase space are separated exponentially with time. The Lyapunov exponent is a useful tool for measuring chaotic systems, in particular, it is usually determines whether a chaotic system is chaotic or hyperchaotic according to the number of positive Lyapunov exponents. According to the given system parameters and initial conditions, the Lyapunov exponent ofa∈−1,1 is simulated by the Jacobi matrix method. The numerical results are shown in Figure 3(a) (the five LE5 is out of plot). From Figure 3(a), we can clearly see that the system has complex dynamic behaviors such as periodic orbit, chaos, and hyperchaos. When a∈−1,−0.03 and a∈0.22,0.3, the system is a periodic orbit, and Figure 4 are the phase portraits of system (2) when a=−1 (the LEs are 0, −0.0424, −0.4394, −0.6696, and −3.4835). When a=0.2, the LEs are 0.0790, −0.0078, −0.0627, −0.1078, and −1.9274, with a positive Lyapunov exponent, so system (2) is in chaotic state (a∈0.3,0.48 and a∈−0.03,0.22). When a∈0.48,1, the system is hyperchaotic, and the typical four-wing hyperchaotic attractor is shown in Figure 1. When a=0.78, the LEs are LE1=0.1712,LE2=0.0907,LE3=0.0107,LE4=0,and LE5=−2.3243, and it can be judged that system (2) is hyperchaotic and has three positive Lyapunov exponents. Figure 3(b) describes the bifurcation diagram of system (2) varying with parameter a. With the increase of a, the system changes from period to chaos. The Kaplan–Yorke dimension of system (2) can be calculated by the following formula:(11)DL=j+∑i=1jLjLj+1=4+0.1712+0.0907+0.0107−02.3243=4.1172,where j is the largest integer satisfying ∑i=1jLj≥0 and ∑i=1jLj<0. It can be seen from DL=4.1172 that the Lyapunov dimension of system (2) is fractional. Therefore, the proposed 5D FWMHS is a real hyperchaotic system with strong complexity.Figure 3
Lyapunov exponents and bifurcation diagram for increasing parametera (the five LE5 is out of plot).
(a)
(b)Figure 4
Periodic phase portraits of system (2) with a=−1: (a) in the x−z plane, (b) in the y−z plane, and (c) in the x−y plane.
(a)
(b)
(c)As an important analytical tool, the Poincare map is used to further study the dynamic characteristics of the 5D HFWMS. Figure5(a) shows the Poincare map of the x−y plane at z=0 and four branches can be seen; Figure 5(b) shows the Poincare map of the x−z plane at y=0 and has many branches; Figure 5(c) shows the Poincare map of the y−z plane at w=0 and the outline of the four wings can be seen, indicating the existence of the four-wing phenomenon. Figure 5 shows that system (2) has a four-wing chaotic attractor with fractal structure.Figure 5
Poincare map of system (2) with parameters a=1: (a) projection on x−y plane with z=0, (b) projection on x−z plane with y=0, and (c) projection on y−z plane with w=0.
(a)
(b)
(c)


#### 2.4.2. Two-Wing Hyperchaotic Attractor

When the system parameters are chosen asa=11,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 0,1,0,0,0, and the numerical results of the Lyapunov exponent varying with the system parameter b is shown in Figure 6(a). Figure 6(b) is a bifurcation diagram corresponding to Figure 6(a). From Figure 6, we can see that system (2) has more complex dynamic behavior in the parameter b∈−10,0 interval, such as quasi-periodicity, period, chaos, and hyperchaos. Table 1 classifies these dynamical behaviors and then makes a detailed analysis of LEs and their dynamical behaviors varying with parameter b as follows:(i)
Whenb∈−4.4,0, the Lyapunov exponents of system (2) at b=−1 are 0.5791, 0.1087, −0.0316, −0.2607, and −4.5443. The system is in a hyperchaotic state, and the corresponding phase portraits are shown in Figure 7(a).(ii)
Whenb∈−5.55,−4.4, the Lyapunov exponents of system (2) at b=−5 are 0, −0.0132, −0.2670, −0.4978, and −2.1754. The system is in a period-5 state, and the corresponding phase portraits are shown in Figure 7(b).(iii)
Whenb∈−7,−6.6 and b∈−10,−7.2, the Lyapunov exponents of system (2) at b=−8 are 0.0279, −0.0271, −0.2093, −0.3481, and −1.9681. The phase portraits shown in Figure 7(c) show that the system is in a quasi-periodic state.(iv)
Whenb∈−6.6,−5.55 and b∈−7.2,−7, the Lyapunov exponents of system (2) at b=−6.15 are 0.0918, −0.0091, −0.16–0.3278, and −2.2332. The system shows a chaotic state, and the corresponding phase portraits are shown in Figure 7(d).Figure 6
Lyapunov exponents and bifurcation diagram for increasing parameterb.
(a)
(b)Table 1
The LEs and dynamical behavior under different parameter range ofb.
b
LE
1
,
LE
2
,
LE
3
,
LE
4
,
LE
5
Dynamic
−
4.4
,
0
+
,
+
,
0
,
−
,
−
Hyperchaotic
−
5.55
,
−
4.4
0
,
−
,
−
,
−
,
−
Periodic
−
6.6
,
−
5.55
+
,
0
,
−
,
−
,
−
Chaotic
−
7
,
−
6.6
0,0
,
−
,
−
,
−
Torus
−
7.2
,
−
7
+
,
0
,
−
,
−
,
−
Chaotic
−
10
,
−
7.2
0,0
,
−
,
−
,
−
Quasi-periodicFigure 7
Simulated phase portraits of system (2) with different b: (a) b=−1, (b) b=−5, (c) b=−8, and (d) b=−6.15.
(a)
(b)
(c)
(d)


#### 2.4.3. Complexity Analysis by Spectral Entropy

The complexity of spectral entropy (SE) is based on the discrete Fourier transformation. The distribution of energy in Fourier transform domain is calculated, and then the SE value is calculated by Shannon entropy, which reflects the disorder of time series in frequency domain [61, 101, 102]. The chaotic diagram using the complexity of SE usually reflects the spatial complexity of chaotic system parameters. In this section, the SE algorithm is used to analyze the complexity of system (2). Figure 8 shows the complexity of SE of system (2) under initial condition 1,−1,1,1,1. It can be seen that Figure 8(a) corresponds well to the maximum Lyapunov exponent in Figure 3. The control parameters a and b of the chaotic system are divided into 101×101 parts, including a∈−1,1 and b∈0,5. Then, calculate the SE of each point a,b in the parameter space, as shown in Figure 8(b). The results show that with the increase of parameter a, the more complex the chaotic system is, and the higher the complexity of the system is mainly concentrated in a∈0.5,1. Figure 9 is the SE diagram of system (2) under initial condition 0,1,0,0,0, where a∈10,20 and b∈−10,0. Figure 9(a) corresponds well to the maximum Lyapunov exponent in Figure 6. Figure 9(b) shows the complexity of SE in the plane of control parameters a and b. The results show that when a=11, with the increase of parameter b, the larger the SE value is, and the higher the complexity of the system is mainly concentrated in b∈−3.8,0.Figure 8
The complexity of SE of system (2): (a) SE versus a when b=1 and (b) in the a−b plane.
(a)
(b)Figure 9
The complexity of SE of system (2): (a) SE versus b when a=11 and (b) in the a−b plane.
(a)
(b)


## 2.1. The 5D HFWMS

Recently, Volos et al. [39] have announced a novel 4D four-wing hyperchaotic system, which is described by(1)x˙=ax+y+yz−cw,y˙=yz−xz,z˙=−z−mxy+b,w˙=x,where a,b,c, and m are all constants and x,y,z,andw are the state variables. System (1) has two positive Lyapunov exponents, showing the four-wing hyperchaotic attractor. Unlike most existing hyperchaotic systems, this hyperchaotic system has a saddle-focus equilibrium and the second equation of the system has no linear term.In this paper, by introducing a flux-controlled memristor to the first equation of system (1), a novel 5D HFWMS is derived by(2)x˙=ax+dWφy+yz−cw,y˙=yz−xz,z˙=−z−mxy+b,w˙=x,φ˙=y,where d is a positive parameter, expressed as memristor strength and Wφ is a memductance function, defined as Wφ=dqφ/dφ. Here, the φ−q characteristic curve of the memristor is given by a smooth continuous cubic monotone increasing nonlinearity [46–49], and then the memductance is given by(3)Wφ=dqφdφ=deφ+nφ3dφ=e+3nφ2,where e and n are two positive constants. This flux-controlled memristor is easier to analyze and implement. At present, many researchers use the special nonlinear dynamic characteristics of this memristor to construct complex chaotic oscillators [54–57].


## 2.2. Equilibria and Stability

The equilibria of system (2) are obtained by setting its right-hand side to zero, that is,(4)ax+de+3nφ2y+yz−cw=0,yz−xz=0,−z−mxy+b=0,x=0,y=0.Through equation (4), we can easily observe that the equilibria of system (2) is a multiline equilibrium point O=x,y,z,wx=y=w=0,z=b,φ=η, where b is an integer and η is an arbitrary real constant.The Jacobian matrix of system (2) at the multiline equilibrium point is(5)JO=ade+3nφ2+zy−c6nφdy−zzy−x00−my−mx−1001000001000.According to the Jacobian matrix (5), we can obtain the characteristic equation of system (2) as follows:(6)λλ+1λ3−a+bλ2+b2+dbe+3dbnφ2+c+abλ−bc=0.Equation (6) can be rewritten to as follows:(7)λλ+1λ3+m1λ2+m2λ+m3=0,where(8)m1=−a+b,m2=b2+dbe+3dbnφ2+c+ab,m3=−bc.From the eigenvalue equation (7), it can be seen that Jacobian matrix (5) has one zero eigenvalue, one negative eigenvalue, and three nonzero eigenvalues. To judge whether system (2) is stable, it is necessary to discriminate the three nonzero eigenvalues. According to the Routh–Hurwitz stability criterion, the need equation (8) satisfies(9)m1>0,m3>0,m1m2−m3>0.If all three conditions in equation (9) are satisfied, the multiline equilibrium point O is stable, otherwise it is unstable. The unstable equilibrium of the system will lead to chaotic behavior. When a=1,b=1,c=0.7,m=1,d=0.2,e=0.1,n=0.01, m1, and m2 are both less than zero, so we can judge that system (2) is unstable.


## 2.3. Symmetry and Dissipativity

The proposed 5D HFWMS has the same symmetry as system (1), and both of them are invariant with respect to z-axis symmetry under coordinate transformation x,y,z,w,φ⟶−x,−y,z,−w,−φ.Furthermore, by calculating(10)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z+∂w˙∂w+∂φ˙∂φ=a+z−1,when a+z<1, system (2) is dissipative and converges exponentially.


## 2.4. Analysis of the 5D HFWMS

Here, the dynamic behavior of the 5D HFWMS is numerically investigated by the use of several tools such as the phase portraits, poincare map, Lyapunov exponentials, and bifurcation diagram.


### 2.4.1. Four-Wing Hyperchaotic Attractor

When the system parameters are selecteda=1,b=1,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 1,−1,1,1,1, the phase portraits of system (2) shown in Figure 1 are obtained by Matlab simulation, which is a typical four-wing chaotic attractor, and the time variations of state equations x,y,z,andw are provided in Figure 2.Figure 1
The four-wing hyperchaotic attractor of system (2): (a) in the x−y plane, (b) in the x−z plane, (c) in the y−z plane, and (d) in the x−y−z space.
(a)
(b)
(c)
(d)Figure 2
The time variation of state equationsx,y,z,andw of system (2): (a) t−x, (b) t−y, (c) t−z, and (d) t−w.
(a)
(b)
(c)
(d)The Lyapunov exponent is used to measure the perturbation caused by initial conditions. If there are slight differences in the system, two adjacent trajectories in the phase space are separated exponentially with time. The Lyapunov exponent is a useful tool for measuring chaotic systems, in particular, it is usually determines whether a chaotic system is chaotic or hyperchaotic according to the number of positive Lyapunov exponents. According to the given system parameters and initial conditions, the Lyapunov exponent ofa∈−1,1 is simulated by the Jacobi matrix method. The numerical results are shown in Figure 3(a) (the five LE5 is out of plot). From Figure 3(a), we can clearly see that the system has complex dynamic behaviors such as periodic orbit, chaos, and hyperchaos. When a∈−1,−0.03 and a∈0.22,0.3, the system is a periodic orbit, and Figure 4 are the phase portraits of system (2) when a=−1 (the LEs are 0, −0.0424, −0.4394, −0.6696, and −3.4835). When a=0.2, the LEs are 0.0790, −0.0078, −0.0627, −0.1078, and −1.9274, with a positive Lyapunov exponent, so system (2) is in chaotic state (a∈0.3,0.48 and a∈−0.03,0.22). When a∈0.48,1, the system is hyperchaotic, and the typical four-wing hyperchaotic attractor is shown in Figure 1. When a=0.78, the LEs are LE1=0.1712,LE2=0.0907,LE3=0.0107,LE4=0,and LE5=−2.3243, and it can be judged that system (2) is hyperchaotic and has three positive Lyapunov exponents. Figure 3(b) describes the bifurcation diagram of system (2) varying with parameter a. With the increase of a, the system changes from period to chaos. The Kaplan–Yorke dimension of system (2) can be calculated by the following formula:(11)DL=j+∑i=1jLjLj+1=4+0.1712+0.0907+0.0107−02.3243=4.1172,where j is the largest integer satisfying ∑i=1jLj≥0 and ∑i=1jLj<0. It can be seen from DL=4.1172 that the Lyapunov dimension of system (2) is fractional. Therefore, the proposed 5D FWMHS is a real hyperchaotic system with strong complexity.Figure 3
Lyapunov exponents and bifurcation diagram for increasing parametera (the five LE5 is out of plot).
(a)
(b)Figure 4
Periodic phase portraits of system (2) with a=−1: (a) in the x−z plane, (b) in the y−z plane, and (c) in the x−y plane.
(a)
(b)
(c)As an important analytical tool, the Poincare map is used to further study the dynamic characteristics of the 5D HFWMS. Figure5(a) shows the Poincare map of the x−y plane at z=0 and four branches can be seen; Figure 5(b) shows the Poincare map of the x−z plane at y=0 and has many branches; Figure 5(c) shows the Poincare map of the y−z plane at w=0 and the outline of the four wings can be seen, indicating the existence of the four-wing phenomenon. Figure 5 shows that system (2) has a four-wing chaotic attractor with fractal structure.Figure 5
Poincare map of system (2) with parameters a=1: (a) projection on x−y plane with z=0, (b) projection on x−z plane with y=0, and (c) projection on y−z plane with w=0.
(a)
(b)
(c)


### 2.4.2. Two-Wing Hyperchaotic Attractor

When the system parameters are chosen asa=11,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 0,1,0,0,0, and the numerical results of the Lyapunov exponent varying with the system parameter b is shown in Figure 6(a). Figure 6(b) is a bifurcation diagram corresponding to Figure 6(a). From Figure 6, we can see that system (2) has more complex dynamic behavior in the parameter b∈−10,0 interval, such as quasi-periodicity, period, chaos, and hyperchaos. Table 1 classifies these dynamical behaviors and then makes a detailed analysis of LEs and their dynamical behaviors varying with parameter b as follows:(i)
Whenb∈−4.4,0, the Lyapunov exponents of system (2) at b=−1 are 0.5791, 0.1087, −0.0316, −0.2607, and −4.5443. The system is in a hyperchaotic state, and the corresponding phase portraits are shown in Figure 7(a).(ii)
Whenb∈−5.55,−4.4, the Lyapunov exponents of system (2) at b=−5 are 0, −0.0132, −0.2670, −0.4978, and −2.1754. The system is in a period-5 state, and the corresponding phase portraits are shown in Figure 7(b).(iii)
Whenb∈−7,−6.6 and b∈−10,−7.2, the Lyapunov exponents of system (2) at b=−8 are 0.0279, −0.0271, −0.2093, −0.3481, and −1.9681. The phase portraits shown in Figure 7(c) show that the system is in a quasi-periodic state.(iv)
Whenb∈−6.6,−5.55 and b∈−7.2,−7, the Lyapunov exponents of system (2) at b=−6.15 are 0.0918, −0.0091, −0.16–0.3278, and −2.2332. The system shows a chaotic state, and the corresponding phase portraits are shown in Figure 7(d).Figure 6
Lyapunov exponents and bifurcation diagram for increasing parameterb.
(a)
(b)Table 1
The LEs and dynamical behavior under different parameter range ofb.
b
LE
1
,
LE
2
,
LE
3
,
LE
4
,
LE
5
Dynamic
−
4.4
,
0
+
,
+
,
0
,
−
,
−
Hyperchaotic
−
5.55
,
−
4.4
0
,
−
,
−
,
−
,
−
Periodic
−
6.6
,
−
5.55
+
,
0
,
−
,
−
,
−
Chaotic
−
7
,
−
6.6
0,0
,
−
,
−
,
−
Torus
−
7.2
,
−
7
+
,
0
,
−
,
−
,
−
Chaotic
−
10
,
−
7.2
0,0
,
−
,
−
,
−
Quasi-periodicFigure 7
Simulated phase portraits of system (2) with different b: (a) b=−1, (b) b=−5, (c) b=−8, and (d) b=−6.15.
(a)
(b)
(c)
(d)


### 2.4.3. Complexity Analysis by Spectral Entropy

The complexity of spectral entropy (SE) is based on the discrete Fourier transformation. The distribution of energy in Fourier transform domain is calculated, and then the SE value is calculated by Shannon entropy, which reflects the disorder of time series in frequency domain [61, 101, 102]. The chaotic diagram using the complexity of SE usually reflects the spatial complexity of chaotic system parameters. In this section, the SE algorithm is used to analyze the complexity of system (2). Figure 8 shows the complexity of SE of system (2) under initial condition 1,−1,1,1,1. It can be seen that Figure 8(a) corresponds well to the maximum Lyapunov exponent in Figure 3. The control parameters a and b of the chaotic system are divided into 101×101 parts, including a∈−1,1 and b∈0,5. Then, calculate the SE of each point a,b in the parameter space, as shown in Figure 8(b). The results show that with the increase of parameter a, the more complex the chaotic system is, and the higher the complexity of the system is mainly concentrated in a∈0.5,1. Figure 9 is the SE diagram of system (2) under initial condition 0,1,0,0,0, where a∈10,20 and b∈−10,0. Figure 9(a) corresponds well to the maximum Lyapunov exponent in Figure 6. Figure 9(b) shows the complexity of SE in the plane of control parameters a and b. The results show that when a=11, with the increase of parameter b, the larger the SE value is, and the higher the complexity of the system is mainly concentrated in b∈−3.8,0.Figure 8
The complexity of SE of system (2): (a) SE versus a when b=1 and (b) in the a−b plane.
(a)
(b)Figure 9
The complexity of SE of system (2): (a) SE versus b when a=11 and (b) in the a−b plane.
(a)
(b)


## 2.4.1. Four-Wing Hyperchaotic Attractor

When the system parameters are selecteda=1,b=1,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 1,−1,1,1,1, the phase portraits of system (2) shown in Figure 1 are obtained by Matlab simulation, which is a typical four-wing chaotic attractor, and the time variations of state equations x,y,z,andw are provided in Figure 2.Figure 1
The four-wing hyperchaotic attractor of system (2): (a) in the x−y plane, (b) in the x−z plane, (c) in the y−z plane, and (d) in the x−y−z space.
(a)
(b)
(c)
(d)Figure 2
The time variation of state equationsx,y,z,andw of system (2): (a) t−x, (b) t−y, (c) t−z, and (d) t−w.
(a)
(b)
(c)
(d)The Lyapunov exponent is used to measure the perturbation caused by initial conditions. If there are slight differences in the system, two adjacent trajectories in the phase space are separated exponentially with time. The Lyapunov exponent is a useful tool for measuring chaotic systems, in particular, it is usually determines whether a chaotic system is chaotic or hyperchaotic according to the number of positive Lyapunov exponents. According to the given system parameters and initial conditions, the Lyapunov exponent ofa∈−1,1 is simulated by the Jacobi matrix method. The numerical results are shown in Figure 3(a) (the five LE5 is out of plot). From Figure 3(a), we can clearly see that the system has complex dynamic behaviors such as periodic orbit, chaos, and hyperchaos. When a∈−1,−0.03 and a∈0.22,0.3, the system is a periodic orbit, and Figure 4 are the phase portraits of system (2) when a=−1 (the LEs are 0, −0.0424, −0.4394, −0.6696, and −3.4835). When a=0.2, the LEs are 0.0790, −0.0078, −0.0627, −0.1078, and −1.9274, with a positive Lyapunov exponent, so system (2) is in chaotic state (a∈0.3,0.48 and a∈−0.03,0.22). When a∈0.48,1, the system is hyperchaotic, and the typical four-wing hyperchaotic attractor is shown in Figure 1. When a=0.78, the LEs are LE1=0.1712,LE2=0.0907,LE3=0.0107,LE4=0,and LE5=−2.3243, and it can be judged that system (2) is hyperchaotic and has three positive Lyapunov exponents. Figure 3(b) describes the bifurcation diagram of system (2) varying with parameter a. With the increase of a, the system changes from period to chaos. The Kaplan–Yorke dimension of system (2) can be calculated by the following formula:(11)DL=j+∑i=1jLjLj+1=4+0.1712+0.0907+0.0107−02.3243=4.1172,where j is the largest integer satisfying ∑i=1jLj≥0 and ∑i=1jLj<0. It can be seen from DL=4.1172 that the Lyapunov dimension of system (2) is fractional. Therefore, the proposed 5D FWMHS is a real hyperchaotic system with strong complexity.Figure 3
Lyapunov exponents and bifurcation diagram for increasing parametera (the five LE5 is out of plot).
(a)
(b)Figure 4
Periodic phase portraits of system (2) with a=−1: (a) in the x−z plane, (b) in the y−z plane, and (c) in the x−y plane.
(a)
(b)
(c)As an important analytical tool, the Poincare map is used to further study the dynamic characteristics of the 5D HFWMS. Figure5(a) shows the Poincare map of the x−y plane at z=0 and four branches can be seen; Figure 5(b) shows the Poincare map of the x−z plane at y=0 and has many branches; Figure 5(c) shows the Poincare map of the y−z plane at w=0 and the outline of the four wings can be seen, indicating the existence of the four-wing phenomenon. Figure 5 shows that system (2) has a four-wing chaotic attractor with fractal structure.Figure 5
Poincare map of system (2) with parameters a=1: (a) projection on x−y plane with z=0, (b) projection on x−z plane with y=0, and (c) projection on y−z plane with w=0.
(a)
(b)
(c)


## 2.4.2. Two-Wing Hyperchaotic Attractor

When the system parameters are chosen asa=11,c=0.7,m=1,d=0.2,e=0.1,andn=0.01 and the initial condition is set to 0,1,0,0,0, and the numerical results of the Lyapunov exponent varying with the system parameter b is shown in Figure 6(a). Figure 6(b) is a bifurcation diagram corresponding to Figure 6(a). From Figure 6, we can see that system (2) has more complex dynamic behavior in the parameter b∈−10,0 interval, such as quasi-periodicity, period, chaos, and hyperchaos. Table 1 classifies these dynamical behaviors and then makes a detailed analysis of LEs and their dynamical behaviors varying with parameter b as follows:(i)
Whenb∈−4.4,0, the Lyapunov exponents of system (2) at b=−1 are 0.5791, 0.1087, −0.0316, −0.2607, and −4.5443. The system is in a hyperchaotic state, and the corresponding phase portraits are shown in Figure 7(a).(ii)
Whenb∈−5.55,−4.4, the Lyapunov exponents of system (2) at b=−5 are 0, −0.0132, −0.2670, −0.4978, and −2.1754. The system is in a period-5 state, and the corresponding phase portraits are shown in Figure 7(b).(iii)
Whenb∈−7,−6.6 and b∈−10,−7.2, the Lyapunov exponents of system (2) at b=−8 are 0.0279, −0.0271, −0.2093, −0.3481, and −1.9681. The phase portraits shown in Figure 7(c) show that the system is in a quasi-periodic state.(iv)
Whenb∈−6.6,−5.55 and b∈−7.2,−7, the Lyapunov exponents of system (2) at b=−6.15 are 0.0918, −0.0091, −0.16–0.3278, and −2.2332. The system shows a chaotic state, and the corresponding phase portraits are shown in Figure 7(d).Figure 6
Lyapunov exponents and bifurcation diagram for increasing parameterb.
(a)
(b)Table 1
The LEs and dynamical behavior under different parameter range ofb.
b
LE
1
,
LE
2
,
LE
3
,
LE
4
,
LE
5
Dynamic
−
4.4
,
0
+
,
+
,
0
,
−
,
−
Hyperchaotic
−
5.55
,
−
4.4
0
,
−
,
−
,
−
,
−
Periodic
−
6.6
,
−
5.55
+
,
0
,
−
,
−
,
−
Chaotic
−
7
,
−
6.6
0,0
,
−
,
−
,
−
Torus
−
7.2
,
−
7
+
,
0
,
−
,
−
,
−
Chaotic
−
10
,
−
7.2
0,0
,
−
,
−
,
−
Quasi-periodicFigure 7
Simulated phase portraits of system (2) with different b: (a) b=−1, (b) b=−5, (c) b=−8, and (d) b=−6.15.
(a)
(b)
(c)
(d)


## 2.4.3. Complexity Analysis by Spectral Entropy

The complexity of spectral entropy (SE) is based on the discrete Fourier transformation. The distribution of energy in Fourier transform domain is calculated, and then the SE value is calculated by Shannon entropy, which reflects the disorder of time series in frequency domain [61, 101, 102]. The chaotic diagram using the complexity of SE usually reflects the spatial complexity of chaotic system parameters. In this section, the SE algorithm is used to analyze the complexity of system (2). Figure 8 shows the complexity of SE of system (2) under initial condition 1,−1,1,1,1. It can be seen that Figure 8(a) corresponds well to the maximum Lyapunov exponent in Figure 3. The control parameters a and b of the chaotic system are divided into 101×101 parts, including a∈−1,1 and b∈0,5. Then, calculate the SE of each point a,b in the parameter space, as shown in Figure 8(b). The results show that with the increase of parameter a, the more complex the chaotic system is, and the higher the complexity of the system is mainly concentrated in a∈0.5,1. Figure 9 is the SE diagram of system (2) under initial condition 0,1,0,0,0, where a∈10,20 and b∈−10,0. Figure 9(a) corresponds well to the maximum Lyapunov exponent in Figure 6. Figure 9(b) shows the complexity of SE in the plane of control parameters a and b. The results show that when a=11, with the increase of parameter b, the larger the SE value is, and the higher the complexity of the system is mainly concentrated in b∈−3.8,0.Figure 8
The complexity of SE of system (2): (a) SE versus a when b=1 and (b) in the a−b plane.
(a)
(b)Figure 9
The complexity of SE of system (2): (a) SE versus b when a=11 and (b) in the a−b plane.
(a)
(b)


## 3. The FPGA-Based Model of the Novel 5D HFWMS

Devices in analog circuits are easy to aging and inflexible, which makes more and more researchers begin to pay attention to digital devices on the FPGA. With the characteristics of high-speed operation, high integration, and free design, FPGA can easily generate chaotic signals. Nowadays, many numerical algorithms are used to solve the nonlinear differential equations of chaotic systems. The Euler algorithm is the simplest of all algorithms, but its accuracy is not high [64, 65]. The Heun algorithm is more sensitive than the Euler method [63]. The Runge–Kutta algorithm is better than other algorithms in operation effect, with high accuracy, stable calculation process, and easy realization. RK-4 is easier to implement than RK-5, so RK-4 is widely used to solve chaotic systems [103]. Equation (12) gives the formula for calculating K1,K2,K3,andK4, which represents the slope value of y0,yi:(12)ti+1=ti+h,K1=fti,yi,K2=fti+h2,yi+h2K1,K3=fti+h2,yi+h2K2,K4=fti+h,yi+hK3.For the designed HFWMS, the initial conditionsx0=1,y0=−1,z0=1,w0=1,andφ0=1 and the step size h=0.01 are given. Five equations in HFWMS are calculated according to the RK-4 algorithm flow chart and equation (12). By using the RK-4 algorithm of Verilog, the designed HFWMS is implemented on FPGA. The central idea of designing a chaotic signal generator with FPGA is to divide the whole system into several functional modules, including RK-4 solving module, data selector module, ys module, and numerical conversion module. The function module contains many arithmetic units, including multiplier, adder, and subtractor. These arithmetic units are created in cooperation with the IP core generator and follow the standard of IEEE 754.The top-level block diagram of the chaotic signal generator based on FPGA using the RK-4 algorithm is shown in Figure10. As can be seen from Figure 10, the design system has three inputs and six outputs. The output signal consists of five 32-bit output signals (X_out,Y_out,Z_out,W_out, and φ_out) and 1 bit flag signal XYZWφ_ready. When the calculation produces X_out,Y_out, Z_out,W_out, and φ_out, the signal XYZWφ_ready will be set to a valid bit. Clock signal (Clk) and Reset are both 1 bit signals, which are used to ensure synchronization between the system and other modules; 32 bit Δh represents step size, which is used to determine the sensitivity of the algorithm.Figure 10
The top-level block diagram of the FPGA-based 5D HFWMS.The second block diagram of the chaotic signal generator based on FPGA using the RK-4 algorithm is displayed in Figure11, which consists of a Multipler (MUX) and 5D HFWMS oscillator. As can be seen from that the MUX unit is used to obtain the initial condition signal at the first operation. These signals were initially defined by the designer, and then obtained as feedback signals by the output signals (X_out,Y_out,Z_out,W_out, and φ_out). Figure 12 is the third block diagram of the chaotic signal generator based on FPGA using the RK-4 algorithm, which consists of three parts: MUX unit, HFWMS oscillator unit, and data processing unit. K1 unit, K2 unit, K3 unit, K4 unit, and ys are important components of the HFWMS oscillator, which are pipelined structures used in the calculation of Runge–Kutta algorithm. ys unit can make the chaotic oscillator produce output signal in a definite clock period. I.C. (initial conditions) are initially defined by the designer. When HFWMS generates a set of output values, the value of XYZWφ_ready is set to a valid bit and the values of xk+1,yk+1,zk+1,wk+1, and φk+1 generated by the oscillator are fed back to MUX as the initial values of the next operation. The data processing unit has two functions: (1) converting 32-bit floating-point signals (X_out,Y_out,Z_out,W_out, and φ_out) generated by the oscillator into 14-bit fixed-point signals and (2) converting signed fixed-point signals into unsigned fixed-point signals. The digital-to-analog converter AN9767 (DAC) converts 14-digit digital signals into analog signals for easy display on the oscilloscope.Figure 11
The second-level block diagram of the FPGA-based 5D HFWMS.Figure 12
The third-level block diagram of the FPGA-based 5D HFWMS.The digital hardware implementation of the 5D hyperchaotic oscillator based on RK-4 has been synthesized on the Xilinx ZYNQ-XC7Z020 FPGA chip. This design is implemented, synthesized, and downloaded using Vivado 2018.3. The parameter statistics of the related resource utilization of the FPGA and the clock speed of each module are calculated. In order to better analyze the experimental results, we convert the experimental data into hexadecimal. Figure13 shows a discrete time series (X_out,Y_out,Z_out,W_out, and φ_out) obtained for the HFWMS oscillator based on FPGA, which corresponds to the x,y,z,w, and φ signals of the continuous chaotic system. Figure 14 are the phase portraits of Y_out andZ_out which are displayed by the 5D HFWMS on the oscilloscope. From Figure 14, it can be seen that several kinds of phase portraits designed based on FPGA are consistent with the Matlab simulation diagrams, which means that the designed HFWMS based on FPGA can be implemented well. Table 2 provides statistical data on resource utilization, chip speed, and performance of the Xilinx ZYNQ-XC7Z020 FPGA chip. The minimum clock period and maximum operating frequency of HFWMS based on FPGA are 6.763 ns and 147.863 MHz.Figure 13
Simulation results of the FPGA-based HFWMS signal generator with the RK-4 algorithm.Figure 14
Implementation platform and exemplificative phase portraits generated by the FPGA implementation of the proposed HFWMS: (a) four-wing hyperchaotic attractor, (b) two-wing hyperchaotic attractor, (c) period-5 state, and (d) quasi-periodic state.
(a)
(b)
(c)
(d)Table 2
Xilinx ZYNQ-XC7Z020 FPGA chip hardware usage statistics of the 5D HFWMS.
Resource
Used
Available
Utilization%
Slice register
26893
106400
25
Number of slice LUTs
23173
53200
43
Number of bonded IOBs
30
125
24
Number of BUFG
1
32
3
Max. clock frequency
147.863 MHz
—
—
Latency
13.53 ns
—
—
Throughput
59.15 Mbit/s
—
—
Power
0.275 W
—
—


### 3.1. Active Control Synchronization and Secure Communication of the Novel 5D HFWMS

#### 3.1.1. Synchronization of the Novel 5D HFWMS by the Active Control Method

Synchronization design is the key to secure communication. Therefore, it is necessary to synchronize the designed chaotic system before realizing secure communication. A synchronization system consists of two parts, one is the master system and the other is the slave system. In this section, the active control method is used to synchronize system (2), and system (2) is set as the master system and rewritten as(13)x˙1=ax1+dWφ1y1+y1z1−cw1,y˙1=y1z1−x1z1,z˙1=−z1−mx1y1+b,w˙1=x1,φ˙1=y1.Here, the slave system uses a 5D hyperchaotic system proposed by Yang and Bai [104], which is described as(14)x˙2=a1y2−x2+u1,y˙2=c1x2+d1y2−x2z2+φ2+u2,z˙2=−b1z2−x22+u3,w˙2=g1y2+f1w2+u4,φ˙2=−r1x2−k1φ2+u5,where a1,b1,c1,d1,g1,f1,r1,andk1 are the system parameters and u=u1,u2,u3,u4,u5T is the active controller of synchronous systems, which can make the master and slave systems gradually synchronize under different initial conditions. Define the errors as(15)e1=x2−x1,e2=y2−y1,e3=z2−z1,e4=w2−w1,e5=φ2−φ1.Then, the error dynamic system can be obtained as(16)e˙1=a−a1e1+a1+dge2−ce4+cw2−ax2−dgy2+a1y1−x1−y1z1−3dnφ12y1+u1,e˙2=c1e1+d1e2+e5+φ1−x2z2+x1z1−y1z1+c1x1+d1y1+u2,e˙3=−b1e3−e3+x22+mx1y1−b+z2−b1z1+u3,e˙4=g1e1+g1y1+f1e4+f1w1+e1−x2+u4,e˙5=−r1e1−k1e5−r1x1−k1φ1−y2+e2+u5.By simplifying the linear term of equation (16), the active controller can be obtained as(17)u1=−cw2+ax2+dgy2−a1y1−x1+y1z1+3dnφ12y1+v1,u2=−φ1+x2z2−x1z1+y1z1−c1x1−d1y1+v2,u3=−x22−mx1y1+b−z2+b1z1+v3,u4=−g1y1−f1w1+x2+v4,u5=r1x1+k1φ1+y2+v5,where v=v1,v2,v3,v4,v5T is the control input. The substitution of (17) in (16) leads to a linear error dynamics equation without the active controller:(18)e˙1=a−a1e1+a1+dge2−ce4+v1,e˙2=c1e1+d1e2+e5+v2,e˙3=−b1e3−e3+v3,e˙4=g1e1+f1e4+e1+v4,e˙5=−r1e1−k1e5+e2+v5.In order to achieve synchronization of different structures, it is necessary to(19)limx⟶∞ei=0,i=1,2,3,4,5.Equation (17) shows that if system (18) tends to be stable over time and under the control of input v=v1,v2,v3,v4,v5T, the error variable e=e1,e2,e3,e4,e5T tends to zero, and then the master system (12) and the slave system (13) realize the synchronization with different structures. To achieve this goal, we define a matrix A to represent the relationship between the error system and the control input, which can be expressed as(20)v=A⋅e.According to the stability criterion, if equation (20) is stable, all eigenvalues of matrix A are negative. Thus, equation (20) can be expressed as(21)v1v2v3v4v5=a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1e1e2e3e4e5,where matrix A is(22)a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1.Therefore, equation (21) can be transformed into(23)v1=a1−a−1x2−x1+cw2−w1−a1+dgy2−y1,v2=−c1x2−x1−d1+1y2−y1−u2−u1,v3=b1z2−z1,v4=−x2−x1−g1y2−y1−f1+1w2−w1,v5=−y2−y1−k1φ2−φ1+r1x2−x1.By substituting equations (23) and (17) into equation (14), the expression of slave system is obtained as follows:(24)x˙2=−x2+2x1+dg+3nφ12y1+y1z1−cw1,y˙2=y1z1−x1z1−y2+y1,z˙2=−z2−mx1y1+b,w˙2=x1−w2+w1,φ˙2=y1−φ2+φ1.(1) Numerical Simulation Results Based on Matlab. The effectiveness of synchronization results of active control for different structures is verified by numerical simulation. Choose the parameters of system (13) as a=1,b=1,c=0.7,m=1,d=0.2,g=0.1,andn=0.01 and initial states as 1,−1,1,1,1; the parameters of system (14) as a1=35,b1=7,c1=35,d1=−5,g1=10.6,f1=1,r1=5,andk1=0.05 and initial states as 0.01,−0.01,0.01,0.01,0.01. Figure 15 shows the time behaviors of the error states. As we can see, the error states converge to 0 in 3 seconds, which means that the two hyperchaotic systems with different structures can achieve synchronization.Figure 15
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on Matlab.(2) Implementation Results Based on FPGA. Because of the aging and temperature change of analog devices, the device values are easy to change. Synchronization requires that the parameters of the transmitter and the receiver be highly matched. This indicates that there are a series of problems in synchronization using analog devices. It is an ideal choice to synchronize the chaotic system with FPGA, which has high reliability because it does not have the problem of temperature drift of components. In this section, according to the principle of active control synchronization, two different 5D hyperchaotic systems with different structures are synchronized on ModelSim. The Verilog design is simulated by Vivado’s built-in simulator. The simulation results of synchronization errors in FPGA are shown in Figure 16. From the graph, it can be seen that all synchronization errors tend to zero. This is the same as the Matlab simulation in Figure 15 of synchronization errors. Therefore, the synchronization of 5D hyperchaotic systems with different structures designed by FPGA can work normally on hardware.Figure 16
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on FPGA.


#### 3.1.2. Chaotic Masking Communication of the Novel 5D HFWMS

The main idea of chaotic masking secure communication design is to superimpose the signal generated by the chaotic system on the useful signal to form a modulation signal, and then use the channel for transmission. The receiver demodulates the modulated signal by the output signal of the chaotic system synchronized with the transmitter and recovers the original useful signal. The application of secure communication studied in this paper is based on active control synchronization between two 5D hyperchaotic systems with different structures. The schematic diagram of the designed secure communication scheme is shown in Figure17, where st is the information signal, mt is the chaotic signal generated by the master system, pt is the mixed signal encrypted by mt to st, nt is the chaotic signal generated by the slave system, and dt is the decrypted signal. The signal transmitted in the channel is a kind of chaotic signal similar to noise, from which it is difficult to obtain useful signals. The controller unit can synchronize the master system and slave system, which is the key of chaotic secure communication. The original useful signal can be recovered effectively by using the signal generated by the synchronized chaotic system of the controller to decrypt the encrypted signal. Set the information signal to be encrypted at the transmitter as(25)st=0.125sin0.1πt.Figure 17
The schematic diagram of chaotic masking communication.The encryption function and decryption function of the chaotic masking communication are, respectively, adopted as(26)pt=st⋅1+kmt,dt=pt1+knt.The error signal between the information signal and the decrypted signal is(27)et=st−dt.(1) Numerical Simulation Results Based on Matlab. Through the encryption function, decryption function, and synchronization-related parameters, the numerical simulation results are shown in Figure 18 with k=10. As can be seen that the signal pt is really difficult to be decrypted; the signal dt generated by the decryption function corresponds well with the information signal, which means that the receiver can recover the information signal very well.Figure 18
(a) Original signalst, (b) transmitted signal pt, (c) recovered signal dt, and (d) error et=st−dt.
(a)
(b)
(c)
(d)(2) Implementation Results Based on FPGA. In order to verify the correctness of the proposed chaotic masking secure communication scheme, ModelSim is selected as the experimental platform of FGPA. According to the scheme shown in Figure 17, the hardware experiment results of the FPGA are shown in Figure 19. From Figure 19, we can see that the effective information signal st can be completely hidden in the chaotic sequence, the encrypted signal has strong concealment, and the received decrypted signal has good restoration quality, which is the same as the pre-encrypted signal.Figure 19
Experimental results of chaotic masking secure communication based on FPGA.


## 3.1. Active Control Synchronization and Secure Communication of the Novel 5D HFWMS

### 3.1.1. Synchronization of the Novel 5D HFWMS by the Active Control Method

Synchronization design is the key to secure communication. Therefore, it is necessary to synchronize the designed chaotic system before realizing secure communication. A synchronization system consists of two parts, one is the master system and the other is the slave system. In this section, the active control method is used to synchronize system (2), and system (2) is set as the master system and rewritten as(13)x˙1=ax1+dWφ1y1+y1z1−cw1,y˙1=y1z1−x1z1,z˙1=−z1−mx1y1+b,w˙1=x1,φ˙1=y1.Here, the slave system uses a 5D hyperchaotic system proposed by Yang and Bai [104], which is described as(14)x˙2=a1y2−x2+u1,y˙2=c1x2+d1y2−x2z2+φ2+u2,z˙2=−b1z2−x22+u3,w˙2=g1y2+f1w2+u4,φ˙2=−r1x2−k1φ2+u5,where a1,b1,c1,d1,g1,f1,r1,andk1 are the system parameters and u=u1,u2,u3,u4,u5T is the active controller of synchronous systems, which can make the master and slave systems gradually synchronize under different initial conditions. Define the errors as(15)e1=x2−x1,e2=y2−y1,e3=z2−z1,e4=w2−w1,e5=φ2−φ1.Then, the error dynamic system can be obtained as(16)e˙1=a−a1e1+a1+dge2−ce4+cw2−ax2−dgy2+a1y1−x1−y1z1−3dnφ12y1+u1,e˙2=c1e1+d1e2+e5+φ1−x2z2+x1z1−y1z1+c1x1+d1y1+u2,e˙3=−b1e3−e3+x22+mx1y1−b+z2−b1z1+u3,e˙4=g1e1+g1y1+f1e4+f1w1+e1−x2+u4,e˙5=−r1e1−k1e5−r1x1−k1φ1−y2+e2+u5.By simplifying the linear term of equation (16), the active controller can be obtained as(17)u1=−cw2+ax2+dgy2−a1y1−x1+y1z1+3dnφ12y1+v1,u2=−φ1+x2z2−x1z1+y1z1−c1x1−d1y1+v2,u3=−x22−mx1y1+b−z2+b1z1+v3,u4=−g1y1−f1w1+x2+v4,u5=r1x1+k1φ1+y2+v5,where v=v1,v2,v3,v4,v5T is the control input. The substitution of (17) in (16) leads to a linear error dynamics equation without the active controller:(18)e˙1=a−a1e1+a1+dge2−ce4+v1,e˙2=c1e1+d1e2+e5+v2,e˙3=−b1e3−e3+v3,e˙4=g1e1+f1e4+e1+v4,e˙5=−r1e1−k1e5+e2+v5.In order to achieve synchronization of different structures, it is necessary to(19)limx⟶∞ei=0,i=1,2,3,4,5.Equation (17) shows that if system (18) tends to be stable over time and under the control of input v=v1,v2,v3,v4,v5T, the error variable e=e1,e2,e3,e4,e5T tends to zero, and then the master system (12) and the slave system (13) realize the synchronization with different structures. To achieve this goal, we define a matrix A to represent the relationship between the error system and the control input, which can be expressed as(20)v=A⋅e.According to the stability criterion, if equation (20) is stable, all eigenvalues of matrix A are negative. Thus, equation (20) can be expressed as(21)v1v2v3v4v5=a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1e1e2e3e4e5,where matrix A is(22)a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1.Therefore, equation (21) can be transformed into(23)v1=a1−a−1x2−x1+cw2−w1−a1+dgy2−y1,v2=−c1x2−x1−d1+1y2−y1−u2−u1,v3=b1z2−z1,v4=−x2−x1−g1y2−y1−f1+1w2−w1,v5=−y2−y1−k1φ2−φ1+r1x2−x1.By substituting equations (23) and (17) into equation (14), the expression of slave system is obtained as follows:(24)x˙2=−x2+2x1+dg+3nφ12y1+y1z1−cw1,y˙2=y1z1−x1z1−y2+y1,z˙2=−z2−mx1y1+b,w˙2=x1−w2+w1,φ˙2=y1−φ2+φ1.(1) Numerical Simulation Results Based on Matlab. The effectiveness of synchronization results of active control for different structures is verified by numerical simulation. Choose the parameters of system (13) as a=1,b=1,c=0.7,m=1,d=0.2,g=0.1,andn=0.01 and initial states as 1,−1,1,1,1; the parameters of system (14) as a1=35,b1=7,c1=35,d1=−5,g1=10.6,f1=1,r1=5,andk1=0.05 and initial states as 0.01,−0.01,0.01,0.01,0.01. Figure 15 shows the time behaviors of the error states. As we can see, the error states converge to 0 in 3 seconds, which means that the two hyperchaotic systems with different structures can achieve synchronization.Figure 15
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on Matlab.(2) Implementation Results Based on FPGA. Because of the aging and temperature change of analog devices, the device values are easy to change. Synchronization requires that the parameters of the transmitter and the receiver be highly matched. This indicates that there are a series of problems in synchronization using analog devices. It is an ideal choice to synchronize the chaotic system with FPGA, which has high reliability because it does not have the problem of temperature drift of components. In this section, according to the principle of active control synchronization, two different 5D hyperchaotic systems with different structures are synchronized on ModelSim. The Verilog design is simulated by Vivado’s built-in simulator. The simulation results of synchronization errors in FPGA are shown in Figure 16. From the graph, it can be seen that all synchronization errors tend to zero. This is the same as the Matlab simulation in Figure 15 of synchronization errors. Therefore, the synchronization of 5D hyperchaotic systems with different structures designed by FPGA can work normally on hardware.Figure 16
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on FPGA.


### 3.1.2. Chaotic Masking Communication of the Novel 5D HFWMS

The main idea of chaotic masking secure communication design is to superimpose the signal generated by the chaotic system on the useful signal to form a modulation signal, and then use the channel for transmission. The receiver demodulates the modulated signal by the output signal of the chaotic system synchronized with the transmitter and recovers the original useful signal. The application of secure communication studied in this paper is based on active control synchronization between two 5D hyperchaotic systems with different structures. The schematic diagram of the designed secure communication scheme is shown in Figure17, where st is the information signal, mt is the chaotic signal generated by the master system, pt is the mixed signal encrypted by mt to st, nt is the chaotic signal generated by the slave system, and dt is the decrypted signal. The signal transmitted in the channel is a kind of chaotic signal similar to noise, from which it is difficult to obtain useful signals. The controller unit can synchronize the master system and slave system, which is the key of chaotic secure communication. The original useful signal can be recovered effectively by using the signal generated by the synchronized chaotic system of the controller to decrypt the encrypted signal. Set the information signal to be encrypted at the transmitter as(25)st=0.125sin0.1πt.Figure 17
The schematic diagram of chaotic masking communication.The encryption function and decryption function of the chaotic masking communication are, respectively, adopted as(26)pt=st⋅1+kmt,dt=pt1+knt.The error signal between the information signal and the decrypted signal is(27)et=st−dt.(1) Numerical Simulation Results Based on Matlab. Through the encryption function, decryption function, and synchronization-related parameters, the numerical simulation results are shown in Figure 18 with k=10. As can be seen that the signal pt is really difficult to be decrypted; the signal dt generated by the decryption function corresponds well with the information signal, which means that the receiver can recover the information signal very well.Figure 18
(a) Original signalst, (b) transmitted signal pt, (c) recovered signal dt, and (d) error et=st−dt.
(a)
(b)
(c)
(d)(2) Implementation Results Based on FPGA. In order to verify the correctness of the proposed chaotic masking secure communication scheme, ModelSim is selected as the experimental platform of FGPA. According to the scheme shown in Figure 17, the hardware experiment results of the FPGA are shown in Figure 19. From Figure 19, we can see that the effective information signal st can be completely hidden in the chaotic sequence, the encrypted signal has strong concealment, and the received decrypted signal has good restoration quality, which is the same as the pre-encrypted signal.Figure 19
Experimental results of chaotic masking secure communication based on FPGA.


## 3.1.1. Synchronization of the Novel 5D HFWMS by the Active Control Method

Synchronization design is the key to secure communication. Therefore, it is necessary to synchronize the designed chaotic system before realizing secure communication. A synchronization system consists of two parts, one is the master system and the other is the slave system. In this section, the active control method is used to synchronize system (2), and system (2) is set as the master system and rewritten as(13)x˙1=ax1+dWφ1y1+y1z1−cw1,y˙1=y1z1−x1z1,z˙1=−z1−mx1y1+b,w˙1=x1,φ˙1=y1.Here, the slave system uses a 5D hyperchaotic system proposed by Yang and Bai [104], which is described as(14)x˙2=a1y2−x2+u1,y˙2=c1x2+d1y2−x2z2+φ2+u2,z˙2=−b1z2−x22+u3,w˙2=g1y2+f1w2+u4,φ˙2=−r1x2−k1φ2+u5,where a1,b1,c1,d1,g1,f1,r1,andk1 are the system parameters and u=u1,u2,u3,u4,u5T is the active controller of synchronous systems, which can make the master and slave systems gradually synchronize under different initial conditions. Define the errors as(15)e1=x2−x1,e2=y2−y1,e3=z2−z1,e4=w2−w1,e5=φ2−φ1.Then, the error dynamic system can be obtained as(16)e˙1=a−a1e1+a1+dge2−ce4+cw2−ax2−dgy2+a1y1−x1−y1z1−3dnφ12y1+u1,e˙2=c1e1+d1e2+e5+φ1−x2z2+x1z1−y1z1+c1x1+d1y1+u2,e˙3=−b1e3−e3+x22+mx1y1−b+z2−b1z1+u3,e˙4=g1e1+g1y1+f1e4+f1w1+e1−x2+u4,e˙5=−r1e1−k1e5−r1x1−k1φ1−y2+e2+u5.By simplifying the linear term of equation (16), the active controller can be obtained as(17)u1=−cw2+ax2+dgy2−a1y1−x1+y1z1+3dnφ12y1+v1,u2=−φ1+x2z2−x1z1+y1z1−c1x1−d1y1+v2,u3=−x22−mx1y1+b−z2+b1z1+v3,u4=−g1y1−f1w1+x2+v4,u5=r1x1+k1φ1+y2+v5,where v=v1,v2,v3,v4,v5T is the control input. The substitution of (17) in (16) leads to a linear error dynamics equation without the active controller:(18)e˙1=a−a1e1+a1+dge2−ce4+v1,e˙2=c1e1+d1e2+e5+v2,e˙3=−b1e3−e3+v3,e˙4=g1e1+f1e4+e1+v4,e˙5=−r1e1−k1e5+e2+v5.In order to achieve synchronization of different structures, it is necessary to(19)limx⟶∞ei=0,i=1,2,3,4,5.Equation (17) shows that if system (18) tends to be stable over time and under the control of input v=v1,v2,v3,v4,v5T, the error variable e=e1,e2,e3,e4,e5T tends to zero, and then the master system (12) and the slave system (13) realize the synchronization with different structures. To achieve this goal, we define a matrix A to represent the relationship between the error system and the control input, which can be expressed as(20)v=A⋅e.According to the stability criterion, if equation (20) is stable, all eigenvalues of matrix A are negative. Thus, equation (20) can be expressed as(21)v1v2v3v4v5=a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1e1e2e3e4e5,where matrix A is(22)a1−a−1−a1−dg0c0−c1−d1−100−100b100−1−e10−f1−10r1−100k1−1.Therefore, equation (21) can be transformed into(23)v1=a1−a−1x2−x1+cw2−w1−a1+dgy2−y1,v2=−c1x2−x1−d1+1y2−y1−u2−u1,v3=b1z2−z1,v4=−x2−x1−g1y2−y1−f1+1w2−w1,v5=−y2−y1−k1φ2−φ1+r1x2−x1.By substituting equations (23) and (17) into equation (14), the expression of slave system is obtained as follows:(24)x˙2=−x2+2x1+dg+3nφ12y1+y1z1−cw1,y˙2=y1z1−x1z1−y2+y1,z˙2=−z2−mx1y1+b,w˙2=x1−w2+w1,φ˙2=y1−φ2+φ1.(1) Numerical Simulation Results Based on Matlab. The effectiveness of synchronization results of active control for different structures is verified by numerical simulation. Choose the parameters of system (13) as a=1,b=1,c=0.7,m=1,d=0.2,g=0.1,andn=0.01 and initial states as 1,−1,1,1,1; the parameters of system (14) as a1=35,b1=7,c1=35,d1=−5,g1=10.6,f1=1,r1=5,andk1=0.05 and initial states as 0.01,−0.01,0.01,0.01,0.01. Figure 15 shows the time behaviors of the error states. As we can see, the error states converge to 0 in 3 seconds, which means that the two hyperchaotic systems with different structures can achieve synchronization.Figure 15
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on Matlab.(2) Implementation Results Based on FPGA. Because of the aging and temperature change of analog devices, the device values are easy to change. Synchronization requires that the parameters of the transmitter and the receiver be highly matched. This indicates that there are a series of problems in synchronization using analog devices. It is an ideal choice to synchronize the chaotic system with FPGA, which has high reliability because it does not have the problem of temperature drift of components. In this section, according to the principle of active control synchronization, two different 5D hyperchaotic systems with different structures are synchronized on ModelSim. The Verilog design is simulated by Vivado’s built-in simulator. The simulation results of synchronization errors in FPGA are shown in Figure 16. From the graph, it can be seen that all synchronization errors tend to zero. This is the same as the Matlab simulation in Figure 15 of synchronization errors. Therefore, the synchronization of 5D hyperchaotic systems with different structures designed by FPGA can work normally on hardware.Figure 16
The trajectories of the synchronization errorse1,e2,e3,e4,ande5 based on FPGA.


## 3.1.2. Chaotic Masking Communication of the Novel 5D HFWMS

The main idea of chaotic masking secure communication design is to superimpose the signal generated by the chaotic system on the useful signal to form a modulation signal, and then use the channel for transmission. The receiver demodulates the modulated signal by the output signal of the chaotic system synchronized with the transmitter and recovers the original useful signal. The application of secure communication studied in this paper is based on active control synchronization between two 5D hyperchaotic systems with different structures. The schematic diagram of the designed secure communication scheme is shown in Figure17, where st is the information signal, mt is the chaotic signal generated by the master system, pt is the mixed signal encrypted by mt to st, nt is the chaotic signal generated by the slave system, and dt is the decrypted signal. The signal transmitted in the channel is a kind of chaotic signal similar to noise, from which it is difficult to obtain useful signals. The controller unit can synchronize the master system and slave system, which is the key of chaotic secure communication. The original useful signal can be recovered effectively by using the signal generated by the synchronized chaotic system of the controller to decrypt the encrypted signal. Set the information signal to be encrypted at the transmitter as(25)st=0.125sin0.1πt.Figure 17
The schematic diagram of chaotic masking communication.The encryption function and decryption function of the chaotic masking communication are, respectively, adopted as(26)pt=st⋅1+kmt,dt=pt1+knt.The error signal between the information signal and the decrypted signal is(27)et=st−dt.(1) Numerical Simulation Results Based on Matlab. Through the encryption function, decryption function, and synchronization-related parameters, the numerical simulation results are shown in Figure 18 with k=10. As can be seen that the signal pt is really difficult to be decrypted; the signal dt generated by the decryption function corresponds well with the information signal, which means that the receiver can recover the information signal very well.Figure 18
(a) Original signalst, (b) transmitted signal pt, (c) recovered signal dt, and (d) error et=st−dt.
(a)
(b)
(c)
(d)(2) Implementation Results Based on FPGA. In order to verify the correctness of the proposed chaotic masking secure communication scheme, ModelSim is selected as the experimental platform of FGPA. According to the scheme shown in Figure 17, the hardware experiment results of the FPGA are shown in Figure 19. From Figure 19, we can see that the effective information signal st can be completely hidden in the chaotic sequence, the encrypted signal has strong concealment, and the received decrypted signal has good restoration quality, which is the same as the pre-encrypted signal.Figure 19
Experimental results of chaotic masking secure communication based on FPGA.


## 4. Conclusion

In this study, a novel 5D continuous time HFWMS with multiline equilibrium and three positive Lyapunov exponents are first introduced. Dynamical analysis is performed in terms of equilibrium points, phase portraits, Poincare map, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy. Then, the four-wing, two-wing, period-5, and quasi-periodic phase portraits of the novel 5D hyperchaotic memristive system are carried out on FPGA, and a discrete time FPGA-based design of the 5D HFWMS is implemented on ModelSim using the RK-4 algorithm. The maximum operating frequency of the designed chaos-based system reaches 147.863 MHz. Finally, an active control synchronization of the 5D hyperchaotic system with different structures and a secure chaotic masking communication application are implemented on the platform of Matlab and FPGA, respectively. All these results justify the successful applications of the novel 5D HFWMS in active control synchronization and various chaos-based embedded secure chaotic masking communication systems. Our future work is to study whether the system has multistability, and to apply the system to image encryption and random number generator.



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*Source: 4047957-2019-11-28.xml*