# Characteristic Analysis of Fractional-Order 4D Hyperchaotic Memristive Circuit

Authors: Jun Mou; Kehui Sun; Huihai Wang; Jingya Ruan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Dynamical behaviors of the 4D hyperchaotic memristive circuit are analyzed with the system parameter. Based on the definitions of fractional-order differential and Adomian decomposition algorithm, the numerical solution of fractional-order 4D hyperchaotic memristive circuit is investigated. The distribution of stable and unstable regions of the fractional-order 4D hyperchaotic memristive circuit is determined, and dynamical characteristics are studied by phase portraits, Lyapunov exponents spectrum, and bifurcation diagram. Complexities are calculated by employing the spectral entropy (SE) algorithm andC0 algorithm. Complexity results are consistent with that of the bifurcation diagrams, and this means that complexity can also reflect the dynamic characteristics of a chaotic system. Results of this paper provide a theoretical and experimental basis for the application of fractional-order 4D hyperchaotic memristive circuit in the field of encryption and secure communication.


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## Body

## 1. Introduction

According to the principle of completeness with variable combination, Professor Chua predicted the existence of memristor in 1971 [1]. In 1976, he expounded the character of memristor, composition principle, and applications [2]. For a long time, the existence of element which satisfied the character of memristor was not discovered, so the study of memristor did not rise to the attention of scientific community and engineering circles. In 2008, the HP laboratory reported the realization of memristor firstly [3, 4], and, since then, the memristor has attracted much attention all over the world. Memristors are often divided into charge-controlled memristor and flux-controlled memristor. Both of them are typical nonlinear elements. It is expected to be an effective memory storage device in computers. It means lower power consumption and less thermal design to deal with, and it is easy to generate a chaotic vibration signal by employing this element. On one hand, we need to prevent the harm of chaos phenomenon in the application. On the other hand, the chaotic vibration signal by employing the memristor can be applied to many fields such as secure communication and aerospace industry. So researchers began to focus on the design and realization of memristive chaotic circuit [5–20]. In these literatures, only one memristor was applied in an independent circuit, and the dynamic characteristics of memristive chaotic system are related to the initial state of memristor, including unique nonlinear physics phenomenon. Zhang and Deng studied double-compound synchronization of six memristor-based Lorenz systems [21]. However, this article focuses on the synchronization method of a net with four Lorenz systems, rather than dynamic behaviors of a memristive circuit. When Bao et al. and Mou et al. applied more than one memristor in a single circuit [22, 23], they found that memristors would affect each other, and the dynamical behaviors of the circuit with more than one memristor become more complex. Compared with ordinary memristive chaotic systems, the memristive hyperchaotic systems have excellent security in communication because the memristive hyperchaotic sequence has better complexity and randomicity [24–26].Fractional calculus is more than 300-year-old topic. It can describe physical phenomena more accurately than that of the integer-order calculus, so fractional calculus has attracted more attention in various areas of applications such as physics, chemistry, bioengineering, signal processing, and control system [27–29]. A very important area of applications is the chaos theory [30–33]. At present, there are three typical methods to solve the fractional-order nonlinear system, such as frequency domain approximation [30], predictor-corrector method (PCM) [34], and Adomian decomposition method (ADM) [35]. The calculation precision of frequency domain approximations is limited. Whether this method accurately reflects the chaos characteristics of a fractional-order nonlinear system was questioned [36, 37]. For PCM, one can obtain more accurate results, but the calculation speed is too slow, and it consumes too much computer resources. Thus, it is unsuitable for engineering practice. Compared with the two solution approaches, ADM is capable of dealing with linear and nonlinear problems in time domain [38, 39]. He et al. concluded the characteristics of this method, such as high accuracy, fast convergence, and less computer resources consumption [40]. Recently, Li et al. proposed a new 4D hyperchaotic memristive circuit [41], which possesses abundant complex dynamics. The main feature of this system is having uncountable infinite number of stable equilibria, which is significantly different from other reported chaotic systems before. However, the article mainly analyzed the equilibrium states, and the analysis of dynamics characteristics of the 4D memristive hyperchaotic system is incomplete. Its fractional-order form has not been studied by now. Thus, it makes a great sense to study the dynamics characteristics of the system and the solution and dynamics of its corresponding fractional-order case. To our knowledge, no one has studied the fractional-order memristive hyperchaotic system. So we will employ ADM algorithm to solve the fractional-order 4D hyperchaotic system.In this paper, we focus on dynamical characteristic of the fractional-order 4D hyperchaotic memristive circuit. It is organized as follows. The dynamical characteristics of integer-order 4D hyperchaotic memristive circuit are investigated in Section2. In Section 3, ADM is introduced briefly, and the iterative algorithm of the fractional-order 4D hyperchaotic memristive circuit is deduced. In Section 4, the distribution of stable and unstable regions of the fractional-order 4D hyperchaotic memristive circuit is determined, and the dynamical characteristics of this system are analyzed. Finally, we summarize the results and indicate future directions.


## 2. 4D Hyperchaotic Memristive Circuit

### 2.1. Model of the 4D Hyperchaotic Memristive Circuit

Figure1 shows the 4D hyperchaotic memristive circuit model, which consists of standard integrators, standard multipliers, linear resistors, linear capacitors, and a nonlinear active memristor. We can use x, y, and z to indicate states of voltages. For the memristive circuit, there is an input from x to y by a flux-controlled memristor, and it is illustrated by W in Figure 1. According to Chua’s definition, a memristor is a passive two-terminal circuit element described by a nonlineari-v characteristic as follows: v=M(z)i or i=W(φ)v, where v,i,z, andφare the voltage, current, charge, and flux associated with the device. M(z) is the memristance defined as follows: M(z)=dφ(z)/dz. W(φ) is the memductance defined as follows: W(φ)=dz(φ)/dφ. Here, we focus on the flux-controlled memristive system described by the following circuit equation: W(φ)=a+3bφ2, where a and b are two positive constant parameters. According to the volt-ampere characteristics of each element and Kirchhoff’s current and voltage law, the differential equation (1) is obtained:(1)Cxx˙=yR1-xR2Cyy˙=-xzR3+yR4-WφxCzz˙=xyR5-zR6φ˙=x,where R is a reference resistor and C is a reference capacitor; then t=RC is the physical time, where t is the dimensionless time. The parameters can be taken as follows: Cx= Cy = Cz =C, R3 = R5 =R, α=R/R1=R/R2, β=R/R4, and γ=R/R6. By employing the normalized operation, (1) becomes(2)x˙=αy-xy˙=-xz+βy-ρWwxz˙=xy-γzw˙=x,where ρ is a positive parameter indicating the strength of the memristor. We should note here that W(w) comes from the memristor and has the same function as the physical memductance mentioned above, but it is dimensionless, which would be convenient for the following discussion.Figure 1
The 4D hyperchaotic memristive circuit.Setting the parametersa=4, b=0.01, α=36, β=20, γ=3, ρ=3, the initial value of (2) is (1, 0, 1, 0), and the time step is t=0.001 s; we get the hyperchaotic attractor as shown in Figure 2. In this case, the Lyapunov exponents of the system are L1 = 0.2566, L2 = 0.0674, and L3 = 0, L4 = −19.2935, and the Lyapunov dimension isDL = 3.017. Obviously the first two positive Lyapunov exponents imply that the 4D memristive circuit is hyperchaotic.Figure 2
Hyperchaotic attractor of the 4D hyperchaotic memristive circuit: (a)x-z and (b) y-w.
(a)
(b)


### 2.2. Dynamic Analysis with Different System Parameter

Fix the initial value of (2) to (1, 0, 1, 0), and change the parameter ρ from 0 to 3.2. Other parameters are the same as mentioned above. We obtain the Lyapunov exponents and its corresponding bifurcation diagram as shown in Figure 3, where the last Lyapunov exponent is not displayed because it is always a big negative number. It shows that the Lyapunov exponents spectrum and bifurcation diagram are consistent. Figure 3(b) shows the routes to chaos of the system (2), and the system transforms into two-scroll hyperchaotic attractor from one-scroll chaotic attractor. When the circuit parameter ρ increases further, the system transforms into three-scroll hyperchaotic attractor from two-scroll hyperchaotic attractor. Obviously, there are two periodic windows in the chaotic region at about ρ=2.55 and ρ=2.79. It indicates that system (2) has abundant dynamical behaviors.Figure 3
Lyapunov exponents spectrum and bifurcation diagram with variation of the parameterρ.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramTo display its dynamics further,y-w phase portraits of different states with different parameter ρ are presented in Figure 4.Figure 4
Phase portraits with different parameterρ.
(a)
Chaotic attractor (ρ=0.5)
(b)
Type 1 limit cycle (ρ=2.55)
(c)
Type 2 limit cycle (ρ=2.79)
(d)
Two-scroll hyperchaotic attractor (ρ=3)
(e)
Three-scroll hyperchaotic attractor (ρ=3.15)


## 2.1. Model of the 4D Hyperchaotic Memristive Circuit

Figure1 shows the 4D hyperchaotic memristive circuit model, which consists of standard integrators, standard multipliers, linear resistors, linear capacitors, and a nonlinear active memristor. We can use x, y, and z to indicate states of voltages. For the memristive circuit, there is an input from x to y by a flux-controlled memristor, and it is illustrated by W in Figure 1. According to Chua’s definition, a memristor is a passive two-terminal circuit element described by a nonlineari-v characteristic as follows: v=M(z)i or i=W(φ)v, where v,i,z, andφare the voltage, current, charge, and flux associated with the device. M(z) is the memristance defined as follows: M(z)=dφ(z)/dz. W(φ) is the memductance defined as follows: W(φ)=dz(φ)/dφ. Here, we focus on the flux-controlled memristive system described by the following circuit equation: W(φ)=a+3bφ2, where a and b are two positive constant parameters. According to the volt-ampere characteristics of each element and Kirchhoff’s current and voltage law, the differential equation (1) is obtained:(1)Cxx˙=yR1-xR2Cyy˙=-xzR3+yR4-WφxCzz˙=xyR5-zR6φ˙=x,where R is a reference resistor and C is a reference capacitor; then t=RC is the physical time, where t is the dimensionless time. The parameters can be taken as follows: Cx= Cy = Cz =C, R3 = R5 =R, α=R/R1=R/R2, β=R/R4, and γ=R/R6. By employing the normalized operation, (1) becomes(2)x˙=αy-xy˙=-xz+βy-ρWwxz˙=xy-γzw˙=x,where ρ is a positive parameter indicating the strength of the memristor. We should note here that W(w) comes from the memristor and has the same function as the physical memductance mentioned above, but it is dimensionless, which would be convenient for the following discussion.Figure 1
The 4D hyperchaotic memristive circuit.Setting the parametersa=4, b=0.01, α=36, β=20, γ=3, ρ=3, the initial value of (2) is (1, 0, 1, 0), and the time step is t=0.001 s; we get the hyperchaotic attractor as shown in Figure 2. In this case, the Lyapunov exponents of the system are L1 = 0.2566, L2 = 0.0674, and L3 = 0, L4 = −19.2935, and the Lyapunov dimension isDL = 3.017. Obviously the first two positive Lyapunov exponents imply that the 4D memristive circuit is hyperchaotic.Figure 2
Hyperchaotic attractor of the 4D hyperchaotic memristive circuit: (a)x-z and (b) y-w.
(a)
(b)


## 2.2. Dynamic Analysis with Different System Parameter

Fix the initial value of (2) to (1, 0, 1, 0), and change the parameter ρ from 0 to 3.2. Other parameters are the same as mentioned above. We obtain the Lyapunov exponents and its corresponding bifurcation diagram as shown in Figure 3, where the last Lyapunov exponent is not displayed because it is always a big negative number. It shows that the Lyapunov exponents spectrum and bifurcation diagram are consistent. Figure 3(b) shows the routes to chaos of the system (2), and the system transforms into two-scroll hyperchaotic attractor from one-scroll chaotic attractor. When the circuit parameter ρ increases further, the system transforms into three-scroll hyperchaotic attractor from two-scroll hyperchaotic attractor. Obviously, there are two periodic windows in the chaotic region at about ρ=2.55 and ρ=2.79. It indicates that system (2) has abundant dynamical behaviors.Figure 3
Lyapunov exponents spectrum and bifurcation diagram with variation of the parameterρ.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramTo display its dynamics further,y-w phase portraits of different states with different parameter ρ are presented in Figure 4.Figure 4
Phase portraits with different parameterρ.
(a)
Chaotic attractor (ρ=0.5)
(b)
Type 1 limit cycle (ρ=2.55)
(c)
Type 2 limit cycle (ρ=2.79)
(d)
Two-scroll hyperchaotic attractor (ρ=3)
(e)
Three-scroll hyperchaotic attractor (ρ=3.15)


## 3. Numerical Solution of Fractional-Order 4D Hyperchaotic Memristive Circuit

### 3.1. Adomian Decomposition Method

For a given fractional-order differential equationD∗t0q(t)=f(x(t)), here x(t)=x1(t),x2(t),…,xn(t)T are variables and D∗t0qt is the Caputo derivative operator of order q [42, 43]. To obtain the following initial value problem, f(x(t)) is separated into three terms [44, 45]:(3)D∗t0qxt=Lxt+Nxt+gtxkt0+=bk,k=0,1,…,m-1.Here,L and N represent linear and nonlinear items, respectively, and g(t)=g1(t),g2(t),…,gn(t)T are constants for autonomous systems, and bk is a specified constant. By applying the operator Jt0q to both sides of (3), the following equation is obtained [46]:(4)x=Jt0qLx+Jt0qNx+Jt0qg+∑k=0m-1bkt-t0kk!.J
t
0
q is Riemann-Liouville fractional integral operator with order q. For t=t0,t1, q⩾0, r⩾0, γ>-1, and real constant C, the fundamental properties of the integral operator Jt0q are described as follows [47]:(5)Jt0qt-t0γ=Γγ+1Γγ+1+qt-t0γ+q,Jt0qC=CΓq+1t-t0q,Jt0qJt0rxt=Jt0q+rxt.Based on ADM, the nonlinear terms of (4) are decomposed according to(6)Aji=1i!didλiNνjiλλ=0νjiλ=∑k=0iλkxjk,where i=0,1,…,∞,j = 1,2,…,n; then the nonlinear terms are expressed as(7)Nx=∑i=0∞Aix0,x1,…,xi.So the solution of (3) x=∑i=0∞xi is derived from(8)x0=Jt0qg+∑k=0m-1bkt-t0kk!x1=Jt0qLx0+Jt0qA0x0x2=Jt0qLx1+Jt0qA1x0,x1⋮xi=Jt0qLxi-1+Jt0qAi-1x0,x1,…,xi-1⋮.


### 3.2. Solution of the Fractional-Order 4D Hyperchaotic Memristive Circuit

The equation of the fractional-order 4D hyperchaotic memristive circuit is(9)D∗t0qx=αy-xD∗t0qy=-xz+βy-ρWwxD∗t0qz=xy-γzD∗t0qw=x,where x, y, z, and w are the state variables, and q(0<q⩽1) is the order of fractional-order differential equation, where W(φ) is the memductance defined as W(w)=a+3bw2, and a, b, α, β, γ, and ρ are the system parameters. According to (5) and (8), the discrete iterative formula of the system (9) is presented by(10)xm+1=xm+αym-xmhqΓq+1+α-xmzm+βym-aρxm-3bρxmwm2+⋯h2qΓ2q+1+⋯ym+1=ym+-xmzm+βym-aρxm-3bρxmwm2hqΓq+1+-xmxmym+⋯h2qΓ2q+1+⋯zm+1=zm+xmym-γzmhqΓq+1+αym-xmym-γxmym-γzm+⋯h2qΓ2q+1+⋯wm+1=wm+xmhqΓq+1+αym-xmh2qΓ2q+1+α-xmzm+βym+⋯h3qΓ3q+1+⋯,where h is iteration step size. Г(·) is Gamma function. Considering the fast convergence of this method, we truncate the first six terms of (10) in this paper. For the computer simulation, the iteration is expressed as follows: (11)C10=xmC20=ymC30=zmC40=wm,C11=αC20-C10C21=-C10C30+βC20-aρC10-3bρC10C402C31=C10C20-γC30C41=C10,C12=αC21-C11C22=-C10C31-C11C30+βC20-aρC11-3bρC11C402+2C10C40C41C32=C11C20+C10C21-γC31C42=C11,C13=αC22-C12C23=-C10C32-C12C30-C11C31Γ2q+1Γq+12+βC22-aρC12-3bρC12C402+2C10C40C42+2C11C40C41+C10C412Γ2q+1Γq+12C33=C10C22+C12C20-γC32+C11C21Γ2q+1Γq+12C43=C12,C14=αC23-C13C24=-C13C30-C10C33-C12C31+C11C32Γ3q+1Γq+1Γ2q+1+βC23-aρC13-3bρC13C402+2C10C40C43+2C12C40C41+2C11C40C42+2C10C41C42Γ3q+1Γq+1Γ2q+1+C11C412Γ3q+1Γq+13C34=C13C20+C10C23+C12C21+C11C22Γ3q+1Γq+1Γ2q+1-γC33C44=C13,C15=αC24-C14C25=-C14C30-C10C34-C13C31+C11C33Γ4q+1Γq+1Γ3q+1-C12C32Γ4q+1Γ2q+12+βC24-aρC14-3bρC14C402+C10C40C44+2C13C40C41+2C11C40C43+2C10C41C43Γ4q+1Γq+1Γ3q+1+2C12C40C42+C10C422Γ4q+1Γ2q+12+C12C412+2C11C41C42Γ4q+1Γ2q+1Γq+12C35=C14C20+C10C24+C13C21+C11C23Γ4q+1Γq+1Γ3q+1+C12C22Γ4q+1Γ2q+12-γC33C45=C14,xm+1ym+1zm+1wm+1=C10C11C12C13C14C15C20C21C22C23C24C25C30C31C32C33C34C35C40C41C42C43C44C44=1hqΓq+1h2qΓ2q+1h3qΓ3q+1h4qΓ4q+1h5qΓ5q+1T,According to (11), the chaotic sequences of the fractional-order 4D hyperchaotic memristive circuit are obtained with initial values, h, q, and appropriate parameters. Then we can analyze the dynamical characteristics of the system by using the chaotic sequences.


## 3.1. Adomian Decomposition Method

For a given fractional-order differential equationD∗t0q(t)=f(x(t)), here x(t)=x1(t),x2(t),…,xn(t)T are variables and D∗t0qt is the Caputo derivative operator of order q [42, 43]. To obtain the following initial value problem, f(x(t)) is separated into three terms [44, 45]:(3)D∗t0qxt=Lxt+Nxt+gtxkt0+=bk,k=0,1,…,m-1.Here,L and N represent linear and nonlinear items, respectively, and g(t)=g1(t),g2(t),…,gn(t)T are constants for autonomous systems, and bk is a specified constant. By applying the operator Jt0q to both sides of (3), the following equation is obtained [46]:(4)x=Jt0qLx+Jt0qNx+Jt0qg+∑k=0m-1bkt-t0kk!.J
t
0
q is Riemann-Liouville fractional integral operator with order q. For t=t0,t1, q⩾0, r⩾0, γ>-1, and real constant C, the fundamental properties of the integral operator Jt0q are described as follows [47]:(5)Jt0qt-t0γ=Γγ+1Γγ+1+qt-t0γ+q,Jt0qC=CΓq+1t-t0q,Jt0qJt0rxt=Jt0q+rxt.Based on ADM, the nonlinear terms of (4) are decomposed according to(6)Aji=1i!didλiNνjiλλ=0νjiλ=∑k=0iλkxjk,where i=0,1,…,∞,j = 1,2,…,n; then the nonlinear terms are expressed as(7)Nx=∑i=0∞Aix0,x1,…,xi.So the solution of (3) x=∑i=0∞xi is derived from(8)x0=Jt0qg+∑k=0m-1bkt-t0kk!x1=Jt0qLx0+Jt0qA0x0x2=Jt0qLx1+Jt0qA1x0,x1⋮xi=Jt0qLxi-1+Jt0qAi-1x0,x1,…,xi-1⋮.


## 3.2. Solution of the Fractional-Order 4D Hyperchaotic Memristive Circuit

The equation of the fractional-order 4D hyperchaotic memristive circuit is(9)D∗t0qx=αy-xD∗t0qy=-xz+βy-ρWwxD∗t0qz=xy-γzD∗t0qw=x,where x, y, z, and w are the state variables, and q(0<q⩽1) is the order of fractional-order differential equation, where W(φ) is the memductance defined as W(w)=a+3bw2, and a, b, α, β, γ, and ρ are the system parameters. According to (5) and (8), the discrete iterative formula of the system (9) is presented by(10)xm+1=xm+αym-xmhqΓq+1+α-xmzm+βym-aρxm-3bρxmwm2+⋯h2qΓ2q+1+⋯ym+1=ym+-xmzm+βym-aρxm-3bρxmwm2hqΓq+1+-xmxmym+⋯h2qΓ2q+1+⋯zm+1=zm+xmym-γzmhqΓq+1+αym-xmym-γxmym-γzm+⋯h2qΓ2q+1+⋯wm+1=wm+xmhqΓq+1+αym-xmh2qΓ2q+1+α-xmzm+βym+⋯h3qΓ3q+1+⋯,where h is iteration step size. Г(·) is Gamma function. Considering the fast convergence of this method, we truncate the first six terms of (10) in this paper. For the computer simulation, the iteration is expressed as follows: (11)C10=xmC20=ymC30=zmC40=wm,C11=αC20-C10C21=-C10C30+βC20-aρC10-3bρC10C402C31=C10C20-γC30C41=C10,C12=αC21-C11C22=-C10C31-C11C30+βC20-aρC11-3bρC11C402+2C10C40C41C32=C11C20+C10C21-γC31C42=C11,C13=αC22-C12C23=-C10C32-C12C30-C11C31Γ2q+1Γq+12+βC22-aρC12-3bρC12C402+2C10C40C42+2C11C40C41+C10C412Γ2q+1Γq+12C33=C10C22+C12C20-γC32+C11C21Γ2q+1Γq+12C43=C12,C14=αC23-C13C24=-C13C30-C10C33-C12C31+C11C32Γ3q+1Γq+1Γ2q+1+βC23-aρC13-3bρC13C402+2C10C40C43+2C12C40C41+2C11C40C42+2C10C41C42Γ3q+1Γq+1Γ2q+1+C11C412Γ3q+1Γq+13C34=C13C20+C10C23+C12C21+C11C22Γ3q+1Γq+1Γ2q+1-γC33C44=C13,C15=αC24-C14C25=-C14C30-C10C34-C13C31+C11C33Γ4q+1Γq+1Γ3q+1-C12C32Γ4q+1Γ2q+12+βC24-aρC14-3bρC14C402+C10C40C44+2C13C40C41+2C11C40C43+2C10C41C43Γ4q+1Γq+1Γ3q+1+2C12C40C42+C10C422Γ4q+1Γ2q+12+C12C412+2C11C41C42Γ4q+1Γ2q+1Γq+12C35=C14C20+C10C24+C13C21+C11C23Γ4q+1Γq+1Γ3q+1+C12C22Γ4q+1Γ2q+12-γC33C45=C14,xm+1ym+1zm+1wm+1=C10C11C12C13C14C15C20C21C22C23C24C25C30C31C32C33C34C35C40C41C42C43C44C44=1hqΓq+1h2qΓ2q+1h3qΓ3q+1h4qΓ4q+1h5qΓ5q+1T,According to (11), the chaotic sequences of the fractional-order 4D hyperchaotic memristive circuit are obtained with initial values, h, q, and appropriate parameters. Then we can analyze the dynamical characteristics of the system by using the chaotic sequences.


## 4. Dynamical Characteristics

### 4.1. Stability Analysis

By setting the left-hand side of (9) to zero, we can calculate the equilibrium points of the system. Obviously, p0=(x,y,z,w)=(0,0,0,n) is the one equilibrium point set for the system, n is a real constant. The Jacobian matrix of the system (9) at equilibrium point set p0 is described as follows:(12)JP0=-αα00-aρ-3ρbn2β0000-γ01000.The characteristic polynomial of (12) is λ(λ+γ)(λ2+(α-β)-αβ+aαρ+3αρbn2)=0, which has root at(13)λ1=0λ2=-γλ2=β-α+j4aαρ+12αρbn2-4αβ-α-β22λ3=β-α-j4aαρ+12αρbn2-4αβ-α-β22.The stability of the equilibrium can be investigated usingTheorem  1: the fractional-order system is asymptotically stable if all the eigenvalues λΨ of the Jacobian matrix JΨ satisfy the condition arg(λi)>(π/2)q [30]. It determines the stable and unstable regions as shown in Figure 5.Figure 5
Stable and unstable regions in fractional-order 4D hyperchaotic memristive circuit.The black region in Figure5 is the stable region that arg(λi)>(π/2)q, while light region is the unstable region that arg(λi)>(π/2)q. When q=1, it is the stability of integer-order case.Here, we consider the simple caseq1 = q2 = q3 = q4 =q, where the fractional-order system has a commensurate order. According toTheorem  2,suppose that the unstable eigenvalues of scroll saddle points are λ3,4=r3,4±jω3,4 [30]. The necessary condition to exhibit the chaotic attractor of (9) is the eigenvalues λ3,4 remaining in the unstable region. The condition for the commensurate derivatives order is q>π/2atan⁡ω3,4/r3,4.


### 4.2. Phase Portraits

Setting the parametersa=4, b=0.01, α=36, β=20, γ=3, ρ=3, and q=0.85, the initial value of (9) is (1, 0, 1, 0). We get the chaotic attractor as shown in Figure 6. In this case, the Lyapunov exponents of the system are L1 = 0.8528, L2 = 0.3133, and L3 = 0, L4 = −59.55 and the Lyapunov dimension is dL=3.02. The fractional-order 4D hyperchaotic memristive circuit is in a hyperchaotic state as shown in Figure 6. Obviously, in this case, the largest Lyapunov exponent is much bigger than that of integer-order system.Figure 6
Chaotic attractor of the circuit with the 4D hyperchaotic memristive circuit.
(a)
x
-
z
(b)
y
-
w


### 4.3. Lyapunov Exponents Spectra and Bifurcation Diagram

According to theLE spectra calculation algorithm [48], we let a=4, b=0.01, α=36, β=20, γ=3, and q=0.85; the initial value of (9) is (1, 0, 1, 0). The Lyapunov exponents spectra and bifurcation diagram of the fractional-order 4D hyperchaotic memristive circuit by changing ρ simultaneously are shown in Figure 7, where the last Lyapunov exponent is not displayed because it is always a big negative number.Figure 7
Lyapunov exponents spectrum and bifurcation diagram with variation of the circuit parameterρ.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramAll of the dynamical behaviors of the system with different parameterρ are summarized in Table 1. It shows that system (9) has abundant dynamical behaviors.Table 1
Dynamical behaviors of the system with differentρ.
ρ
Sign of Lyapunov exponents
Dynamical behaviors
0~2.03
+ 0 0 −
Chaos
2.04
0 0 −  −
Period 2 cycle
2.05~2.34
+ 0 0 −
chaos
2.35
+ + 0 −
Hyperchaos
2.36~2.45
+ 0 0 −
Chaos
2.46
+ + 0 −
Hyperchaos
2.47~2.51
+ 0 0 −
Chao
2.52
+ + 0 −
Hyperchaos
2.53~2.55
+ 0 0 −
Chaos
2.56
+ + 0 −
Hyperchaos
2.57~2.65
+ 0 0 −
Chaos
2.66~3.08
+ + 0 −
Hyperchaos
3.09
+ 0 0 −
chaos
3.1~3.2
0 − − −
Limit cycleTo display its dynamics further,y-w phase portraits of different states with different parameter ρ are presented in Figure 8. With the increase of the parameter ρ, the system transforms from hyperchaotic attractor to limit cycle. Obviously, there is a periodic window in the chaotic region when ρ = (2.0394–2.0407).Figure 8
Phase portrait with different circuit parameterρ.
(a)
Chaotic attractor (ρ=0.1)
(b)
Period 2 cycle (ρ=2.04)
(c)
Hyperchaotic attractor (ρ=3)
(d)
Limit cycle (ρ=3.15)For above circuit parameters andρ=3, the Lyapunov exponents spectra and bifurcation diagram of the fractional-order 4D hyperchaotic memristive circuit by changing q simultaneously are shown in Figure 9, where the last Lyapunov exponent is not displayed because it is always a big negative number. The Lyapunov exponents spectra and the bifurcation diagram match very well. Both of them show that the system has complex behaviors. Obviously, once the system is in hyperchaotic state, with the increase of the parameterq, the largest Lyapunov exponent becomes smaller and smaller until in the q=1. So the fractional-order hyperchaotic system has a higher complexity compared to its corresponding integer-order system.Figure 9
Lyapunov exponents spectrum and bifurcation diagram with variation of the circuit parameterq.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramAll of the dynamical behaviors of the system with different parameterq are summarized in Table 2. It shows that system (9) has abundant and complex dynamical behaviors.Table 2
Dynamical behaviors of the system with differentq.
q
Signs of Lyapunov exponents
Dynamical behaviors
0.75~0.778
0 − − −
Sink
0.779~0.799
0 − − −
Type 1 periodic orbit
0.8~0.801
+ 0 −  −
Four-scroll chaos
0.802
0 − − −
Type 1 periodic orbit
0.803~0.804
+ 0 −  −
Four-scroll chaos
0.805~0.806
0 − − −
Type 2 periodic orbit
0.807
+ 0 −  −
Four-scroll chaos
0.808~0.809
0 − − −
Type 2 periodic orbit
0.81~0.814
+ 0 −  −
Four-scroll chaos
0.815~0.818
+ + 0 −
Three scroll hyperchaos
0.819
+ 0 −  −
Four-scroll chaos
0.82~0.822
+ + 0 −
Three-scroll hyperchaos
0.823~0.825
+ 0 −  −
Four-scroll chaos
0.826~0.829
+ + 0 −
Three-scroll hyperchaos
0.83
+ 0 −  −
Four-scroll chaos
0.831~0.84
+ + 0 −
Three-scroll hyperchaos
0.841
+ 0 −  −
Four-scroll chaos
0.842~0.85
+ + 0 −
Three-scroll hyperchaos
0.851
+ 0 0 −
Two-scroll chaos
0.852~1
+ + 0 −
Two-scroll hyperchaosTo further display its dynamics,y-w phase portraits of different states with different parameter q are presented in Figure 10. The route to limit cycle from sink can be observed. With the increase of the parameter q, the system transforms from chaotic attractor to hyperchaotic attractor. Obviously, there are several periodic windows in the chaotic region.Figure 10
Phase portrait with different circuit parameterq.
(a)
Sink (q=0.76)
(b)
Type 1 limit cycle (q=0.78)
(c)
Four-scroll chaotic attractor (q=0.801)
(d)
Type 2 limit cycle (q=0.806)
(e)
Three-scroll hyperchaotic attractor (q=0.816)
(f)
Two-scroll chaotic attractor (q=0.851)
(g)
Two-scroll hyperchaotic attractor (q=1)


### 4.4. Spectral Entropy (SE) andC0 Complexity

Complexity measure is an important reference to measure dynamics of a chaotic system. If a chaotic system is used in information security, it can reflect the security of the system to some extent. At present, there are several algorithms for measuring the complexity of chaotic sequences, including intensive statistical [49], multiscale entropy (MSE) [50], spectral entropy (SE) algorithm [51], andC0 complexity arithmetic [52]. Among them,C0 and SE complexity algorithms have less parameters, faster calculation speed, and higher accuracy. So we employ the SE algorithm to measure the complexity of the fractional-order 4D hyperchaotic system.Leta=4, b=0.01, α=36, β=20, γ=3, and q=0.85; the initial value of (9) is (1, 0, 1, 0). According to the arithmetic spectral entropy (SE) andC0 complexity arithmetic, the SE andC0 complexity of the fractional-order 4D hyperchaotic memristive circuit by changing q and ρ simultaneously are shown in Figure 11.Figure 11
SE andC0 complexity of the fractional-order 4D hyperchaotic memristive circuit by changing q and ρ.
(a)
SE complexity
(b)
C
0 complexityFigures7, 9, and 11 show that the Lyapunov exponents spectrum, bifurcation diagram, and complexity are consistent. Figures 11(a) and 11(b) show that the change trends of SE complexity andC0 complexity are consistent, but the values are different. It is mainly because of the differences defined by the SE andC0 definitions, which mean that complexity can also reflect the dynamic characteristics of a chaotic system.


## 4.1. Stability Analysis

By setting the left-hand side of (9) to zero, we can calculate the equilibrium points of the system. Obviously, p0=(x,y,z,w)=(0,0,0,n) is the one equilibrium point set for the system, n is a real constant. The Jacobian matrix of the system (9) at equilibrium point set p0 is described as follows:(12)JP0=-αα00-aρ-3ρbn2β0000-γ01000.The characteristic polynomial of (12) is λ(λ+γ)(λ2+(α-β)-αβ+aαρ+3αρbn2)=0, which has root at(13)λ1=0λ2=-γλ2=β-α+j4aαρ+12αρbn2-4αβ-α-β22λ3=β-α-j4aαρ+12αρbn2-4αβ-α-β22.The stability of the equilibrium can be investigated usingTheorem  1: the fractional-order system is asymptotically stable if all the eigenvalues λΨ of the Jacobian matrix JΨ satisfy the condition arg(λi)>(π/2)q [30]. It determines the stable and unstable regions as shown in Figure 5.Figure 5
Stable and unstable regions in fractional-order 4D hyperchaotic memristive circuit.The black region in Figure5 is the stable region that arg(λi)>(π/2)q, while light region is the unstable region that arg(λi)>(π/2)q. When q=1, it is the stability of integer-order case.Here, we consider the simple caseq1 = q2 = q3 = q4 =q, where the fractional-order system has a commensurate order. According toTheorem  2,suppose that the unstable eigenvalues of scroll saddle points are λ3,4=r3,4±jω3,4 [30]. The necessary condition to exhibit the chaotic attractor of (9) is the eigenvalues λ3,4 remaining in the unstable region. The condition for the commensurate derivatives order is q>π/2atan⁡ω3,4/r3,4.


## 4.2. Phase Portraits

Setting the parametersa=4, b=0.01, α=36, β=20, γ=3, ρ=3, and q=0.85, the initial value of (9) is (1, 0, 1, 0). We get the chaotic attractor as shown in Figure 6. In this case, the Lyapunov exponents of the system are L1 = 0.8528, L2 = 0.3133, and L3 = 0, L4 = −59.55 and the Lyapunov dimension is dL=3.02. The fractional-order 4D hyperchaotic memristive circuit is in a hyperchaotic state as shown in Figure 6. Obviously, in this case, the largest Lyapunov exponent is much bigger than that of integer-order system.Figure 6
Chaotic attractor of the circuit with the 4D hyperchaotic memristive circuit.
(a)
x
-
z
(b)
y
-
w


## 4.3. Lyapunov Exponents Spectra and Bifurcation Diagram

According to theLE spectra calculation algorithm [48], we let a=4, b=0.01, α=36, β=20, γ=3, and q=0.85; the initial value of (9) is (1, 0, 1, 0). The Lyapunov exponents spectra and bifurcation diagram of the fractional-order 4D hyperchaotic memristive circuit by changing ρ simultaneously are shown in Figure 7, where the last Lyapunov exponent is not displayed because it is always a big negative number.Figure 7
Lyapunov exponents spectrum and bifurcation diagram with variation of the circuit parameterρ.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramAll of the dynamical behaviors of the system with different parameterρ are summarized in Table 1. It shows that system (9) has abundant dynamical behaviors.Table 1
Dynamical behaviors of the system with differentρ.
ρ
Sign of Lyapunov exponents
Dynamical behaviors
0~2.03
+ 0 0 −
Chaos
2.04
0 0 −  −
Period 2 cycle
2.05~2.34
+ 0 0 −
chaos
2.35
+ + 0 −
Hyperchaos
2.36~2.45
+ 0 0 −
Chaos
2.46
+ + 0 −
Hyperchaos
2.47~2.51
+ 0 0 −
Chao
2.52
+ + 0 −
Hyperchaos
2.53~2.55
+ 0 0 −
Chaos
2.56
+ + 0 −
Hyperchaos
2.57~2.65
+ 0 0 −
Chaos
2.66~3.08
+ + 0 −
Hyperchaos
3.09
+ 0 0 −
chaos
3.1~3.2
0 − − −
Limit cycleTo display its dynamics further,y-w phase portraits of different states with different parameter ρ are presented in Figure 8. With the increase of the parameter ρ, the system transforms from hyperchaotic attractor to limit cycle. Obviously, there is a periodic window in the chaotic region when ρ = (2.0394–2.0407).Figure 8
Phase portrait with different circuit parameterρ.
(a)
Chaotic attractor (ρ=0.1)
(b)
Period 2 cycle (ρ=2.04)
(c)
Hyperchaotic attractor (ρ=3)
(d)
Limit cycle (ρ=3.15)For above circuit parameters andρ=3, the Lyapunov exponents spectra and bifurcation diagram of the fractional-order 4D hyperchaotic memristive circuit by changing q simultaneously are shown in Figure 9, where the last Lyapunov exponent is not displayed because it is always a big negative number. The Lyapunov exponents spectra and the bifurcation diagram match very well. Both of them show that the system has complex behaviors. Obviously, once the system is in hyperchaotic state, with the increase of the parameterq, the largest Lyapunov exponent becomes smaller and smaller until in the q=1. So the fractional-order hyperchaotic system has a higher complexity compared to its corresponding integer-order system.Figure 9
Lyapunov exponents spectrum and bifurcation diagram with variation of the circuit parameterq.
(a)
Lyapunov exponents spectrum
(b)
Bifurcation diagramAll of the dynamical behaviors of the system with different parameterq are summarized in Table 2. It shows that system (9) has abundant and complex dynamical behaviors.Table 2
Dynamical behaviors of the system with differentq.
q
Signs of Lyapunov exponents
Dynamical behaviors
0.75~0.778
0 − − −
Sink
0.779~0.799
0 − − −
Type 1 periodic orbit
0.8~0.801
+ 0 −  −
Four-scroll chaos
0.802
0 − − −
Type 1 periodic orbit
0.803~0.804
+ 0 −  −
Four-scroll chaos
0.805~0.806
0 − − −
Type 2 periodic orbit
0.807
+ 0 −  −
Four-scroll chaos
0.808~0.809
0 − − −
Type 2 periodic orbit
0.81~0.814
+ 0 −  −
Four-scroll chaos
0.815~0.818
+ + 0 −
Three scroll hyperchaos
0.819
+ 0 −  −
Four-scroll chaos
0.82~0.822
+ + 0 −
Three-scroll hyperchaos
0.823~0.825
+ 0 −  −
Four-scroll chaos
0.826~0.829
+ + 0 −
Three-scroll hyperchaos
0.83
+ 0 −  −
Four-scroll chaos
0.831~0.84
+ + 0 −
Three-scroll hyperchaos
0.841
+ 0 −  −
Four-scroll chaos
0.842~0.85
+ + 0 −
Three-scroll hyperchaos
0.851
+ 0 0 −
Two-scroll chaos
0.852~1
+ + 0 −
Two-scroll hyperchaosTo further display its dynamics,y-w phase portraits of different states with different parameter q are presented in Figure 10. The route to limit cycle from sink can be observed. With the increase of the parameter q, the system transforms from chaotic attractor to hyperchaotic attractor. Obviously, there are several periodic windows in the chaotic region.Figure 10
Phase portrait with different circuit parameterq.
(a)
Sink (q=0.76)
(b)
Type 1 limit cycle (q=0.78)
(c)
Four-scroll chaotic attractor (q=0.801)
(d)
Type 2 limit cycle (q=0.806)
(e)
Three-scroll hyperchaotic attractor (q=0.816)
(f)
Two-scroll chaotic attractor (q=0.851)
(g)
Two-scroll hyperchaotic attractor (q=1)


## 4.4. Spectral Entropy (SE) andC0 Complexity

Complexity measure is an important reference to measure dynamics of a chaotic system. If a chaotic system is used in information security, it can reflect the security of the system to some extent. At present, there are several algorithms for measuring the complexity of chaotic sequences, including intensive statistical [49], multiscale entropy (MSE) [50], spectral entropy (SE) algorithm [51], andC0 complexity arithmetic [52]. Among them,C0 and SE complexity algorithms have less parameters, faster calculation speed, and higher accuracy. So we employ the SE algorithm to measure the complexity of the fractional-order 4D hyperchaotic system.Leta=4, b=0.01, α=36, β=20, γ=3, and q=0.85; the initial value of (9) is (1, 0, 1, 0). According to the arithmetic spectral entropy (SE) andC0 complexity arithmetic, the SE andC0 complexity of the fractional-order 4D hyperchaotic memristive circuit by changing q and ρ simultaneously are shown in Figure 11.Figure 11
SE andC0 complexity of the fractional-order 4D hyperchaotic memristive circuit by changing q and ρ.
(a)
SE complexity
(b)
C
0 complexityFigures7, 9, and 11 show that the Lyapunov exponents spectrum, bifurcation diagram, and complexity are consistent. Figures 11(a) and 11(b) show that the change trends of SE complexity andC0 complexity are consistent, but the values are different. It is mainly because of the differences defined by the SE andC0 definitions, which mean that complexity can also reflect the dynamic characteristics of a chaotic system.


## 5. Conclusions

In this paper, the dynamical characteristics of integer-order 4D hyperchaotic memristive circuit are analyzed firstly. Based on Adomian decomposition method (ADM), the numerical solution of a fractional-order 4D hyperchaotic memristive system is investigated and all parameters values of the system are determined. The equilibrium points are obtained, and the distribution of stable and unstable regions of the system is determined. Dynamical characteristics are studied by phase portraits, Lyapunov exponents spectrum, bifurcation diagram, SE complexity, andC0 complexity. We found that the dynamical characteristics of fractional-order 4D hyperchaotic memristive system are more complex than that of the integer-order system, and the SE andC0 complexity could reflect the dynamical characteristics quite well. According to the variation of the system dynamics, it is found that the fractional-order hyperchaotic system has better application prospect than its corresponding integer-order chaotic system in the field of chaotic secure communication. Next, we will try to study its implementation in hardware.



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*Source: 2313768-2017-07-10.xml*