# Analysis and Implementation of a New Switching Memristor Scroll Hyperchaotic System and Application in Secure Communication

Authors: Ping Liu; Rui Xi; Pengbo Ren; Jialin Hou; Xiang Li
Journal: Complexity
 (2018)

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## Abstract

This paper proposed a novel switching scroll hyperchaotic system based on a memristor device and explored its application to secure communication. The new system could be switched between the double-scroll chaotic system and multiscroll one by switchS1 and switch S2. We gave the construction process of the novel system, its numerical simulations, and dynamical properties, firstly. Moreover, the memristive circuit implementation of the new switching system was presented and the results were also in agreement with those of numerical simulation. Finally, the new switching memristive system was applied to secure communication by means of the drive-response synchronization with chaotic masking. When the voice signal is a rising waveform, it is encrypted by the double-scroll memristive system. When the voice signal is a falling waveform, the multiscroll memristive system works. The voice signal is completely submerged in the chaotic signal and could not be distinguished at all. Security analyses show that it is a successful application to secure communication.


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## Body

## 1. Introduction

Chaotic systems based on memristor have widely attracted attention recently. It has many applications such as in secure communication [1–4], neural network [5–9], and chemical route [10]. A lot of researches on memristive chaotic (hyperchaotic) system have been reported, for example, global synchronization [11], state estimation [12], time delay [13], and adaptive synchronization [14] of the memristive chaotic system. On the other hand, scroll chaotic systems have been explored in many papers extensively. For instance, Ma et al. introduced simulation and circuit implementation of 12-scroll chaotic system in [15]. Chen et al. investigated the generation of grid multiscroll chaotic attractors in [16]. In [17], Chen et al. studied the fractional-order multiscroll chaotic system. And in [18-19], García-Martínez et al. and Liu et al. discussed the multiscroll hyperchaotic system and its application to secure communication, but the memristor has not been introduced to chaotic system to produce multiscroll.With respect to the chaotic system based on memrister, its application to secure communication has been studied in many works, such as in [1–4], since memristors are nonlinear elements with memory function, which are different from resistors, capacitors, and inductors. So the applications to secure communication based on chaotic system with memristor have become a hot topic. Meanwhile, researches on application of the scroll chaotic system have also existed. Whether the memristive chaotic (hyperchaotic) system or the scroll system, application to secure communication has been very common. Till now, secure communication based on the switching memristor scroll hyperchaotic system has not been explored. Thus, it has a great significance to investigate the switching memristive scroll hyperchaotic system and its application to secure communication.The rest of the paper was organized as follows. Section2 introduced the construction, numerical simulations, and dynamic analysis of the switching scroll hyperchaotic system based on a memristor. The circuit implementation was given in Section 3. In Section 4, application to voice encryption was discussed using the drive-response synchronization with chaotic masking and some conclusions are given in the last.


## 2. System Construction and Dynamic Analysis

### 2.1. System Construction

The Chua’s circuit is composed of a linear resistor, a linear inductor, two linear capacitors, and a nonlinear Chua’s diode as shown in Figure1.Figure 1
Chua’s circuit diagram.Based on Figure1, the differential equation [20] could be obtained as
(1)dxdt=ay−x+fx,dydt=x−y+z,dzdt=−by,where fx=−5x−1.5x+1+x−1/7, when taking system parameters a=10 and b=15 with initial values x0=y0=z0=0.1, we can get the projection on the x−z phase plane of the double-scroll attractors shown in Figure 2.Figure 2
A projection of the double-scroll attractor onto thex,z plane.In Figure1, the resistor R and the Chua’s diode are replaced by a magnetically controlled memristor M and a linear function, respectively. So, we can get a new hyperchaotic memristive circuit demonstrated by Figure 3. The memristor is the fourth kind of basic passive circuit element which is proposed by Professor Chua. There are two mathematical definitions of memristor, which are charge-controlled memristor and flux-controlled memristor, respectively [21]. The expression of a charge-controlled memristor is
(2)vt=Mqtit,Mqt=dφqdq.Figure 3
The new hyperchaotic memristive circuit.The flux-controlled memristor could be expressed as(3)it=Wφtvt,Wφ=dqφdφ.And basic model of the memristor can be found in [22]. The definition of memristor was extended in [23],
(4)yt=gz,u,tut,dzdt=fu,z,t,where ut and yt are the input signal and output signal of the memristor, respectively. z is the system state variable. The input of the magnetically controlled memristor is voltage u, the output is the current i following through the memristor, and flux ϕ is the state variable.According to the definition of memristor, we obtain a novel expression as(5)i=Wϕu,dϕdt=u2−u1−cϕ,where Wϕ is called memory conductance, let Wϕ=0.1ϕ2. According to the Kirchhoff’s law and component parameter constraints, circuit dynamic equation could be obtained as
(6)C1dv1dt=WϕV2−V1+gV1,C2dv2dt=WϕV1−V2+Ii,LdIidt=−V2,RCdϕdt=V2−V1−cϕ,where τ0=RC is the time constant. Dimensionless time τ=t/τ0 could be obtained by the scaling time and state space. The dimensional state variable x=V1, y=V1, z=Ri, w=ϕ, a=C/C1, C=C2, b=cR2/L, fw=Wϕ/R, and gx=gV1/R, so we can obtain
(7)dxdt=afwy−x+gx,dydt=fwx−y+z,dzdt=−by,dwdt=y−x−cw,where fw=dw2, gx=0.9x, and a, b, c, and d are system parameters. The simulation results are given in Figure 4 by taking initial values x0=1, y0=z0=w0=0, and parameters a=10, b=60, c=10, and d=0.1. Then, based on (7), the N-scroll is realized in y direction by using the step function.Figure 4
Projections of the double-scroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)There are two forms of step function, when(8)hy=A∑i=1Msgn[y+2i−1A+∑i=1Msgny−2i−1A,M≥1,2M+1 scrolls could be generated in y direction.When(9)hy=A−sgny+∑i=0N−1sgn[y+2iA+∑i=0N−1sgny−2iA,N≥0,2N scrolls could be generated in y direction.N-scroll could be obtained by a reasonable set of step function parameters. For example, whenM=1, A=12,
(10)hy=12sgny−12+sgny+12.The 2 × 3 multiscroll hyperchaotic system based on a memristor device is obtained as(11)dxdt=afwy−x−hy+gx,dydt=fwx−y+hy+z,dzdt=−by−hy,dwdt=y−x−hy−cw.When system parameters area=10, b=60, c=10, and d=0.1 with initial values x0=3 and y0=z0=w0=0, simulation results are presented in Figure 5.Figure 5
Projections of the 2 × 3 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)Similarly, 2 × 5 and 2 × 6 multiscroll hyperchaotic system could be also obtained by takingM=2, A=12, and N=3, A=12, and
(12)hy=12sgny+12+sgny−12+sgny+36+sgny−36,hy=12sgny+sgny−24+sgny+24+sgny−48+sgny+48,respectively.When selecting system parametersa=10, b=60, c=10, and d=0.1 with initial values x0=3 and y0=z0=w0=0, the numerical simulations of 2 × 5 and 2 × 6 multiscroll hyperchaotic system are shown in Figures 6 and 7.Figure 6
Projections of the 2 × 5 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)Figure 7
Projections of the 2 × 6 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)


### 2.2. Dynamic Analysis

#### 2.2.1. The Double-Scroll Memristive System Is Taken as an Example to Analyse the Dynamic Properties

(1) Lyapunov Exponents. When selecting system parameters a=10, b=60, c=10, and d=0.1 with initial values x0=1 and y0=z0=w0=0, we can get the Lyapunov exponents diagram as shown in Figure 8.Figure 8
Lyapunov exponents diagram.From Figure8, we can see that the chaotic system has two positive Lyapunov exponents. So, the system is a hyperchaotic system and has two scrolls.(2) Bifurcation Diagram. Lyapunov exponents and bifurcation diagram of the system due to the variation of parameter b is displayed in Figure 9. We could see that when 35<b<43 and 54<b<70, the system shows chaotic behavior.Figure 9
Lyapunov exponents and bifurcation diagram with respect to parameterb. (a) Lyapunov exponents diagram. (b) Bifurcation diagram.
(a)
(b)(3) Dissipation. Generation of chaotic behavior is decided by whether the system has a dissipative structure or not [24]. The dissipative formula of the system is
(13)∇V=∂ẋ∂x+∂ẏ∂y+∂ż∂z+∂ẇ∂w=0.9a−c<0,so the system is dissipative, converging at an exponential rate e0.9a−c until it becomes 0. When t→∞, every volume element which contains trajectories of the system shrinks to zero with the rate of exponential convergent.(4) Equilibrium Points and Stability. The Jacobian matrix of the system is
(14)9−w2w202wy−x0.1w2−0.1w210.2wx−y0−6000−110−10.The equilibrium points of the system could be calculated by making (7) as 0. We could get three equilibrium points as
(15)S0=0,0,0,0,S1=30,0,−27,−3S2=−30,0,27,3.The Jacobian matrix at pointS0 becomes
(16)900000100−6000−110−10.Eigenvalues at pointS0 are λ1=−10, λ2=9, λ3=7.746i, and λ4=−7.746i. The Jacobian matrix at point S1 and S2 becomes
(17)0901800.9−0.91−180−6000−110−10.The EquilibriaS1 and S2 could have eigenvalues λ1=−5.9483+13.6659i, λ2=−5.9483−13.6659i, λ3=0.4983+6.9549i, and λ4=0.4983−6.9549i, which are called saddle points of index 2 since the two complex conjugate eigenvalues have positive real parts[25, 26]. It is clear that S0 is the first type saddle point since the real eigenvalue is positive [25, 26]. It is noticed that the scrolls are generated only around the equilibria of saddle points of index 2 [25, 26]. Moreover, equilibria S1 and S2 correspond to the two saturated plateaus, which are responsible for generating the two scrolls in the double-scroll attractor. However, the equilibrium point S0 corresponds to the saturated slope and is responsible for connecting these two symmetrical scrolls.


#### 2.2.2. The Dynamic Analysis of 2 × 3 Multiscroll Hyperchaotic System

The dynamic analysis of 2 × 3 multiscroll hyperchaotic system is introduced briefly as follows. Let(18)dxdt=dydt=dzdt=dwdt=0.Equation of equilibrium points could be obtained as
(19)afwy−x−hy+gx=0,fwx−y+hy+z=0,−by−hy=0,y−x−hy−cw=0.From (19), we could get the coordinate of system equilibrium points in the y-axis as y=hy, that is, y=0, ±12, ±24, x=±30. The figures of equilibrium points and chaotic attractors of the multiscroll hyperchaotic system are demonstrated in Figure 10. Therefore, the number of equilibrium points in the y-axis direction is 5 × 2. The solid points represent the second type saddle point, which correspond to the six saturated plateaus and are responsible for generating the six scrolls in the 2 × 3-scroll attractors. The hollow points represent the first type saddle point, which correspond to the saturated slope and are responsible for connecting these six symmetrical scrolls.Figure 10
Equilibrium points and attractors of the 2 × 3 multiscroll hyperchaotic system. (a) Equilibrium points. (b) 2 × 3 multiscroll attractors.
(a)
(b)


## 2.1. System Construction

The Chua’s circuit is composed of a linear resistor, a linear inductor, two linear capacitors, and a nonlinear Chua’s diode as shown in Figure1.Figure 1
Chua’s circuit diagram.Based on Figure1, the differential equation [20] could be obtained as
(1)dxdt=ay−x+fx,dydt=x−y+z,dzdt=−by,where fx=−5x−1.5x+1+x−1/7, when taking system parameters a=10 and b=15 with initial values x0=y0=z0=0.1, we can get the projection on the x−z phase plane of the double-scroll attractors shown in Figure 2.Figure 2
A projection of the double-scroll attractor onto thex,z plane.In Figure1, the resistor R and the Chua’s diode are replaced by a magnetically controlled memristor M and a linear function, respectively. So, we can get a new hyperchaotic memristive circuit demonstrated by Figure 3. The memristor is the fourth kind of basic passive circuit element which is proposed by Professor Chua. There are two mathematical definitions of memristor, which are charge-controlled memristor and flux-controlled memristor, respectively [21]. The expression of a charge-controlled memristor is
(2)vt=Mqtit,Mqt=dφqdq.Figure 3
The new hyperchaotic memristive circuit.The flux-controlled memristor could be expressed as(3)it=Wφtvt,Wφ=dqφdφ.And basic model of the memristor can be found in [22]. The definition of memristor was extended in [23],
(4)yt=gz,u,tut,dzdt=fu,z,t,where ut and yt are the input signal and output signal of the memristor, respectively. z is the system state variable. The input of the magnetically controlled memristor is voltage u, the output is the current i following through the memristor, and flux ϕ is the state variable.According to the definition of memristor, we obtain a novel expression as(5)i=Wϕu,dϕdt=u2−u1−cϕ,where Wϕ is called memory conductance, let Wϕ=0.1ϕ2. According to the Kirchhoff’s law and component parameter constraints, circuit dynamic equation could be obtained as
(6)C1dv1dt=WϕV2−V1+gV1,C2dv2dt=WϕV1−V2+Ii,LdIidt=−V2,RCdϕdt=V2−V1−cϕ,where τ0=RC is the time constant. Dimensionless time τ=t/τ0 could be obtained by the scaling time and state space. The dimensional state variable x=V1, y=V1, z=Ri, w=ϕ, a=C/C1, C=C2, b=cR2/L, fw=Wϕ/R, and gx=gV1/R, so we can obtain
(7)dxdt=afwy−x+gx,dydt=fwx−y+z,dzdt=−by,dwdt=y−x−cw,where fw=dw2, gx=0.9x, and a, b, c, and d are system parameters. The simulation results are given in Figure 4 by taking initial values x0=1, y0=z0=w0=0, and parameters a=10, b=60, c=10, and d=0.1. Then, based on (7), the N-scroll is realized in y direction by using the step function.Figure 4
Projections of the double-scroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)There are two forms of step function, when(8)hy=A∑i=1Msgn[y+2i−1A+∑i=1Msgny−2i−1A,M≥1,2M+1 scrolls could be generated in y direction.When(9)hy=A−sgny+∑i=0N−1sgn[y+2iA+∑i=0N−1sgny−2iA,N≥0,2N scrolls could be generated in y direction.N-scroll could be obtained by a reasonable set of step function parameters. For example, whenM=1, A=12,
(10)hy=12sgny−12+sgny+12.The 2 × 3 multiscroll hyperchaotic system based on a memristor device is obtained as(11)dxdt=afwy−x−hy+gx,dydt=fwx−y+hy+z,dzdt=−by−hy,dwdt=y−x−hy−cw.When system parameters area=10, b=60, c=10, and d=0.1 with initial values x0=3 and y0=z0=w0=0, simulation results are presented in Figure 5.Figure 5
Projections of the 2 × 3 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)Similarly, 2 × 5 and 2 × 6 multiscroll hyperchaotic system could be also obtained by takingM=2, A=12, and N=3, A=12, and
(12)hy=12sgny+12+sgny−12+sgny+36+sgny−36,hy=12sgny+sgny−24+sgny+24+sgny−48+sgny+48,respectively.When selecting system parametersa=10, b=60, c=10, and d=0.1 with initial values x0=3 and y0=z0=w0=0, the numerical simulations of 2 × 5 and 2 × 6 multiscroll hyperchaotic system are shown in Figures 6 and 7.Figure 6
Projections of the 2 × 5 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)Figure 7
Projections of the 2 × 6 multiscroll hyperchaotic attractors onto they,w and x,y plane. (a) y,w plane. (b) x,y plane.
(a)
(b)


## 2.2. Dynamic Analysis

### 2.2.1. The Double-Scroll Memristive System Is Taken as an Example to Analyse the Dynamic Properties

(1) Lyapunov Exponents. When selecting system parameters a=10, b=60, c=10, and d=0.1 with initial values x0=1 and y0=z0=w0=0, we can get the Lyapunov exponents diagram as shown in Figure 8.Figure 8
Lyapunov exponents diagram.From Figure8, we can see that the chaotic system has two positive Lyapunov exponents. So, the system is a hyperchaotic system and has two scrolls.(2) Bifurcation Diagram. Lyapunov exponents and bifurcation diagram of the system due to the variation of parameter b is displayed in Figure 9. We could see that when 35<b<43 and 54<b<70, the system shows chaotic behavior.Figure 9
Lyapunov exponents and bifurcation diagram with respect to parameterb. (a) Lyapunov exponents diagram. (b) Bifurcation diagram.
(a)
(b)(3) Dissipation. Generation of chaotic behavior is decided by whether the system has a dissipative structure or not [24]. The dissipative formula of the system is
(13)∇V=∂ẋ∂x+∂ẏ∂y+∂ż∂z+∂ẇ∂w=0.9a−c<0,so the system is dissipative, converging at an exponential rate e0.9a−c until it becomes 0. When t→∞, every volume element which contains trajectories of the system shrinks to zero with the rate of exponential convergent.(4) Equilibrium Points and Stability. The Jacobian matrix of the system is
(14)9−w2w202wy−x0.1w2−0.1w210.2wx−y0−6000−110−10.The equilibrium points of the system could be calculated by making (7) as 0. We could get three equilibrium points as
(15)S0=0,0,0,0,S1=30,0,−27,−3S2=−30,0,27,3.The Jacobian matrix at pointS0 becomes
(16)900000100−6000−110−10.Eigenvalues at pointS0 are λ1=−10, λ2=9, λ3=7.746i, and λ4=−7.746i. The Jacobian matrix at point S1 and S2 becomes
(17)0901800.9−0.91−180−6000−110−10.The EquilibriaS1 and S2 could have eigenvalues λ1=−5.9483+13.6659i, λ2=−5.9483−13.6659i, λ3=0.4983+6.9549i, and λ4=0.4983−6.9549i, which are called saddle points of index 2 since the two complex conjugate eigenvalues have positive real parts[25, 26]. It is clear that S0 is the first type saddle point since the real eigenvalue is positive [25, 26]. It is noticed that the scrolls are generated only around the equilibria of saddle points of index 2 [25, 26]. Moreover, equilibria S1 and S2 correspond to the two saturated plateaus, which are responsible for generating the two scrolls in the double-scroll attractor. However, the equilibrium point S0 corresponds to the saturated slope and is responsible for connecting these two symmetrical scrolls.


### 2.2.2. The Dynamic Analysis of 2 × 3 Multiscroll Hyperchaotic System

The dynamic analysis of 2 × 3 multiscroll hyperchaotic system is introduced briefly as follows. Let(18)dxdt=dydt=dzdt=dwdt=0.Equation of equilibrium points could be obtained as
(19)afwy−x−hy+gx=0,fwx−y+hy+z=0,−by−hy=0,y−x−hy−cw=0.From (19), we could get the coordinate of system equilibrium points in the y-axis as y=hy, that is, y=0, ±12, ±24, x=±30. The figures of equilibrium points and chaotic attractors of the multiscroll hyperchaotic system are demonstrated in Figure 10. Therefore, the number of equilibrium points in the y-axis direction is 5 × 2. The solid points represent the second type saddle point, which correspond to the six saturated plateaus and are responsible for generating the six scrolls in the 2 × 3-scroll attractors. The hollow points represent the first type saddle point, which correspond to the saturated slope and are responsible for connecting these six symmetrical scrolls.Figure 10
Equilibrium points and attractors of the 2 × 3 multiscroll hyperchaotic system. (a) Equilibrium points. (b) 2 × 3 multiscroll attractors.
(a)
(b)


## 2.2.1. The Double-Scroll Memristive System Is Taken as an Example to Analyse the Dynamic Properties

(1) Lyapunov Exponents. When selecting system parameters a=10, b=60, c=10, and d=0.1 with initial values x0=1 and y0=z0=w0=0, we can get the Lyapunov exponents diagram as shown in Figure 8.Figure 8
Lyapunov exponents diagram.From Figure8, we can see that the chaotic system has two positive Lyapunov exponents. So, the system is a hyperchaotic system and has two scrolls.(2) Bifurcation Diagram. Lyapunov exponents and bifurcation diagram of the system due to the variation of parameter b is displayed in Figure 9. We could see that when 35<b<43 and 54<b<70, the system shows chaotic behavior.Figure 9
Lyapunov exponents and bifurcation diagram with respect to parameterb. (a) Lyapunov exponents diagram. (b) Bifurcation diagram.
(a)
(b)(3) Dissipation. Generation of chaotic behavior is decided by whether the system has a dissipative structure or not [24]. The dissipative formula of the system is
(13)∇V=∂ẋ∂x+∂ẏ∂y+∂ż∂z+∂ẇ∂w=0.9a−c<0,so the system is dissipative, converging at an exponential rate e0.9a−c until it becomes 0. When t→∞, every volume element which contains trajectories of the system shrinks to zero with the rate of exponential convergent.(4) Equilibrium Points and Stability. The Jacobian matrix of the system is
(14)9−w2w202wy−x0.1w2−0.1w210.2wx−y0−6000−110−10.The equilibrium points of the system could be calculated by making (7) as 0. We could get three equilibrium points as
(15)S0=0,0,0,0,S1=30,0,−27,−3S2=−30,0,27,3.The Jacobian matrix at pointS0 becomes
(16)900000100−6000−110−10.Eigenvalues at pointS0 are λ1=−10, λ2=9, λ3=7.746i, and λ4=−7.746i. The Jacobian matrix at point S1 and S2 becomes
(17)0901800.9−0.91−180−6000−110−10.The EquilibriaS1 and S2 could have eigenvalues λ1=−5.9483+13.6659i, λ2=−5.9483−13.6659i, λ3=0.4983+6.9549i, and λ4=0.4983−6.9549i, which are called saddle points of index 2 since the two complex conjugate eigenvalues have positive real parts[25, 26]. It is clear that S0 is the first type saddle point since the real eigenvalue is positive [25, 26]. It is noticed that the scrolls are generated only around the equilibria of saddle points of index 2 [25, 26]. Moreover, equilibria S1 and S2 correspond to the two saturated plateaus, which are responsible for generating the two scrolls in the double-scroll attractor. However, the equilibrium point S0 corresponds to the saturated slope and is responsible for connecting these two symmetrical scrolls.


## 2.2.2. The Dynamic Analysis of 2 × 3 Multiscroll Hyperchaotic System

The dynamic analysis of 2 × 3 multiscroll hyperchaotic system is introduced briefly as follows. Let(18)dxdt=dydt=dzdt=dwdt=0.Equation of equilibrium points could be obtained as
(19)afwy−x−hy+gx=0,fwx−y+hy+z=0,−by−hy=0,y−x−hy−cw=0.From (19), we could get the coordinate of system equilibrium points in the y-axis as y=hy, that is, y=0, ±12, ±24, x=±30. The figures of equilibrium points and chaotic attractors of the multiscroll hyperchaotic system are demonstrated in Figure 10. Therefore, the number of equilibrium points in the y-axis direction is 5 × 2. The solid points represent the second type saddle point, which correspond to the six saturated plateaus and are responsible for generating the six scrolls in the 2 × 3-scroll attractors. The hollow points represent the first type saddle point, which correspond to the saturated slope and are responsible for connecting these six symmetrical scrolls.Figure 10
Equilibrium points and attractors of the 2 × 3 multiscroll hyperchaotic system. (a) Equilibrium points. (b) 2 × 3 multiscroll attractors.
(a)
(b)


## 3. The Circuit Implementation of the Switching Scroll Hyperchaotic System

### 3.1. The Circuit Implementation of the Double-Scroll Hyperchaotic System

Circuit diagram of the double-scroll hyperchaotic system is displayed in Figure3. With respect to the memristor shown in Figure 11(a), its internal circuit structure could be obtained in [27], shown in Figure 11(b). Circuit implementation results are shown in Figure 12, which are in agreement with numerical simulation results in Figure 4.Figure 11
The memristor and its internal circuit. (a) The memristor. (b) Its internal circuit.
(a)
(b)Figure 12
Circuit implementation of the double-scroll hyperchaotic system. (a)y,w plane. (b) x,y plane.
(a)
(b)Symbolic function circuit is designed by the saturated output voltage of the operational amplifier to achieve step function. The saturated voltage of the operational amplifier in Figure13 is E.
(20)Vout=−R3R2×R5R4×E×sgnVin+V5+sgnVin−V8.Figure 13
Step function circuit.The step function circuit designed by the method is easy to expand and achieve, that is, more complex function could be realized by increasing the number of corresponding comparison circuit. In the following part, we choose the 2 × 3 scroll system as an example to discuss the multiscroll system. Based on Figure13, step function is connected between the capacitor C1 and the ground to get the circuit diagram of the multiscroll hyperchaotic system shown in Figure 14, and its circuit simulation results are displayed in Figure 15 which are in accordance with the numerical simulation results.Figure 14
Circuit diagram of the multiscroll hyperchaotic system.Figure 15
Circuit implementation of the multiscroll hyperchaotic system. (a)y,w plane. (b) x,y plane.
(a)
(b)


## 3.1. The Circuit Implementation of the Double-Scroll Hyperchaotic System

Circuit diagram of the double-scroll hyperchaotic system is displayed in Figure3. With respect to the memristor shown in Figure 11(a), its internal circuit structure could be obtained in [27], shown in Figure 11(b). Circuit implementation results are shown in Figure 12, which are in agreement with numerical simulation results in Figure 4.Figure 11
The memristor and its internal circuit. (a) The memristor. (b) Its internal circuit.
(a)
(b)Figure 12
Circuit implementation of the double-scroll hyperchaotic system. (a)y,w plane. (b) x,y plane.
(a)
(b)Symbolic function circuit is designed by the saturated output voltage of the operational amplifier to achieve step function. The saturated voltage of the operational amplifier in Figure13 is E.
(20)Vout=−R3R2×R5R4×E×sgnVin+V5+sgnVin−V8.Figure 13
Step function circuit.The step function circuit designed by the method is easy to expand and achieve, that is, more complex function could be realized by increasing the number of corresponding comparison circuit. In the following part, we choose the 2 × 3 scroll system as an example to discuss the multiscroll system. Based on Figure13, step function is connected between the capacitor C1 and the ground to get the circuit diagram of the multiscroll hyperchaotic system shown in Figure 14, and its circuit simulation results are displayed in Figure 15 which are in accordance with the numerical simulation results.Figure 14
Circuit diagram of the multiscroll hyperchaotic system.Figure 15
Circuit implementation of the multiscroll hyperchaotic system. (a)y,w plane. (b) x,y plane.
(a)
(b)


## 4. Applications to Secure Communication

### 4.1. Drive-Response Synchronization of the Switching Systems

The synchronization of the switching systems is briefly stated as follows. The drive system from (11) is
(21)ẋ1=afw1y1−x1−hy1+gx1,ẏ1=fw1x1−y1+hy1+z1,ż1=−by1−hy1,ẇ1=y1−x1−hy1−cw1,where fw1=dw12, gx1=0.9x1, a, b, c, and d are system parameters, a=10, b=60, c=10, and d=0.1, and initial values x10=1 and y10=z10=w10=0.And the response system is(22)ẋ2=afw2y2−x2−hy2+gx2+u1,ẏ2=fw2x2−y2+hy2+z2+u2,ż2=−by2−hy2+u3,ẇ2=y2−x2−hy2−cw2+u4,where fw2=dw22, gx2=0.9x2, a, b, c, and d are system parameters, a=10, b=60, c=10, and d=0.1, and initial values x20=1 and y20=z20=w20=0.The state error is defined as(23)e=P−εQ,where ε is diagonal matrix, P=x1,y1,z1,w1T, and Q=x2,y2,z2,w2T. ε=diagε1,ε2,ε3,ε4 is scaling function matrix (if ε=diag1,1,1,1, ε=diag−1,−1,−1,−1, ε=diagα,α,α,α (α is a constant), and ε=diagα1,α2,α3,α4, respectively, then the synchronization is called as complete synchronization, antisynchronization, projective synchronization, and modified projective synchronization). In this paper, we choose ε=diag−1,−1,−1,−1, that is to say, the synchronization type is antisynchronization.The synchronization controller of the response system is designed as(24)u1=−w22y2−x2−hy2−w12y1−x1−hy1−9x2−9x1−μ1e1,u2=−0.1w22x2−y2+hy2−z2−0.1w12x1−y1+hy1−z1−μ2e2,u3=60y2−hy2+60y1−hy1−μ3e3,u4=−y2+x2+hy2+10w4−y1+x1+hy1+10w1−μ4e4.According to the systems (21), (22), (23), and controller (24), we could obtain the final expression of the error system as
(25)ė1=−μ1e1,ė2=−μ2e2,ė3=−μ3e3,ė4=−μ4e4.Letμ1=2, μ2=9, μ3=7, and μ4=4, it is obvious that all roots of the error system have negative real parts. When t→∞, the error system (25) converge to 0 and therefore the synchronization between the drive system and the response system is realized.Next, we illustrate the validity of the proposed controller by MATLAB 2010a. Selecting initial valuesx10=3, y10=0, z10=0, w10=0, x20=3, y20=1, z20=2, and w20=2, we could get those of the error system as e10=6, e20=1, e30=2, and e40=2. Keeping the system parameters, the initial values, the scale factors, and synchronization controllers unchanged, simulation results are illustrated in Figure 16. One can see that the error dynamic system is stable asymptotically by using the designed controllers, which implies that the drive system and the response system could achieve synchronization well.Figure 16
The state trajectories of the error system. (a) State trajectories ofe1. (b) State trajectories of e2. (c) State trajectories of e3. (d) State trajectories of e4.
(a)
(b)
(c)
(d)


### 4.2. Drive-Response Circuit Synchronization of the Switching Systems

Drive-response synchronous circuit of the multiscroll hyperchaotic system is demonstrated in Figure17.Figure 17
Drive-response synchronous circuit of the multiscroll hyperchaotic system.Thex1−T and y1−T waveforms of the drive system and the x2−T and y2−T waveforms of the response system are, respectively, shown in Figures 18(a) and 18(b). The x1−x2 waveform at the time of synchronization is shown in Figure 19. From the simulation results, we could see that voltage signal waveforms of the drive and response system are exactly the same, and the synchronous phase diagram of the corresponding state is a straight line through the origin point with 45 degrees, indicating that the corresponding state variables in the two circuits achieve a good synchronization. When it is applied to the secure communication, the synchronization method is simple and the circuit implementation is convenient.Figure 18
Thex1−T and y1−T waveforms and the x2−T and y2−T waveforms.
(a)
(b)Figure 19
Thex1−x2 waveform at the time of synchronization.The drive-response synchronous circuit of the double-scroll hyperchaotic system is similar to that of the multiscroll one and will be omitted here.


### 4.3. Application to Secure Communication

In the circuit implementation, the double-scroll and multiscroll circuits are encapsulated, respectively, so as to simplify the circuit structure.


#### 4.3.1. Description of Encryption and Decryption Algorithm

In this subsection, the synchronization circuit above is applied to secure communications. A voice signalmt=0.50.1sint+1 carrying the message to be transmitted could be masked by the chaotic sequence x1t, which is a key sequence from the double-scroll and multiscroll hyperchaotic system. And
(26)x1=x1d,2nπ≤t≤2nπ+π2,x1m,2nπ+3π2≤t<2n+1π,n=0,1,2,….The chaotic synchronous circuit given above could be applied to extract the message at the receiver. Some strategies could be used to make the actual transmitted signalst as broadband as possible, that is, to make its detection through spectral techniques difficult. In general, three strategies are proposed in chaotic secure communications [28]. One is signal masking, where st=x1t+αmt; the second is modulation, st=x1t+mt; the third is a combination of masking and modulation, st=x1t+αmt. Here, chaotic masking is used to encryption. The transmitted signal is st=x1t+αmt and injected into the transmitter and, simultaneously, transmitted to the receiver. By the above synchronous circuit, a chaotic receiver is then derived to recover the voice signal at the receiving end, that is,
(27)mt=1αlimt→∞x2t+st=1αlimt→∞x2t+x1t+αmt=mt.The switching encryption circuit is shown in Figure20. System 1 and system 2 represent the double-scroll and multiscroll hyperchaotic circuit, respectively. When the voice signal is a rising waveform, switch S1 connects to k1, and switch S2 connects to k3. At this time, x1d is used as the key sequence in the encryption, that is to say, the double-scroll hyperchaotic system works. When the voice signal is a falling waveform, switch S1 connects to k2, and switch S2 connects to k4. At this point, x1m is used to encrypt the voice signal, and the multiscroll hyperchaotic system does. With the voice waveform changes, the connection mode of the switch S1 and switch S2 is different. Finally, the switching between the double-scroll and the multiscroll hyperchaotic system is realized. In the decryption process, the chaotic signal generated by the response system is used to decrypt the signal. The schematic diagram is shown in Figure 21. The key sequence generated by the drive system is used to cover the voice signal and that generated by the response system works in removing the cover of the encrypted signal.Figure 20
The switching encryption circuit.Figure 21
Encryption and decryption flow chart.Results of encryption and decryption are displayed in Figure22. We can know that the voice signal is completely covered by the chaotic signal, and the original shape could not be seen at all after the voice signal and the chaotic signal generated by the driving system is superimposed. The cracker could not get any information of the voice signal from the channel [29-30]. It has proved that the switching encryption method has pretty good privacy.Figure 22
Encrypted and decrypted waveforms. (a) Original voice signal. (b) Encrypted voice signal. (c) Decrypted voice signal.
(a)
(b)
(c)Moreover, the decrypted voice signal and the original voice signal are identical, indicating that the original signal could be recovered well by the chaotic signal generated by the response system. This proves the effectiveness and reliability of the encryption method.


#### 4.3.2. Security Analyses

(1) Key Space Analysis. The size of key space is the total number of different keys used in the encryption process. The key space should be large enough to resist attacks. In the encryption scheme, thirteen key parameters are used, which are x0, y0, z0, and w0 from the double-scroll hyperchaotic system, ẋ0, ẏ0, ż0, and ẇ0 from the multiscroll hyperchaotic system, scaling function matrix ε=diag−1,1,−1,−1, synchronization controller u, and system parameters a, b, c, and d. If the precision is chosen as 10−14, the total key space is (1014)14, which is obviously larger than that in [31].(2) Key Sensitivity Analysis. In order to test the sensitivity of the encryption algorithm, key parameter x0=1 is changed to x0=1+10−4 and others unchanged in the decryption, the result is shown in Figure 23. We could see that the encryption scheme in the paper is so sensitive to key parameters that a small change can lead to a completely different result.Figure 23
Wrong decryption result.(3) Encryption Speed. For a good encryption algorithm, running speed is an important reference aspect. MATLAB 2014a is used to run the program that realizes the proposed algorithm in a personal computer with a Pentium 4 CPU 3.0 GHz, 4.0 GB RAM, 500 GB hard disk, and Microsoft Windows 7 operating system. The encryption algorithm in the paper is compared with that in [32] and [33]; results between different algorithms are shown in Table 1. We could see that the speed of our encryption algorithm is much faster.Table 1
Comparison of encryption speed between different algorithms.
Algorithms
Encryption speed (MB/s)
Algorithm of [32]
4.80
Algorithm of [33]
6.21
The proposed algorithm
7.91


## 4.1. Drive-Response Synchronization of the Switching Systems

The synchronization of the switching systems is briefly stated as follows. The drive system from (11) is
(21)ẋ1=afw1y1−x1−hy1+gx1,ẏ1=fw1x1−y1+hy1+z1,ż1=−by1−hy1,ẇ1=y1−x1−hy1−cw1,where fw1=dw12, gx1=0.9x1, a, b, c, and d are system parameters, a=10, b=60, c=10, and d=0.1, and initial values x10=1 and y10=z10=w10=0.And the response system is(22)ẋ2=afw2y2−x2−hy2+gx2+u1,ẏ2=fw2x2−y2+hy2+z2+u2,ż2=−by2−hy2+u3,ẇ2=y2−x2−hy2−cw2+u4,where fw2=dw22, gx2=0.9x2, a, b, c, and d are system parameters, a=10, b=60, c=10, and d=0.1, and initial values x20=1 and y20=z20=w20=0.The state error is defined as(23)e=P−εQ,where ε is diagonal matrix, P=x1,y1,z1,w1T, and Q=x2,y2,z2,w2T. ε=diagε1,ε2,ε3,ε4 is scaling function matrix (if ε=diag1,1,1,1, ε=diag−1,−1,−1,−1, ε=diagα,α,α,α (α is a constant), and ε=diagα1,α2,α3,α4, respectively, then the synchronization is called as complete synchronization, antisynchronization, projective synchronization, and modified projective synchronization). In this paper, we choose ε=diag−1,−1,−1,−1, that is to say, the synchronization type is antisynchronization.The synchronization controller of the response system is designed as(24)u1=−w22y2−x2−hy2−w12y1−x1−hy1−9x2−9x1−μ1e1,u2=−0.1w22x2−y2+hy2−z2−0.1w12x1−y1+hy1−z1−μ2e2,u3=60y2−hy2+60y1−hy1−μ3e3,u4=−y2+x2+hy2+10w4−y1+x1+hy1+10w1−μ4e4.According to the systems (21), (22), (23), and controller (24), we could obtain the final expression of the error system as
(25)ė1=−μ1e1,ė2=−μ2e2,ė3=−μ3e3,ė4=−μ4e4.Letμ1=2, μ2=9, μ3=7, and μ4=4, it is obvious that all roots of the error system have negative real parts. When t→∞, the error system (25) converge to 0 and therefore the synchronization between the drive system and the response system is realized.Next, we illustrate the validity of the proposed controller by MATLAB 2010a. Selecting initial valuesx10=3, y10=0, z10=0, w10=0, x20=3, y20=1, z20=2, and w20=2, we could get those of the error system as e10=6, e20=1, e30=2, and e40=2. Keeping the system parameters, the initial values, the scale factors, and synchronization controllers unchanged, simulation results are illustrated in Figure 16. One can see that the error dynamic system is stable asymptotically by using the designed controllers, which implies that the drive system and the response system could achieve synchronization well.Figure 16
The state trajectories of the error system. (a) State trajectories ofe1. (b) State trajectories of e2. (c) State trajectories of e3. (d) State trajectories of e4.
(a)
(b)
(c)
(d)


## 4.2. Drive-Response Circuit Synchronization of the Switching Systems

Drive-response synchronous circuit of the multiscroll hyperchaotic system is demonstrated in Figure17.Figure 17
Drive-response synchronous circuit of the multiscroll hyperchaotic system.Thex1−T and y1−T waveforms of the drive system and the x2−T and y2−T waveforms of the response system are, respectively, shown in Figures 18(a) and 18(b). The x1−x2 waveform at the time of synchronization is shown in Figure 19. From the simulation results, we could see that voltage signal waveforms of the drive and response system are exactly the same, and the synchronous phase diagram of the corresponding state is a straight line through the origin point with 45 degrees, indicating that the corresponding state variables in the two circuits achieve a good synchronization. When it is applied to the secure communication, the synchronization method is simple and the circuit implementation is convenient.Figure 18
Thex1−T and y1−T waveforms and the x2−T and y2−T waveforms.
(a)
(b)Figure 19
Thex1−x2 waveform at the time of synchronization.The drive-response synchronous circuit of the double-scroll hyperchaotic system is similar to that of the multiscroll one and will be omitted here.


## 4.3. Application to Secure Communication

In the circuit implementation, the double-scroll and multiscroll circuits are encapsulated, respectively, so as to simplify the circuit structure.


### 4.3.1. Description of Encryption and Decryption Algorithm

In this subsection, the synchronization circuit above is applied to secure communications. A voice signalmt=0.50.1sint+1 carrying the message to be transmitted could be masked by the chaotic sequence x1t, which is a key sequence from the double-scroll and multiscroll hyperchaotic system. And
(26)x1=x1d,2nπ≤t≤2nπ+π2,x1m,2nπ+3π2≤t<2n+1π,n=0,1,2,….The chaotic synchronous circuit given above could be applied to extract the message at the receiver. Some strategies could be used to make the actual transmitted signalst as broadband as possible, that is, to make its detection through spectral techniques difficult. In general, three strategies are proposed in chaotic secure communications [28]. One is signal masking, where st=x1t+αmt; the second is modulation, st=x1t+mt; the third is a combination of masking and modulation, st=x1t+αmt. Here, chaotic masking is used to encryption. The transmitted signal is st=x1t+αmt and injected into the transmitter and, simultaneously, transmitted to the receiver. By the above synchronous circuit, a chaotic receiver is then derived to recover the voice signal at the receiving end, that is,
(27)mt=1αlimt→∞x2t+st=1αlimt→∞x2t+x1t+αmt=mt.The switching encryption circuit is shown in Figure20. System 1 and system 2 represent the double-scroll and multiscroll hyperchaotic circuit, respectively. When the voice signal is a rising waveform, switch S1 connects to k1, and switch S2 connects to k3. At this time, x1d is used as the key sequence in the encryption, that is to say, the double-scroll hyperchaotic system works. When the voice signal is a falling waveform, switch S1 connects to k2, and switch S2 connects to k4. At this point, x1m is used to encrypt the voice signal, and the multiscroll hyperchaotic system does. With the voice waveform changes, the connection mode of the switch S1 and switch S2 is different. Finally, the switching between the double-scroll and the multiscroll hyperchaotic system is realized. In the decryption process, the chaotic signal generated by the response system is used to decrypt the signal. The schematic diagram is shown in Figure 21. The key sequence generated by the drive system is used to cover the voice signal and that generated by the response system works in removing the cover of the encrypted signal.Figure 20
The switching encryption circuit.Figure 21
Encryption and decryption flow chart.Results of encryption and decryption are displayed in Figure22. We can know that the voice signal is completely covered by the chaotic signal, and the original shape could not be seen at all after the voice signal and the chaotic signal generated by the driving system is superimposed. The cracker could not get any information of the voice signal from the channel [29-30]. It has proved that the switching encryption method has pretty good privacy.Figure 22
Encrypted and decrypted waveforms. (a) Original voice signal. (b) Encrypted voice signal. (c) Decrypted voice signal.
(a)
(b)
(c)Moreover, the decrypted voice signal and the original voice signal are identical, indicating that the original signal could be recovered well by the chaotic signal generated by the response system. This proves the effectiveness and reliability of the encryption method.


### 4.3.2. Security Analyses

(1) Key Space Analysis. The size of key space is the total number of different keys used in the encryption process. The key space should be large enough to resist attacks. In the encryption scheme, thirteen key parameters are used, which are x0, y0, z0, and w0 from the double-scroll hyperchaotic system, ẋ0, ẏ0, ż0, and ẇ0 from the multiscroll hyperchaotic system, scaling function matrix ε=diag−1,1,−1,−1, synchronization controller u, and system parameters a, b, c, and d. If the precision is chosen as 10−14, the total key space is (1014)14, which is obviously larger than that in [31].(2) Key Sensitivity Analysis. In order to test the sensitivity of the encryption algorithm, key parameter x0=1 is changed to x0=1+10−4 and others unchanged in the decryption, the result is shown in Figure 23. We could see that the encryption scheme in the paper is so sensitive to key parameters that a small change can lead to a completely different result.Figure 23
Wrong decryption result.(3) Encryption Speed. For a good encryption algorithm, running speed is an important reference aspect. MATLAB 2014a is used to run the program that realizes the proposed algorithm in a personal computer with a Pentium 4 CPU 3.0 GHz, 4.0 GB RAM, 500 GB hard disk, and Microsoft Windows 7 operating system. The encryption algorithm in the paper is compared with that in [32] and [33]; results between different algorithms are shown in Table 1. We could see that the speed of our encryption algorithm is much faster.Table 1
Comparison of encryption speed between different algorithms.
Algorithms
Encryption speed (MB/s)
Algorithm of [32]
4.80
Algorithm of [33]
6.21
The proposed algorithm
7.91


## 4.3.1. Description of Encryption and Decryption Algorithm

In this subsection, the synchronization circuit above is applied to secure communications. A voice signalmt=0.50.1sint+1 carrying the message to be transmitted could be masked by the chaotic sequence x1t, which is a key sequence from the double-scroll and multiscroll hyperchaotic system. And
(26)x1=x1d,2nπ≤t≤2nπ+π2,x1m,2nπ+3π2≤t<2n+1π,n=0,1,2,….The chaotic synchronous circuit given above could be applied to extract the message at the receiver. Some strategies could be used to make the actual transmitted signalst as broadband as possible, that is, to make its detection through spectral techniques difficult. In general, three strategies are proposed in chaotic secure communications [28]. One is signal masking, where st=x1t+αmt; the second is modulation, st=x1t+mt; the third is a combination of masking and modulation, st=x1t+αmt. Here, chaotic masking is used to encryption. The transmitted signal is st=x1t+αmt and injected into the transmitter and, simultaneously, transmitted to the receiver. By the above synchronous circuit, a chaotic receiver is then derived to recover the voice signal at the receiving end, that is,
(27)mt=1αlimt→∞x2t+st=1αlimt→∞x2t+x1t+αmt=mt.The switching encryption circuit is shown in Figure20. System 1 and system 2 represent the double-scroll and multiscroll hyperchaotic circuit, respectively. When the voice signal is a rising waveform, switch S1 connects to k1, and switch S2 connects to k3. At this time, x1d is used as the key sequence in the encryption, that is to say, the double-scroll hyperchaotic system works. When the voice signal is a falling waveform, switch S1 connects to k2, and switch S2 connects to k4. At this point, x1m is used to encrypt the voice signal, and the multiscroll hyperchaotic system does. With the voice waveform changes, the connection mode of the switch S1 and switch S2 is different. Finally, the switching between the double-scroll and the multiscroll hyperchaotic system is realized. In the decryption process, the chaotic signal generated by the response system is used to decrypt the signal. The schematic diagram is shown in Figure 21. The key sequence generated by the drive system is used to cover the voice signal and that generated by the response system works in removing the cover of the encrypted signal.Figure 20
The switching encryption circuit.Figure 21
Encryption and decryption flow chart.Results of encryption and decryption are displayed in Figure22. We can know that the voice signal is completely covered by the chaotic signal, and the original shape could not be seen at all after the voice signal and the chaotic signal generated by the driving system is superimposed. The cracker could not get any information of the voice signal from the channel [29-30]. It has proved that the switching encryption method has pretty good privacy.Figure 22
Encrypted and decrypted waveforms. (a) Original voice signal. (b) Encrypted voice signal. (c) Decrypted voice signal.
(a)
(b)
(c)Moreover, the decrypted voice signal and the original voice signal are identical, indicating that the original signal could be recovered well by the chaotic signal generated by the response system. This proves the effectiveness and reliability of the encryption method.


## 4.3.2. Security Analyses

(1) Key Space Analysis. The size of key space is the total number of different keys used in the encryption process. The key space should be large enough to resist attacks. In the encryption scheme, thirteen key parameters are used, which are x0, y0, z0, and w0 from the double-scroll hyperchaotic system, ẋ0, ẏ0, ż0, and ẇ0 from the multiscroll hyperchaotic system, scaling function matrix ε=diag−1,1,−1,−1, synchronization controller u, and system parameters a, b, c, and d. If the precision is chosen as 10−14, the total key space is (1014)14, which is obviously larger than that in [31].(2) Key Sensitivity Analysis. In order to test the sensitivity of the encryption algorithm, key parameter x0=1 is changed to x0=1+10−4 and others unchanged in the decryption, the result is shown in Figure 23. We could see that the encryption scheme in the paper is so sensitive to key parameters that a small change can lead to a completely different result.Figure 23
Wrong decryption result.(3) Encryption Speed. For a good encryption algorithm, running speed is an important reference aspect. MATLAB 2014a is used to run the program that realizes the proposed algorithm in a personal computer with a Pentium 4 CPU 3.0 GHz, 4.0 GB RAM, 500 GB hard disk, and Microsoft Windows 7 operating system. The encryption algorithm in the paper is compared with that in [32] and [33]; results between different algorithms are shown in Table 1. We could see that the speed of our encryption algorithm is much faster.Table 1
Comparison of encryption speed between different algorithms.
Algorithms
Encryption speed (MB/s)
Algorithm of [32]
4.80
Algorithm of [33]
6.21
The proposed algorithm
7.91


## 5. Conclusions

In this paper, a new switching scroll hyperchaotic system was proposed, which consisted of the double-scroll and multiscroll hyperchaotic system. Based on a memristor device, the double-scroll hyperchaotic system was taken as an example to analyse the dynamic properties, such as Lyapunov exponents, bifurcation diagram, dissipation and equilibrium points, and stability. Furthermore, circuit implementation of the system was introduced in detail and the results are consistent with those of numerical simulations. At last, the switching scroll hyperchaotic system was successfully applied to voice encryption and showed its effectiveness.



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*Source: 3497640-2018-07-02.xml*