# Chaos in a System with an Absolute Nonlinearity and Chaos Synchronization

Authors: Victor Kamdoum Tamba; Karthikeyan Rajagopal; Viet-Thanh Pham; Duy Vo Hoang
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

A system with an absolute nonlinearity is studied in this work. It is noted that the system is chaotic and has an adjustable amplitude variable, which is suitable for practical uses. Circuit design of such a system has been realized without any multiplier and experimental measurements have been reported. In addition, an adaptive control has been applied to get the synchronization of the system.


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## Body

## 1. Introduction

Although chaos in dynamic systems has been investigated for many years [1–4], new systems with chaos still attract the attention of numerous researches [5–11]. Finding new chaotic systems and investigating chaos control and chaos synchronization methodologies are attractive topics [12–15] due to the applications of chaos in various areas such as waveforms of chaotic radar [16], image encryption [17], secure image transmission [18], video encryption design [19], and S-box construction [20].Previous studies suggest that absolute function is effective to design chaotic systems [21, 22]. It is worth noting that an absolute term is not a quadratic nonlinearity and can be implemented with diodes and operational amplifiers [22]. By using an absolute term, one of the most elementary chaotic systems was introduced by Linz and Sprott [21]. Such a system was also realized by a circuit [22]. Jerk systems with absolute nonlinearities were presented in [23]. Authors investigated the synchronization of a chaotic system, which includes only four terms and an absolute-value nonlinearity [24]. In addition, absolute-value term was explored to propose a hyperchaotic circuit without any multiplier [25]. Huang and Liu introduced a fractional-order chaotic system with the presence of an absolute term [26]. Bao et al. designed a memristor-based system with four line equilibria by implementing three absolute terms [27]. It is interesting that adjustable amplitude of chaotic attractor was obtained with absolute terms [28].The aim of this work is to study a simple system with chaos. There is only one nonlinear term, an absolute nonlinearity, in such system. It is noted that the system exhibits variable chaotic attractors, which have been rarely investigated in Sprott’s systems with absolute-value nonlinearity and six terms. Dynamics, circuit, and synchronization of such a system with an absolute nonlinearity are presented in the next sections.


## 2. The System with an Absolute Term and Its Dynamics

Absolute function has been applied to construct different systems with chaotic behavior [27, 28]. In this work, by using an absolute nonlinearity, we study a six-term system described by(1)x˙=ay+bz,y˙=1+z,z˙=-cx-z.System (1) has three positive parameters (a,b,c>0). We have found that system (1) displays different behavior when varying the parameter a.We have changed the parametera for plotting the bifurcation diagram and the maximum Lyapunov exponents (presented in Figures 1 and 2). As shown in Figures 1 and 2, system (1) is periodic for a<6.67. From Figures 1 and 2, we also observe a period doubling route to chaos, which is illustrated further in Figure 3. For a>6.67, chaotic dynamics can be seen. For a=6.8, b=4, and c=1, chaos in system (1) is presented in Figure 4. Chaos in this case is verified by the Lyapunov exponents of the system L1=0.1046>0, L2=0, and L3=-1.1048.Figure 1
Presentation of the bifurcation diagram fora∈5.5,7, while b=4 and c=1.Figure 2
Presentation of maximum Lyapunov exponents fora∈5.5,7, while b=4 and c=1.Figure 3
Different periodical dynamics observed inx-z plane for b=4 and c=1; initial conditions (x(0),y(0),z(0))=(0,0,0) when (a) a=5.75, (b) a=6.25, (c) a=6.625, and (d) a=6.665.
(a)
(b)
(c)
(d)Figure 4
Phase portrait observed in (a)x-y plane, (b) x-z plane, (c) y-z plane, and (d) x-y-z space for a=6.8, b=4, and c=1 and initial conditions (x(0),y(0),z(0))=(0,0,0).
(a)
(b)
(c)
(d)Interestingly, we can change the amplitude of the variablex easily by adding a control parameter (kx) into system (1):(2)x˙=ay+bz,y˙=1+z,z˙=-cx-z+kx.As shown in Figure 5, chaotic attractors are adjusted by using the control parameter kx. When increasing kx, the average value of the variable x is increased (see Figure 6).Figure 5
Variable chaotic attractor when varying the control parameterkx: black color for kx=0, blue color for kx=-2, and red color for kx=2.Figure 6
Average values of three variables (x,y,z) for kx∈[-3,3].Moreover, the amplitudes of three variables(x,y,z) are changed simultaneously by introducing a control parameter (kxyz) into system (1) as follows:(3)x˙=ay+bz,y˙=kxyz+z,z˙=-cx-z.As illustrated in Figure 7, chaotic attractors are reduced and enlarged when varying the control parameter kxyz. It is worth noting that Sprott has discovered various systems with absolute-value nonlinearity and six terms [3]. However, there are few systems displaying controllable chaotic attractors, which have received significant attention recently [29–31].Figure 7
Controllable chaotic attractors in (a)x-y plane and (b) x-z plane when varying the control parameter kxyz: black color for kxyz=1, blue color for kxyz=2, and red color for kxyz=0.5.
(a)
(b)


## 3. Circuit Design for the System

The numerical approach is vital for investigation of the dynamics of theoretical chaotic models [32–35]. By using this method, the dynamical behaviors of such models can be characterized in terms of their parameters. However, to explore their feasibilities, the electronic circuit implementation of these theoretical models is needed [36–39]. Moreover, the physical realization of theoretical chaotic models is relevant in many engineering applications [40–42]. In this section, we design and implement an electronic circuit to illustrate the feasibility of system (1). The electronic circuit diagram for system (1) is depicted in Figure 8.Figure 8
Electronic circuit design of system (1) (a) and the circuit realization of the absolute value function (b).
(a)
(b)The circuit diagram of Figure8 consists of operational amplifiers associated with resistors and capacitors exploited to implement the basic operations such as integration, addition, and subtraction. The nonlinear term of the model is implemented by absolute-value circuit of Figure 8(b). The bias is provided by a 15 Volts DC symmetry source. By applying Kirchhoff’s laws into the circuit of Figure 8, we obtain the following state equations:(4)dVxdt=VyRaC+VzRbCdVydt=VDCRC+VzRCdVzdt=-VxRcC-VzRC,where Vx, Vy, and Vz are the output voltages of the operational amplifiers OP_1, OP_2, and OP_3, respectively. In order to compare system (4) with theoretical model (1), the following settings of variables and parameters, Vx=x×1V, Vy=y×1V, Vz=z×1V, t=τRC, a=R/Ra, b=R/Rb, and c=R/Rc, are adopted. With the following values of parameters, a=6.8, b=4, and c=1 (for which system (1) displays chaotic behavior), the values of circuit components are selected as follows: C=10nF, R=10kΩ, Ra=1.47kΩ, Rb=2.5kΩ, and Rc=10kΩ.As shown in Figure9, the circuit has been implemented and experimental measurements have been recorded. Details of the real circuit are presented in Figure 10. The experimental phase portraits of the circuit in (Vx,Vy), (Vx,Vz), and (Vy,Vz) planes obtained with an oscilloscope are shown in Figure 11.Figure 9
Implemented circuit was measured by using an oscilloscope.Figure 10
Real circuit implemented by using electronic components.Figure 11
Experimental phase portraits of the real circuit in (a) (Vx,Vy) plane, (b) (Vx,Vz) plane, and (c) (Vy,Vz) plane with Ra=1.47kΩ, Rb=2.5kΩ, and Rc=10kΩ.
(a)
(b)
(c)From Figure11, one can see that the experimental chaotic phase portraits agree with those obtained from the numerical simulations. This means that the proposed electronic circuit emulates well the dynamics of theoretical model (1).


## 4. Synchronization for the System with Unknown Parameters

It is well known that, in practical situations, some or all of the system parameters cannot be exactly known in advance. Also, most parameters values are characterized by uncertainties related to the modeling errors or experimental conditions (temperature, external electric and magnetic fields, etc.) that can destroy or even break the synchronization [43–45]. Therefore, it is essential to consider the synchronization problem of chaotic systems in the presence of unknown system parameters. In this section, we design an adaptive control scheme [43] to synchronize two identical structures of system (1) with unknown parameters.


### 4.1. Design of the Slave System

We will assume that all the state variables and parameters of the master system (1) are accessible to measurements and those of slave system are unknown. Based on the concept of adaptive method, the following theorem is formulated.Theorem 1.
Let system (1) be the master system rewritten in the following form:(5)X˙m=fXm+FXmθ,where(6)Xmxm,ym,zmT,fXm=0,1+zm,-zmT,FXm=ymzm000000-xm,θ=a,b,cT;then the slave system(7)X˙s=fXs+FXsθ^t+ucan synchronize with the master system (5), with the control function u designed as(8)u=-e+fXs-fXm+FXs-FXmθ^tand the update law of the estimations of the unknown parameters determined by(9)θ^˙t=-FTXme,where e=Xs-Xm is the error system and θ^=(a^,b^,c^)T are the estimations of the corresponding parameters of the slave system (7).Proof.
The error dynamical system can be expressed as(10)e˙=X˙s-X˙m=-e+FXmθ^t-θ.Choose the storage Lyapunov function as(11)Ve,θt=12eTe+12θ^t-θTθ^t-θ.Then, the time derivative of V(e,θ(t)) along the trajectory is(12)V˙=12e˙Te+eTe˙+12θ^˙tTθ^t-θ+θ^t-θTθ^˙t=12-eTe+θ^t-θTFTXme-eTe+eTFXmθ^t-θ+12θ^˙tTθ^t-θ+θ^t-θTθ^˙t=-eTe.So V˙ is negative semidefinite, and since V is positive definite, it follows that e∈L∞ and θ∈L∞. Thus e˙∈L∞, and, according to (10), it can be obtained that(13)∫0te2dt=∫0teTedt≤-1l∫0tV˙dt=1lV0-Vt≤1lV0.
SinceV(0)≤∞ and e∈L2, according to Barbalat’s lemma, we have e(t)→0 as t→∞; that is, the error dynamical system (10) will be stabilized at the zero equilibrium asymptotically. Thus, according to the Lyapunov stability theorem, the adaptive synchronization with unknown parameters between the drive system (5) and the response system (7) is achieved under the controller defined in (8) and parameters update law determined by (9). This completes the proof.


### 4.2. Numerical Verifications

For numerical verification, the master system is defined as in (5) with parameters a, b, and c. According to Theorem 1, the slave system is described as follows:(14)x˙s=a^ym+b^zm-xs-xm,y˙s=1+zm-ys-ym,z˙s=-c^xm-zm-zs-zm,where(15)a^˙=-ymxs-xm,b^˙=-zmys-ym,c^˙=xmzs-zm.The numerical computations are obtained using the standard fourth-order Runge-Kutta integration algorithm with a time stepΔt=0.001; initial conditions on parameters are being selected randomly as follows: a^(0)=1.20, b^(0)=0.80, and c^(0)=0.25. The master system’s parameters are chosen as a=6.8, b=4, and c=1 in order to ensure the chaotic behavior. The synchronization errors and the graph of parameters estimations are shown in Figures 12 and 13, respectively.Figure 12
Error dynamics showing transition to synchronized states of the system and its observer. (a)ex=xs-xm, (b) ey=ys-ym, and (c) ez=zs-zm. The initial conditions are chosen randomly as xm(0)=0.01, ym(0)=0.02, and zm(0)=0.03 and xs(0)=0.8, ys(0)=0.4, and zs(0)=0.5, respectively, for the drive and response systems. The master system parameters are a=6.8, b=4, and c=1 to ensure the chaotic behavior.
(a)
(b)
(c)Figure 13
Graphs of estimation of unknown parameters of slave system. (a)a^, (b) b^, and (c) c^. The initial conditions are selected randomly as a^(0)=1.2, b^(0)=0.8, and c^(0)=0.25.
(a)
(b)
(c)Numerical simulations (see Figures12 and 13) show that the adaptive synchronization between master system (5) and slave system (7) with unknown parameters is achieved successfully and the error signals approach asymptotically zero. Obviously, these results may be exploited in engineering applications such as communication, image processing, physics, and mechatronics.


## 4.1. Design of the Slave System

We will assume that all the state variables and parameters of the master system (1) are accessible to measurements and those of slave system are unknown. Based on the concept of adaptive method, the following theorem is formulated.Theorem 1.
Let system (1) be the master system rewritten in the following form:(5)X˙m=fXm+FXmθ,where(6)Xmxm,ym,zmT,fXm=0,1+zm,-zmT,FXm=ymzm000000-xm,θ=a,b,cT;then the slave system(7)X˙s=fXs+FXsθ^t+ucan synchronize with the master system (5), with the control function u designed as(8)u=-e+fXs-fXm+FXs-FXmθ^tand the update law of the estimations of the unknown parameters determined by(9)θ^˙t=-FTXme,where e=Xs-Xm is the error system and θ^=(a^,b^,c^)T are the estimations of the corresponding parameters of the slave system (7).Proof.
The error dynamical system can be expressed as(10)e˙=X˙s-X˙m=-e+FXmθ^t-θ.Choose the storage Lyapunov function as(11)Ve,θt=12eTe+12θ^t-θTθ^t-θ.Then, the time derivative of V(e,θ(t)) along the trajectory is(12)V˙=12e˙Te+eTe˙+12θ^˙tTθ^t-θ+θ^t-θTθ^˙t=12-eTe+θ^t-θTFTXme-eTe+eTFXmθ^t-θ+12θ^˙tTθ^t-θ+θ^t-θTθ^˙t=-eTe.So V˙ is negative semidefinite, and since V is positive definite, it follows that e∈L∞ and θ∈L∞. Thus e˙∈L∞, and, according to (10), it can be obtained that(13)∫0te2dt=∫0teTedt≤-1l∫0tV˙dt=1lV0-Vt≤1lV0.
SinceV(0)≤∞ and e∈L2, according to Barbalat’s lemma, we have e(t)→0 as t→∞; that is, the error dynamical system (10) will be stabilized at the zero equilibrium asymptotically. Thus, according to the Lyapunov stability theorem, the adaptive synchronization with unknown parameters between the drive system (5) and the response system (7) is achieved under the controller defined in (8) and parameters update law determined by (9). This completes the proof.


## 4.2. Numerical Verifications

For numerical verification, the master system is defined as in (5) with parameters a, b, and c. According to Theorem 1, the slave system is described as follows:(14)x˙s=a^ym+b^zm-xs-xm,y˙s=1+zm-ys-ym,z˙s=-c^xm-zm-zs-zm,where(15)a^˙=-ymxs-xm,b^˙=-zmys-ym,c^˙=xmzs-zm.The numerical computations are obtained using the standard fourth-order Runge-Kutta integration algorithm with a time stepΔt=0.001; initial conditions on parameters are being selected randomly as follows: a^(0)=1.20, b^(0)=0.80, and c^(0)=0.25. The master system’s parameters are chosen as a=6.8, b=4, and c=1 in order to ensure the chaotic behavior. The synchronization errors and the graph of parameters estimations are shown in Figures 12 and 13, respectively.Figure 12
Error dynamics showing transition to synchronized states of the system and its observer. (a)ex=xs-xm, (b) ey=ys-ym, and (c) ez=zs-zm. The initial conditions are chosen randomly as xm(0)=0.01, ym(0)=0.02, and zm(0)=0.03 and xs(0)=0.8, ys(0)=0.4, and zs(0)=0.5, respectively, for the drive and response systems. The master system parameters are a=6.8, b=4, and c=1 to ensure the chaotic behavior.
(a)
(b)
(c)Figure 13
Graphs of estimation of unknown parameters of slave system. (a)a^, (b) b^, and (c) c^. The initial conditions are selected randomly as a^(0)=1.2, b^(0)=0.8, and c^(0)=0.25.
(a)
(b)
(c)Numerical simulations (see Figures12 and 13) show that the adaptive synchronization between master system (5) and slave system (7) with unknown parameters is achieved successfully and the error signals approach asymptotically zero. Obviously, these results may be exploited in engineering applications such as communication, image processing, physics, and mechatronics.


## 5. Conclusions

By using an absolute nonlinearity, we have introduced a six-term system with chaos. Dynamics of the system with only one absolute nonlinearity have been investigated. One interesting finding is that thex variable can be adjusted with a control parameter. In addition, it is simple to implement this chaotic system because we do not need any analog multiplier. Adaptive synchronization between such two chaotic systems has been reported and these results should be exploited further for practical applications.



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