Chaos in a System with an Absolute Nonlinearity and Chaos Synchronization

1Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Cameroon 2Department of Electrical and Communication Engineering, The PNG University of Technology, Lae, Morobe, Papua New Guinea 3Modeling Evolutionary Algorithms Simulation and Artificial Intelligent, Faculty of Electrical & Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

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.

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 System (1) has three positive parameters (, ,  > 0).We have found that system (1) displays different behavior when varying the parameter .
We have changed the parameter  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  < 6.67.From Figures 1 and 2, we also observe a period doubling route to chaos, which is illustrated further in Figure 3.For  > 6.67, chaotic dynamics can be seen.For  = 6.8,  = 4, and  = 1, chaos in system (1) is presented in Figure 4. Chaos in this case is verified by the Lyapunov exponents of the system  1 = 0.1046 > 0,  2 = 0, and  3 = −1.1048.
Interestingly, we can change the amplitude of the variable  easily by adding a control parameter (  ) into system (1): As shown in Figure 5, chaotic attractors are adjusted by using the control parameter   .When increasing   , the average value of the variable  is increased (see Figure 6).Moreover, the amplitudes of three variables (, , ) are changed simultaneously by introducing a control parameter (  ) into system (1) as follows: (3) As illustrated in Figure 7, chaotic attractors are reduced and enlarged when varying the control parameter   .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][30][31].

Circuit Design for the System
The numerical approach is vital for investigation of the dynamics of theoretical chaotic models [32][33][34][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][37][38][39].Moreover, the physical realization of theoretical chaotic models is relevant in many engineering applications [40][41][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.
The circuit diagram of Figure 8 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: Advances in Mathematical Physics where   ,   , and   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,   = ×1,   = ×1,   =  × 1,  = ,  = /  ,  = /  , and  = /  , are adopted.With the following values of parameters,  = 6.8,  = 4, and  = 1 (for which system (1) displays chaotic behavior), the values of circuit components are selected as follows:  = 10 nF,  = 10 kΩ,   = 1.47 kΩ,   = 2.5 kΩ, and   = 10 kΩ.
As shown in Figure 9, 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 (  ,   ), (  ,   ), and (  ,   ) planes obtained with an oscilloscope are shown in Figure 11.
From Figure 11, 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).

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][44][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.

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.

Advances in Mathematical Physics
Theorem 1.Let system (1) be the master system rewritten in the following form: where then the slave system can synchronize with the master system (5), with the control function  designed as and the update law of the estimations of the unknown parameters determined by where  =   −  is the error system and θ = (â, b, ĉ)  are the estimations of the corresponding parameters of the slave system (7).
Proof.The error dynamical system can be expressed as Choose the storage Lyapunov function as Then, the time derivative of (, ()) along the trajectory is So V is negative semidefinite, and since  is positive definite, it follows that  ∈  ∞ and  ∈  ∞ .Thus ė ∈  ∞ , and, according to (10), it can be obtained that Since (0) ≤ ∞ and  ∈  2 , according to Barbalat's lemma, we have ‖()‖ → 0 as  → ∞; 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.

Numerical Verifications.
For numerical verification, the master system is defined as in (5) with parameters , , and .According to Theorem 1, the slave system is described as follows: The numerical computations are obtained using the standard fourth-order Runge-Kutta integration algorithm with a time step Δ = 0.001; initial conditions on parameters are being selected randomly as follows: â(0) = 1.20, b(0) = 0.80, and ĉ(0) = 0.25.The master system's parameters are chosen as  = 6.8,  = 4, and  = 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.
Numerical simulations (see Figures 12 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.

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 the  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.

Figure 8 :
Figure 8: Electronic circuit design of system (1) (a) and the circuit realization of the absolute value function (b).

Figure 9 :
Figure 9: Implemented circuit was measured by using an oscilloscope.

Figure 10 :Figure 11 :
Figure 10: Real circuit implemented by using electronic components.