Drive-Response Synchronization of a Fractional-Order Hyperchaotic System and Its Circuit Implementation

. A novel fractional-order hyperchaotic system is proposed; the theoretical analysis and numerical simulation of this system are studied. Based on the stability theory of fractional calculus, we propose a novel drive-response synchronization scheme. In order to achieve this synchronization control, the Adams-Bashforth-Moulton algorithm is studied. And then, a drive-response synchronization controller is designed to realize the synchronization of the drive and response system, and the simulation results are given. At last, the fractional oscillator circuit of the new fractional-order hyperchaotic system is designed based on the EWB software, and it is verified that the simulation results of the fractional-order oscillator circuit are consistent with the numerical simulation results through circuit simulation.


Introduction
Fractional-order calculus is a mathematical theory which studies the characteristic and application of the arbitraryorder differential and integral operator.It has the same history as the integer-order calculus and can be dated back to the 17th century [1,2].It has attracted more researchers' interest and has more broad application prospects due to its unique advantages.But until the last 20 years, the fractional-order calculus theory has been applied to the practical projects; it has been applied to the chaos system, electromagnetism, signal processing, mechanical engineering, robot control, and so on.
Since hyperchaos was firstly reported by Rossler [3], there have been considerable achievements in the study of hyperchaos.Hyperchaotic systems have more than one positive Lyapunov exponent, and the strange attractor is usually unstable in more than one direction.Hyperchaotic system is a high-dimensional chaotic system.The fractionalorder differential operator is introduced into hyperchaotic system which can reflect the hyperchaotic system with complicated nonlinear dynamic characteristics.Fractionalorder hyperchaotic system implementation and application have been attracted more researchers' interest and in-depth study.In recent years, the fractional-order chaotic dynamical systems began to attract more researchers' attention, such as the fractional-order Chua system, the fractional-order Lorenz system, Chen's system, and Liu's system and so on [4][5][6][7][8][9].
In this paper, a novel fractional-order hyperchaotic system is presented.Based on the stability theory of fractional calculus, a novel drive-response synchronization scheme is proposed.Section 4 studied the Adams-Bashforth-Moulton algorithm to achieve this synchronization scheme.The fractional oscillator circuit of the new hyperchaotic system is designed based on the EWB software in Section 5, and the simulation results are given to demonstrate that the fractional-order oscillator is a hyperchaotic system.Finally, conclusions end this paper.
By calculating, this system has two positive Lyapunov exponents: LE 1 = 0.7980 and LE 1 = 0.1841, and the negative Lyapunov exponents are LE 3 = −0.9656and LE 4 = −2.5164.According to chaos theory, there are two positive Lyapunov exponents implying that its dynamics is expanded in more than one direction simultaneously and that the system is a hyperchaotic system.
Based on the Lyapunov exponents, we can calculate the Hausdorff dimension (Lyapunov dimension) of the nonlinear autonomous system.
In this paper, the following simulations are all performed by using  = 0.9, (, , ) = (2.5, 4, 4).By simulations, we have obtained the 2D phase portraits of the fractional-order system as shown in Figure 1 and the time domain waveform of the () as shown in Figure 2.These figures clearly show that the fractional-order hyperchaotic system exhibits chaotic behaviors.

The Theory of the Fractional Drive-Response Synchronization
In this section, the drive-response method is used to realize synchronization control of the fractional-order system, assuming that the fractional order chaotic system is studied in the form of the driving equation where 0 <  < 1,  ∈   is the column vector,  ∈  × ,  ∈  × ,  ∈  1× , and  ∈  ×1 are the continuous matrixes, and  ∈ ,  :  1 →   is the nonlinear function.
It is worth noting that we must select the appropriate driving signal to drive the response system; generally we select the linear term which can be written as follows: The fractional-order response system is defined where ỹ = x, x ∈   are the column vectors and  = [ 1 ,  2 , . . .,   ]  are the synchronization control parameters.
The error system is defined as () = () − x().By ( 3), (4), and ( 5), the error equation of the two fractional order chaotic systems can be obtained as follows: Obviously, the system (6) and the following system (7) have the same stability: where () = −  ê() is the feedback control signal; it determines the synchronization effect of the fractional-order system.Based on the stability theory of the fractional-order system, we propose a novel synchronization theorem [10,11].
Theorem 1.If the fractional order linear system (7) is asymptotically stable, the necessary and sufficient condition is that all the eigenvalues   of the coefficient matrix (  −     ) are satisfied: where arg(  ) is the explement of the eigenvalues   .
Proof.If the coefficient matrix  = (  −     ) of the system (7) has  different eigenvalues   ( = 1, . . ., ), there exists a nonsingular transformation matrix, which makes the system (7) convert into where () = ê(), Λ =  −1 = diag{ 1 ,  2 , . . .,   },  max is the maximum singular value of ; and  min is the minimum singular value of , hence The analytical solution of the system (9) can be expressed as the function  , () as follows: If ‖()‖ → 0 is established, it is necessary that  ,1 (    ) → 0, and this is equivalent to that all the eigenvalues  ncj are satisfied: Then, the proof is completed.

Circuit Implementation of the Fractional-Order Hyperchaotic System
In the circuit design, we use linear resistor, capacitor, linear operational amplifier (LM741), analog multiplier (AD633), and the fractional unit circuit structure in the fractionalorder hyperchaotic circuits.The linear operational amplifier is used for addition and subtraction, and the analog multiplier is used to realize the nonlinear term.Then, we designed the circuit of fractional-order system (6) with  = 0.9, as shown in Figure 4.

Conclusions
In this paper, we proposed a new fractional-order hyperchaotic system and analyzed the basic properties of this new system.Based on the stability theory of fractional calculus, we proposed a novel drive-response synchronization scheme and studied the Adams-Bashforth-Moulton algorithm.The drive-response synchronization controller is designed to realize synchronization based on the discrete form of the two systems by MATLAB software.And based on the fractionalorder unit circuit, we designed the fractional-order oscillator circuit of this system by EWB software.The simulation results of this circuit are consistent with the numerical simulation results.This fractional-order hyperchaotic circuit can be used for the other electronic oscillator and the controller can be used for application in the chaos control due to its simple construction.

Figure 2 :
Figure 2: The time domain waveform of the ().

Figure 4 :
Figure 4: The circuit realization of the fractional-order system.

Figure 6 :
Figure 6: The circuit experimental phase diagram of the hyperchaotic system.