Circuit Implementation of a New Fractional-Order Hyperchaotic System

In the paper, some basic dynamic properties of a new fractional-order hyperchaotic system were firstly investigated, such as the phase trajectory, bifurcation diagram, Lyapunov exponent etc. Then the paper designed an analog circuit to implement the fractional-order hyperchaotic system. The results from the circuit were consistent with those from the numerical analysis, and thus the hyperchaotic characteristics of the new fractional-order system are verified physically. It is very important to implement fractional-order hyperchaotic systems with more complicated dynamics for theoretical research and practical application. The new hyperchaotic system will provide a new model for engineering applications


Introduction
In the last century, Rӧssler proposed the first hyperchaotic system [1].Since then, the hyperchaotic system has attracted more and more attention owning to its abundant and complex dynamic characteristics.Extensive and in-depth researches have been carried out, such as the modified hyperchaotic system, circuit implementation of hyperchaotic systems [2][3][4], especially in synchronous control [5][6][7].
In 1995, Hartley studied the fractional-order Chua system in which the chaotic attractor was found [8].Soon chaotic attractors in the fractional-order Lorenz system were also confirmed [9] and then fractional-order systems have been investigated and developed in more and more fields [10][11][12][13].
At present, most researchers focus on the low-dimensional chaotic system, while there are few researches on the higher-dimensional and hyperchaotic system.However, many real systems in nature and soci-ety often are higher-dimensional and hyperchaotic.Therefore, it is very interesting to study fractional-order hyperchaotic systems with more complicated dynamics for theoretical research and practical application.
For the fractional-order differential equation, numerical methods used commonly are the frequency domain approximation or Adams-Bash-Moulton algorithm.In this paper, with the frequency domain approximation, we study some basic dynamical characteristics of a four-dimensional hyperchaotic system, such as the phase trajectory, Lyapunov exponent and bifurcation diagram.And then we design a circuit to implement the four-dimensional fractional-order hyperchaotic system.These will provide technical support for engineering applications.
When the parameters are chosen as: .07,  = 5,  = 0.4, the system is hyperchaotic and the hyperchaotic strange attractor is shown in Fig. 1.

Hyperchaotic System
Basing on the integer-order system mentioned above, we introduce a fractional-order hyperchaotic H-M system: where , , ,  are state variables and , , , , ℎ, ,  are system parameters.
We utilize the Riemann-Liouville definition of fractional-order differentiation: Reference [1] provides an approximation of 1/ 0.9 with an error of about 2 dB as follows: 1  0.9 ≈ 2.2675(+1.292)(+215.4)(+0.01292)(+2.154)(+359.4) .(5) We use the approximation to analyze the system (2) by using the Matlab.Fig. 2   In order to further study the chaotic characteristics of the system (2), we obtain a bifurcation diagram by altering parameter ℎ, which is shown in Fig. 3.It can be seen from the figure that the system discretization is most obvious when ℎ is in the range of 5-12.
In addition, we further consider the Lyapunov exponents of the system (2), which is shown in Fig. 4. The result shows that the biggest two Lyapunov exponents are positive, which satisfies the necessary conditions for the hyperchaotic system.

Circuit Design of the Fractional-Order System
In this section, utilizing the frequency domain approximation, we design the circuit of the system (2) by using the resistance, capacitance, analog operational amplifier LF347BN and multiplier AD633JN.The amplitude of voltage operational amplifier is ±13.5V that is significantly less than the range of the system.Therefore, we make the voltage to be one tenth of the value that is a normal value in the system.We choose  1 =  4 =  This circuit is designed to implement the system (2), and the experimental results obtained by the Multisim are shown in Fig. 6.Obviously, the experimental results are consistent with the results of the numerical analysis, which physically proves the system (2).

Conclusions
In this paper, we proposed a fractional-order hyperchaotic H-M system based on the integer-order H-M system.Some basic dynamical characteristics of the proposed system, such as the phase trajectory, Lyapunov exponent, and bifurcation diagram, were analyzed respectively by using the Matlab.The results show that the system has more complicated dynamic characteristics, which will enrich the research content of the fractional-order hyperchaotic system.What's more, an analog circuit was designed to realize the four-dimensional fractional-order hyperchaotic system, and the experimental results are in agreement with the numerical analysis, which further confirms the existence of the four-dimensional fractional-order hyperchaotic system.The work in this paper may offer a model for engineering applications.

Figure 4
Figure 4 Lyapunov exponents of the system (2) Fig. 5.This circuit is designed to implement the system (2), and the experimental results obtained by the Multisim are shown in Fig.6.Obviously, the experimental results are consistent with the results of the numerical analysis, which physically proves the system (2).

Figure 5
Figure 5 Circuit implementation of the system (2)

Figure 6
Figure 6 Experimental results by using the Multisim.(a) x − y phase plane, (b) x − z phase plane.