Analysis and Circuit Implementation for the Fractional-order Chen System

The paper firstly analyzes the fractional-order Chen system by using function approximation in frequency domain and time domain respectively, gives its Lyapunov exponents and bifurcation diagrams, and shows its chaotic characteristics from numerical analysis point of view. Then, based on fractional-order frequency domain approximation, two different analog circuits are designed to implement the fractional-order Chen system, and the results observed by oscillation are coincided with those obtained by numerical simulation. That is, the chaotic characteristics of the fractional-order Chen system are verified by numerical analysis and analog circuit.


Introduction
In recent decades, many fractional-order systems are found to show chaos, such as the fractional-order Lorenz system [1], the fractional-order Chen system [2], the fractional-order Chua's circuit [3], and so on [4][5].The research on fractional-order chaotic dynamics has begun to attract more and more interest [6][7][8].Because of the difficulty of computing fractional calculus, some approximation methods, such as time-domain approximation method and frequency-domain approximation method, are always adopted to analyze the fractional-order systems when studying them.Generally speaking, the timedomain approximation method is more reliable to show the dynamics of the system [9][10], and the frequency-domain approximation method is more practical to design the circuit of the system from an engineer's viewpoint [11][12].
In this paper, three different function approximation methods are firstly used to investigate dynamics of the fractional-order Chen system, and the corresponding Lyapunov exponents and bifurcation diagrams for the fractionalorder Chen system are also given.Interestingly, the chaotic dynamics are all found when using three different approximation methods to analyze the system, and the characteristics obtained by the three methods are almost same.Then, two analog circuits are designed to implement the fractional-order Chen system based on the two different frequency-domain approximations.All the results in this paper indicate that the chaotic characteristics exit in the fractional-order Chen system.

Analysis of the fractional-order Chen system
The fractional-order Chen system is described by Where a , b , and c is system parameters,  ,  , and  is the fractional order.
Generally speaking, computing a fractional differential equation is a complex and difficult work, therefore, we adopt approximation methods to analyze the fractional-order systems numerically, such as the time-domain approximation methods and the frequency-domain approximation methods.In this paper, the three approximation methods will be used to show the chaotic characteristics of the fractional-order Chen system.

Analysis based on time domain approximation method
When fixing 35 a  , 3 b  ,        , and varying c , based on the timedomain approximation method, advised Adams-Bashforth-Moulton method, the fractional-order Chen system is investigated.And the corresponding Lyapunov exponents and bifurcation diagrams are given, as shown in figure 1(a) and 1(b), respectively.Generally speaking, when the system's biggest Lyapunov exponents is large than zero, and the points in the corresponding bifurcation diagram are dense, the chaotic attractor will be found to exit in this system.Therefore, From the Lyapunov exponents and bifurcation diagrams in figure 1(a) and 1(b), a conclusion can be obtained that chaos exit in the fractionalorder Chen system when selecting a certain range of parameters.Furthermore, when selecting 35 a  , 3 b  , and , c 28 attractor can be observed by using advised Adams-Bashforth-Moulton method, as shown in figure 1(c).b  ,        , and varying c , the fractional- order Chen system is investigated based on the frequency-domain approximation method.The approximation function is And the corresponding Lyapunov exponents and bifurcation diagrams are also given, as shown in figure 2(a) and 2(b), respectively.Based on Lyapunov exponents and bifurcation diagrams in figure 2(a) and 2(b), a conclusion can be obtained that chaos exit in the fractional-order Chen system when selecting a certain range of parameters.Furthermore, when selecting 35 a  , 3 b  ,        , and 28 c  , a chaotic attractor can be observed by using approximation function(2), as shown in figure 2(c).b  ,        , and varying c , based on another the frequency-domain approximation method, the fractional-order Chen system is investigated again.The approximation function is 0.9 and the corresponding Lyapunov exponents and bifurcation diagrams are also given, as shown in figure 3(a) and 3(b), respectively.Based on Lyapunov exponents and bifurcation diagrams in figure 3(a) and 3(b), a conclusion can be obtained that chaos exit in the fractional-order Chen system when selecting a certain range of parameters.Furthermore, when selecting 35 a  , 3 b  ,        , and 28 c  , a chaotic attractor can be observed by using approximation function(3), as shown in figure 3(c).Through analyzing all the above figures, we can observe these Lyapunov exponents, bifurcation diagrams and chaotic attractors are almost same.That is, the chaotic characteristics can be found in the fractional-order Chen system when using the above three different approximation.

Circuit implementation for the fractional-order Chen system
According to the numerical analysis, we can draw the conclusion that the chaotic characteristics indeed exist in the fractional-order Chen system.However, it seems insufficient to show complex dynamics of the fractionalorder Chen system just by using numerical analysis.So, based on the fractionalorder frequency domain approximation functions ( 2) and ( 3), two analog circuits are designed to implement the system.One analog circuit is shown in Fig. 4, in the circuit, we select LF347N and AD633 as the amplifier (the outline gain is 0.1) and the multiplier, respectively.Based on the selected system parameters and approximation function, resistors and capacitors in the circuit are 1 1uF , In addition, the resistors 1 R , 3 R are adjusted according to the system parameters c .The result observed in the oscilloscope is shown in Fig. 5, and the chaotic attractor can be obviously found to be same as the one in Fig. 2(c).Fig. 4 Analog circuit of the fractional-order Chen system Fig. 5 The circuit implementation result in oscilloscope The other circuit is shown in Fig. 6, Resistors and capacitors in the circuit are 1 1uF , In addition, the resistors 1 R , 3 R are also adjusted according to the system parameters 28 c  .The observed result is shown in Fig. (7), and it also shows that the system is chaotic.That is, the results from both the circuits not only prove the chaotic characteristics of the fractional-order Chen system, but also show that they are coincident with numerical analysis.

Conclusions
In this paper, using the frequency-domain approximation method and timedomain approximation method, we investigate fractional-order Chen system, and find its chaotic dynamics within a certain range of parameters.That is, chaos exists in the fractional-order Chen system.Especially, by analyzing all the Lyapunov exponent diagrams, bifurcation diagrams, and chaotic attractors, we can found that the dynamics obtained by the three different approximations are almost same for the fractional-order Chen system.In addition, based on fractional-order frequency domain approximation, two different analog circuits are designed to implement this system, and the results from circuit experiment are well consistent with numerical simulation, physically verifying chaotic characteristics of the fractional-order Chen system.

Fig. 1 2 . 2
Fig.1 Chaotic analysis of the fractional-order Chen system based on the time-domain method

Fig. 2
Chaotic analysis of the fractional-order Chen system based on the approximation function(2) When fixing 35 a  , 3

Fig. 3
Chaotic analysis of the fractional-order Chen system based on the approximation function(3)

Fig. 6
Fig. 6 Analog circuit of the fractional-order Chen system