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1、 - 1 - 中国中国科技论文在线科技论文在线 Circuit simulation and implementation for synchronization between fractional-order modified Liu chaotic systems# MA Tiedong, GUO Dong* (College of Automation, Chongqing University, Chongqing 400044) 5 Foundations: Specialized Research Fund for the Doctoral Program of Higher E
2、ducation(Grant: 20100191120025) Brief author introduction:MA Tiedong(1978-), male, associate professor, major research direction： Abstract: For a novel modified fractional-order Liu chaotic system, this paper studies the circuit simulation for this system and its synchronization. Based on the approx
3、imation theory of fractional-order operator, an electronic circuit is designed to describe the dynamic behavior of the modified fractional-order system firstly, and then the synchronization between two identical fractional-order Liu systems is designed by given circuit. Our results are supported by
4、numerical 10 simulation and circuit implementation. Key words: Chaos; Fractional-order Liu system; Synchronization; Circuit simulation 0 Introduction Recently, the study of fractional differential equations has attracted lots of attention in the past few decades. In fractional-order version of many
5、famous chaotic system, chaotic attractor are 15 found, such as Lorenz system 1, Chua system 2, Chen system 3, L system 4, Liu system 5, and so on. It has useful application in many field of science like engineering, physic, mathematical biology, psychological and life sciences. Chaos synchronization
6、, another important topic in nonlinear science, has extensive applications in vast areas of physics and engineering science. Therefore, the synchronization of 20 fractional-order chaotic systems starts to attract increasing attention due to its potential applications in secure communication and cont
7、rol processing 6-8. During the past two decades, a variety of approaches which include OGY method 9, linear and nonlinear feedback control 10, active control 11,12, adaptive control 13 and sliding mode control 14 have been proposed for synchronization of chaotic systems. Among these synchronization
8、methods, the active control 25 approach is easy to implement and has developed to be an effective one. In this paper, we apply the active control to obtain the generalized synchronization of the fractional-order Liu chaotic systems. In addition, the circuit implementation can verify the chaotic char
9、acteristics of the chaotic systems physically, provide support for the application of chaos, and promote their technological 30 application in the future. So the circuit implementation of the chaotic systems has also attracted more and more attention for engineering applications. Especially for thos
10、e fractional-order attractors, the circuit implementations for them are more important 15-19. This paper firstly does numerical simulation and circuit simulation to study a modified Liu chaotic attractor reported by Alireza K et al. 20. Then design an active controller to achieve 35 generalized sync
11、hronization of two fractional-order systems. Finally, an electronic circuit is designed to realize the synchronization of fractional-order systems. The experimental circuit results are obtained in accordance with numerical simulation. 1 Fractional-order modified Liu system and circuit implement Frac
12、tional calculus plays an important role in modern science. There are many definitions of a 40 - 2 - 中国中国科技论文在线科技论文在线 fractional-order differential system 21. The commonly used definitions for the general fractional calculus are the Riemann-Liouville (RL) definition 22 and Caputo definition 23. The R
13、L definition is described by: 1 0 ( )1( ) ()() n t nn d f tdf d dtndtt + = , (1) where 1nn = (4) 70 Let 0.98=and the chaotic trajectories of the modified fractional-order Liu system are shown in Fig. 1. -2.5-2-1.5-1-0.500.5 -4 -2 0 2 4 x1 x2 -4-2024 -2 -1 0 1 2 3 x2 x3 (a) Projection on 1 x- 2 x pla
14、ne (b) Projection on 2 x- 3 x plane Fig. 1 The chaotic trajectories of fractional-order system (3) 75 1.2 Circuit design In this paper, an electronic circuit model of chain shape unit is designed as shown in Fig.2 to realize the approximations of 1/ s with 0.1 0.9= and 0.98=. We take 0.98= for examp
15、le to show the design process of circuit unit for the transfer function. We can obtain an transfer function approximation of 0.98 1/ s with an error of 1 dB in Ref.2 as follow: 80 0.98 1.2974(1125) 1/ (1423)(0.01125) s s ss + = + (5) The corresponding circuit unit is displayed in Fig.3 to implement
16、Eq.(5) in the Laplace domain. Fig. 2 The chain ship unit of 1/ s 85 Fig. 3 The chain ship unit of 0.98 1/ s Using the circuitry theory in the Laplace domain, we can obtain the transfer function ( )H s between A and B in Fig.3 as follows: - 4 - 中国中国科技论文在线科技论文在线 12 1122 00 12 1212 01122 ( ) 11 11 ()() / () 1 , (1/)(1/) RR H s sRCsR C CC sCC CCRR CsRCsR C =+ + + = + (
