A Switching Piecewise-Linear Control Approach for Generating of

Abstract—This work presents a method to generate a multi- scroll chaotic attractor by switched fractional system. Based on two piecewise-liear oliear fuctios, sub-models are constructed with the system order less than one. The switched system can demonstrate n-scroll chaotic attractors if its parameters ad switchig rule are suitably chose. By analyzing the lyapunov exponent and the stability of the system, It’s illustrated that chaotic attractors can be obtained with system orders as low as 0.9. Finally, Numerical examples are provided to verify the effectiveness of the proposed scheme.

I. INTRODUCTION RACTIONAL calculus is a mathematical topic with more than 300 years old history, but its application to physics and engineering have been attracted lots of attention only in recent years. It has been found that many systems can be described by fractional differential equations. For example, dielectric-polarization [1], electrode-electrolyte polarization1 [2], electromagnetic waves [3], and finance [4] have been known to display fractional order dynamics. There are many fractional order systems can exhibit chaos with proper parameters, such as the fractional order Duffing system [5], the fractional order Chua system [6] and the fractional order Lorenz system [7]. Recently, exploiting chaotic dynamics in high-tech and industrial engineering applications has attracted more and more interest, in which much attention has been focused on effectively creating chaos. In 1986, Chua found the Chua’s double-scrolls circuit [8-10], by adding breakpoints in the PWL characteristic function of the Chua’s circuit, more and more n-scroll chaotic attractors have been presented. In 2001, Tang et al. proposed a smooth sine-function approach for creating n-scroll chaotic attractor [11,12], with a systematical circuit realization that can physically produce up to as many

Manuscript received MAY 8, 2010.Research is partially supported by National Natural Science Foundation of China (60904050), and China Postdoctoral Science Foundation Funded Project (20090450997).

Mao Wang is with the Space Control and Inertial Technology Research Center, Harbin Institute of TechnologyHarbin 150001, Heilongjiang Province, PR China. E-mail:wangmao0451@sina.com.

Guanghui Sun is with the Space Control and Inertial Technology Research Center, Harbin Institute of TechnologyHarbin 150001, Heilongjiang Province, PR China. E-mail:guanghuisun@hit.edu.cn.

Lilian Huang is with College of Information and Communication Engineering, Harbin Engineering University. Harbin 150001, Heilongjiang Province, PR China. E-mail: lilian_huang@163.com.

as ten scrolls visible on the oscilloscope. Nowadays, generating multi-scroll chaotic attractors is no longer a very difficult task. In this paper, we introduce two piecewise-linear in our fractional sub-models. With the proper parameters and switching rule, the multi-scroll fractional chaotic attractor can present with the system order less than 0.9. The rest of this paper is organized as follows. Section 2 summarizes the basic concepts in fractional calculus and fractional differential equations. The system description and dynamical analysis are presented in Section 3. The numerical simulation and parameter analysis can be found in section 4. Finally, some conclusions are given in Section 5.

II. . BASIC CONCEPTS FOR FRACTIONAL CALCULUS

A. Definition Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator. Some of the existing definitions are described [13]. The Riemann-Liouvill definition is the simplest one, which is shown as follows:

11d()()()d()ntrtnrnfDftdnrttDDWWW󰀅󰀆 *󰀅󰀅³ (1)

where1nrn󰀅󰀊󰀊, ()*x is the Gamma function. It’s

unpractical that the R-L definition requires the knowledge of the non-integer order derivatives of the function at t=0. But the problem does not exist in the Caputo definition of the fractional derivative. The Caputo definition of derivative, which is sometimes called smooth fractional derivative, is described as:

()101(),1d()()()dd(),dnt

n

nn

fdnn

ftnt

tftnt

DD

D

WWD

DW

D󰀅󰀆

­󰀁󰀁󰀁󰀁󰀁󰀁󰀅󰀊󰀊°

*󰀅󰀅°

®

°󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁󰀁 °¯

³ (2)

The Laplace transform of the Caputo fractional derivative is: ^`11()0d()()(0)dnkk

k

ftLsLftsf

t

DDD

D

󰀅󰀅󰀅 ­½

󰀅®¾

¯¿

¦ (3)

where 1nnND󰀅󰀊d󰀗. It is clear that only integer order derivatives of function f appear in the Laplace transform of the Caputo fractional derivative. For zero initial conditions,