PDF《Rssler systems with uncertain》

Chin.

Phys. B Vol. 20, No.

8 (2011)

080507 Complete synchronization of double-delayed R¨ ossler systems with uncertain parameters? Sang Jin-Yu( )a) , Yang Ji( )b) , and Yue Li-Juan( )a)? a)School of Physics, Northeast Normal University, Changchun 130024, China b)Department of Basic Course, Aviation University of Airforce, Changchun 130022, China (Received

25 January 2011;

revised manuscript received

16 March 2011) In this paper, we investigate complete synchronization of double-delayed R¨ ossler systems with uncertain parameters as the master system is in chaotic synchronization. The uncertain parameters can be nonlinearly expressed in the system. The analysis and proof are given by means of the Lyapunov stability theorem. Based on theoretical analysis, some su?cient conditions of complete synchronization are proved. In order to validate the proposed scheme, numerical simulations are performed and the numerical results show that our scheme is very e?ective. Keywords: complete synchronization, double-delayed R¨ ossler system, uncertain parameters PACS: 05.45.Xt, 05.45.Pq DOI: 10.1088/1674-1056/20/8/080507 1. Introduction In recent years, the synchronization of chaotic sys- tems has become an area of active research, partly due to its potential application in secure communication. Since the synchronization of chaotic dynamical sys- tems was observed by Pecora and Carroll[1,2] in 1990, di?erent methods of chaotic synchronization have been proposed in interacting chaotic systems both theoretically and experimentally, such as the drive- response method, the variable feedback method,[3] the adaptive method,[4,5] the pulse method,[6,7] and the linear observer based method.[8,9] However, most of the research has focused on chaotic systems with cer- tain parameters. In fact, we can hardly ever obtain parameters of chaotic dynamical systems in communi- cation, so the study on the synchronization of chaotic systems with uncertain parameters has some practi- cal value.[10,11] Recently, chaotic time-delayed systems have been suggested as good candidates for secure communication and they have been paid more atten- tion because the dimensions of their chaotic dynam- ics can be raised by increasing the delay time su?- ciently. Otherwise, chaotic time-delayed systems can be considered as a special cases of spatiotemporal sys- tems. It extends complete synchronization to in?nite- dimensional systems.[12?16] In the present paper, we consider the complete synchronization in double-delayed R¨ ossler systems with uncertain parameters. The rest of this paper is organized as follows. In Section 2, we describe a double-delayed R¨ ossler system. In Section 3, we theoretically analyse the complete synchronization of double-delayed R¨ ossler systems with uncertain param- eters on the basis of the Lyapunov stability theory. In Section 4, numerical simulations are presented to con- ?rm the validity of the proposed theoretical approach in Section 3. Finally, conclusions are drawn in Section 5. 2. Description of the system A double-delayed R¨ ossler system is written as[17] ? ? ? ? ? ? ? B x1 = ?x2 ? x3 + ax1(t ? τ1) + bx1(t ? τ2), B x2 = x1 + ax2, B x3 = 0.2 + x1x3 ? cx3. (1) In Eq. (1), x1, x2, and x3 are state variables;

a, b, and c are the related parameters;

τ1 and τ2 are delay parameters. The system is chaotic over a wide param- eter range. The chaotic attractor for a = 0.2, b = 0.5, c = 5.7, and τ1 = 1, τ2 =

2 is given in Fig. 1. ?Project supported by the National Natural Science Foundation of China (Grant No. 10847110). ?Corresponding author. E-mail: [email protected] c