Function Synchronization of the Fractional-Order Chaotic System

Abstract:

Function synchronization is an important type of chaos synchronization because of enhancing the security of communication. In order to obtain the better conformances of function synchronization, a method of the fractional-order chaotic system is presented, which based on the stability theory of the fractional order system. This method need construct a parameter matrix and a coupled matrix using fractional-order chaotic drive system at first, and then the chaotic response system is set up with these matrixes. The synchronization error function between drive system and response system is satisfied with the asymptotic stability. Function synchronization of the fractional-order Rikitake chaotic system is selected as a typical example. Numerical simulation results demonstrate the validity of the presented method. This method not only has better synchronization conformances, but also can be applied in the chaotic secure communications.

• Periodical: Advanced Materials Research (Volumes 631-632)
• Pages: 1220-1225
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.631-632.1220
• Online since: January 2013
• Authors: J.W. Fan, N. Zhao, Y. Gao, H.L. Lan
• Keywords: Chaos Synchronization, Fractional-Order Chaotic System, Rikitake System

References

[1] X. J. Wu, H. Wang and H. T. Lu, Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication , Nonlinear Analysis: Real World Applications 13, (2012)1441.

DOI: 10.1016/j.nonrwa.2011.11.008

[2] I. Grigorenko and E. Grigorenko , Chaotic dynamics of the fractional Lorenz system, Physics Review Letters, 91(2003) 3411.

DOI: 10.1103/physrevlett.91.034101

[3] C. G. Li and G. R. Chen, Chaos in the fractional order Chen system and its control, Chaos, Solitons & Fractals, 22(2004) 549.

DOI: 10.1016/j.chaos.2004.02.035

[4] S. T. Mohammad and H. Mohammad, Synchronization of chaotic fractional-order systems via active silding mode controller, Physica A , 387 (2008)57.

[5] X. J. Wu, J. Li and G. R. Chen, Chaos in the fractional order unified system and its synchronization, Journal of the Franklin Institute, 345 (2008) 392.

DOI: 10.1016/j.jfranklin.2007.11.003

[6] L. P. Chen, Y. Chai and R. C. Wu, Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems, Physics Letters A , 375(2011) (2099).

DOI: 10.1016/j.physleta.2011.04.015

[7] C. Yin, S. M. Zhong and W. F. Chen, Design of sliding mode controller for a class of fractional-order chaotic systems, Commun Nonlinear Sci Numer Simulat 17( 2012)356.

DOI: 10.1016/j.cnsns.2012.04.019

[8] I. Podlubny , Fractional Differential Equations, New York, (1999).

[9] K. Diethelm, Ford N J and Freed A D , A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29(2002) 3.

[10] D. Matignon , Stability results for fractional differential equations with applications to control processing IEEE-SMC Proc. Computational Engineering in Systems and Application Multi-Conference (IMACS vol 2), Lille, France, 1996, p.963–968.

[11] X. J. Wu and H. Wang, A new chaotic system with fractional order and its projective synchronization, Nonlinear dynamics 61( 2010)407-417.

DOI: 10.1007/s11071-010-9658-x