Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

doi:10.1007/s10483-012-1571-6

Applied Mathematics and Mechanics (English Edition) ›› 2012, Vol. 33 ›› Issue (5): 567-582.doi: https://doi.org/10.1007/s10483-012-1571-6

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Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

Gui-tian HE, Mao-kang LUO

Department of Mathematics, Sichuan University, Chengdu 610064, P. R. China

Received:2011-10-30 Revised:2012-02-05 Online:2012-05-10 Published:2012-05-10
Contact: Mao-kang LUO, Professor, Ph.D., E-mail: makaluo@scu.edu.cn E-mail:makaluo@scu.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11171238) and the Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (No. IRTO0742)

Abstract

Abstract:

With the increasingly deep studies in physics and technology, the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research. In this paper, the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated. With the stability criterion of linear fractional systems, the synchronization of a fractional non-autonomous system is obtained. Specifically, an effective singly active control is proposed and used to synchronize a fractional order Duffing system. The numerical results demonstrate the effectiveness of the proposed methods.

2010 MSC Number:

Gui-tian HE;Mao-kang LUO. Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control. Applied Mathematics and Mechanics (English Edition), 2012, 33(5): 567-582.

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References

[1] Podlubny, I. Fractional Differential Equations, Academic Press, New York (1999)

[2] Kilbas, A. A., Sarivastava, H. M., and Trujillo, J. J. Theory and Applications of Fractional DifferentialEquations, Elsevier, New York (2006)

[3] Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press,London (2010)

[4] Sheu, L. J., Chen, H. K., Chen, J. H., and Tam, L. M. Chaotic dynamics of the fractionallydamped Duffing equation. Chaos, Solitons and Fractals, 32(4), 1459-1468 (2007)

[5] Ge, Z. M. and Ou, C. Y. Chaos synchronization of fractional order modified Duffing systems withparameters excited by a chaotic signal. Chaos, Solitons and Fractals, 35(2), 705-717 (2008)

[6] Yu, Y. G., Li, H. X., Wang, S., and Yu, J. Z. Dyanmic analysis of a fractional-order Lorenz chaoticsystem. Chaos, Solitons and Fractals, 42(2), 1181-1189 (2009)

[7] Wang, Z. H. and Hu, H. Y. Stability of a linear oscillator with damping force of the fractionalorderderivative. Science in China Series G: Physics, Mechanics and Astronomy, 53(2), 345-352(2010)

[8] Xin, G. and Yu, J. B. Chaos in the fractional order periodically forced complex Duffing's oscillators.Chaos, Solitons and Fractals, 24(4), 1097-1104 (2005)

[9] Chen, J. H. and Chen, W. C. Chaotic dynamics of the fractionally damped van der Pol equation.Chaos, Solitons and Fractals, 35(1), 188-198 (2008)

[10] Ahn, C. K. Generalized passivity-based chaos synchronization. Applied Mathematics and Mechanics(English Edition), 31(8), 1009-1018 (2010) DOI 10.1007/s10483-010-1336-6

[11] Chai, Y., Lü, L., and Zhao, H. Y. Lag synchronization between discrete chaotic systems withdiverse structure. Applied Mathematics and Mechanics (English Edition), 31(6), 733-738 (2010)DOI 10.1007/s10483-010-1307-7

[12] Liu, Y. and Lü, L. Synchronization of N different coupled chaotic systems with ring and chainconnections. Applied Mathematics and Mechanics (English Edition), 29(10), 1299-1308 (2008)DOI 10.1007/s10483-008-1005-y

[13] Luo, A. C. J. and Min, F. H. Synchronization dynamics of two different dynamical systems. Chaos,Solitons and Fractals, 44(6), 362-380 (2011)

[14] Luo, A. J. L. A theory for synchronization of dynamical systems. Communications in NonlinearScience and Numerical Simulation, 14(5), 1901-1951 (2009)

[15] Habib, D. and Antonio, L. Adaptive unknown-input observers-based synchronization of chaoticsystems for telecommunication. IEEE Transactions on Circuits Systems, 58(4), 800-812 (2011)

[16] Olga, I. M., Alexey, A. K., and Alexander, E. H. Generalized synchronization of chaos for securecommunication: remarkable stability to noise. Physics Letters A, 374(29), 2925-2931 (2010)

[17] Wang, X. Y., He, Y. J., and Wang, M. J. Chaos control of a fractional order modified coupleddynamos system. Nonlinear Analysis, 71(12), 6126-6134 (2009)

[18] Sachin, B. and Varsha, D. G. Synchronization of different fractional order chaotic systems usingactive control. Communications in Nonlinear Science and Numerical Simulation, 15(11), 3536-3546 (2010)

[19] Wu, X. J. and Lu, Y. Generalized projective synchronization of the fractional-order Chen hyperchaoticsystem. Nonlinear Dynamics, 57(1-2), 25-35 (2009)

[20] Matouk, A. E. Chaos, feedback control and synchronization of a fractional-order modified autonomousvan der Pol-Duffing circuit. Communications in Nonlinear Science and Numerical Simulation,16(2), 975-986 (2011)

[21] Abel, A. and Schwarz, W. Chaos communications — principles, schemes, and system analysis.Proceedings of the IEEE, 90(5), 691-710 (2002)

[22] Hu, N. Q. and Wen, X. S. The application of Duffing oscillator in characteristic signal detectionof early fault. Journal of Sound and Vibration, 268(5), 917-931 (2003)

[23] Nadakuditi, R. R. and Silverstein, J. W. Fundamental limit of sample generalized eigenvaluebased detection of signals in noise using relatively few signal-bearing and noise-only samples.IEEE Transactions on Industrial Electronics, 4(3), 468-480 (2010)

[24] Zhao, Z., Wang, F. L., Jia, M. X., and Wang, S. Intermittent-chaos-and-cepstrum-analysis-basedearly fault detection on shuttle valve of hydraulic tube tester. IEEE Transactions on IndustrialElectronics, 56(7), 2764-2770 (2009)

[25] Diethelm, K. and Ford, N. J. Analysis of fractional differential equations. Journal of MathematicalAnalysis and Applications, 265(2), 229-248 (2002)

[26] Li, C. P. and Zhang, F. R. A survey on the stability of fractional differential equations. EuropeanPhysical Journal Special Topics, 193(1), 27-47 (2011)

[27] Sabattier, J., Moze, M., and Farges, C. LMI stability conditions for fractional order system.Computers and Mathematics with Applications, 59(5), 1594-1609 (2010)

[28] Thavazoei, M. S. and Haeri, M. A note on the stability of fractional order system. Mathematicsand Computers in Simulation, 79(5), 1566-1576 (2009)

[29] Diethelm, K. and Ford, N. J. Multi-order fractional differential equations and their numericalsolution. Applied Mathematics and Computation, 154(3), 621-640 (2004)

[30] Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Y. Algorithms for the fractional calculus: aselection of numerical method. Computer Methods in Applied Mechanics and Engineering, 194(6-8), 743-773 (2005)

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

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