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.
Similar content being viewed by others
References
Podlubny, I. Fractional Differential Equations, Academic Press, New York (1999)
Kilbas, A. A., Sarivastava, H. M., and Trujillo, J. J. Theory and Applications of Fractional Differential Equations, Elsevier, New York (2006)
Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010)
Sheu, L. J., Chen, H. K., Chen, J. H., and Tam, L. M. Chaotic dynamics of the fractionally damped Duffing equation. Chaos, Solitons and Fractals, 32(4), 1459–1468 (2007)
Ge, Z. M. and Ou, C. Y. Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos, Solitons and Fractals, 35(2), 705–717 (2008)
Yu, Y. G., Li, H. X., Wang, S., and Yu, J. Z. Dyanmic analysis of a fractional-order Lorenz chaotic system. Chaos, Solitons and Fractals, 42(2), 1181–1189 (2009)
Wang, Z. H. and Hu, H. Y. Stability of a linear oscillator with damping force of the fractionalorder derivative. Science in China Series G: Physics, Mechanics and Astronomy, 53(2), 345–352 (2010)
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)
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)
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
Chai, Y., Lü, L., and Zhao, H. Y. Lag synchronization between discrete chaotic systems with diverse structure. Applied Mathematics and Mechanics (English Edition), 31(6), 733–738 (2010) DOI 10.1007/s10483-010-1307-7
Liu, Y. and Lü, L. Synchronization of N different coupled chaotic systems with ring and chain connections. Applied Mathematics and Mechanics (English Edition), 29(10), 1299–1308 (2008) DOI 10.1007/s10483-008-1005-y
Luo, A. C. J. and Min, F. H. Synchronization dynamics of two different dynamical systems. Chaos, Solitons and Fractals, 44(6), 362–380 (2011)
Luo, A. J. L. A theory for synchronization of dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 14(5), 1901–1951 (2009)
Habib, D. and Antonio, L. Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication. IEEE Transactions on Circuits Systems, 58(4), 800–812 (2011)
Olga, I. M., Alexey, A. K., and Alexander, E. H. Generalized synchronization of chaos for secure communication: remarkable stability to noise. Physics Letters A, 374(29), 2925–2931 (2010)
Wang, X. Y., He, Y. J., and Wang, M. J. Chaos control of a fractional order modified coupled dynamos system. Nonlinear Analysis, 71(12), 6126–6134 (2009)
Sachin, B. and Varsha, D. G. Synchronization of different fractional order chaotic systems using active control. Communications in Nonlinear Science and Numerical Simulation, 15(11), 3536–3546 (2010)
Wu, X. J. and Lu, Y. Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dynamics, 57(1–2), 25–35 (2009)
Matouk, A. E. Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol-Duffing circuit. Communications in Nonlinear Science and Numerical Simulation, 16(2), 975–986 (2011)
Abel, A. and Schwarz, W. Chaos communications — principles, schemes, and system analysis. Proceedings of the IEEE, 90(5), 691–710 (2002)
Hu, N. Q. and Wen, X. S. The application of Duffing oscillator in characteristic signal detection of early fault. Journal of Sound and Vibration, 268(5), 917–931 (2003)
Nadakuditi, R. R. and Silverstein, J. W. Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples. IEEE Transactions on Industrial Electronics, 4(3), 468–480 (2010)
Zhao, Z., Wang, F. L., Jia, M. X., and Wang, S. Intermittent-chaos-and-cepstrum-analysis-based early fault detection on shuttle valve of hydraulic tube tester. IEEE Transactions on Industrial Electronics, 56(7), 2764–2770 (2009)
Diethelm, K. and Ford, N. J. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2), 229–248 (2002)
Li, C. P. and Zhang, F. R. A survey on the stability of fractional differential equations. European Physical Journal Special Topics, 193(1), 27–47 (2011)
Sabattier, J., Moze, M., and Farges, C. LMI stability conditions for fractional order system. Computers and Mathematics with Applications, 59(5), 1594–1609 (2010)
Thavazoei, M. S. and Haeri, M. A note on the stability of fractional order system. Mathematics and Computers in Simulation, 79(5), 1566–1576 (2009)
Diethelm, K. and Ford, N. J. Multi-order fractional differential equations and their numerical solution. Applied Mathematics and Computation, 154(3), 621–640 (2004)
Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Y. Algorithms for the fractional calculus: a selection of numerical method. Computer Methods in Applied Mechanics and Engineering, 194(6–8), 743–773 (2005) 583–592 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
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)
Rights and permissions
About this article
Cite this article
He, Gt., Luo, Mk. Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control. Appl. Math. Mech.-Engl. Ed. 33, 567–582 (2012). https://doi.org/10.1007/s10483-012-1571-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-012-1571-6