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Chaos synchronization for a class of nonlinear oscillators with fractional order. (English) Zbl 1187.34066

Summary: The chaos synchronization problem of the fractional-order Qi oscillators coupled in a master-slave pattern is examined by applying three different kinds of methods: the nonlinear feedback method, the one-way coupling method and the method based on the state observer. Suitable synchronization conditions are derived by using the Lyapunov stability theory, and most importantly, a sufficient and necessary synchronization condition is presented. Results of numerical simulations validate the effectiveness and applicability of the proposed schemes.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
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[1] Butzer, P. L.; Westphal, U., An Introduction to Fractional Calculus (2000), World Scientific: World Scientific Singapore · Zbl 0987.26005
[2] Kenneth, S. M.; Bertram, R., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley-Interscience Publication: Wiley-Interscience Publication US · Zbl 0789.26002
[3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[4] Pldlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York
[5] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst. I, 42, 485-490 (1995)
[6] Song, L.; Xu, S. Y.; Yang, J. Y., Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simul., 15, 616-628 (2010) · Zbl 1221.93234
[7] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)
[8] Torvik, P. J.; Bagley, R. L., On the appearance of the fractional derivative in the behavior of real materials, Trans. ASME, 51, 294-298 (1984) · Zbl 1203.74022
[9] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. Circuits Syst. I, 43, 485-490 (1995)
[10] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91, 034101 (2003)
[11] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals, 22, 3, 549-554 (2004) · Zbl 1069.37025
[12] Lu, J. G., Chaotic dynamics of the fractional order Lü system and its synchronization, Phys. Lett. A, 354, 4, 305-311 (2006)
[13] Li, C.; Chen, G., Chaos and hyperchaos in the fractional order Rössler equations, Physica A, 341, 55-61 (2004)
[14] Chua, L. O.; Itah, M., Chaos synchronization in Chua’s circuits, J. Circuits Syst. Comput., 3, 93-108 (1993)
[15] Chua, L. O.; Yang, T.; Zhong, G. Q., Adaptive synchronization of Chua’s oscillators, Int. J. Bifur. Chaos, 6, 189-201 (1996)
[16] Chen, G. R.; Dong, X., From Chaos to Order (1998), World Scientific: World Scientific Singapore
[17] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[18] Wu, X. J.; Li, J.; Chen, G. R., Chaos in the fractional order unified system and its synchronization, J. Franklin Inst., 345, 392-401 (2008) · Zbl 1166.34030
[19] Peng, G. J., Synchronization of the fractional order chaotic systems, Phys. Lett. A, 363, 426-432 (2007) · Zbl 1197.37040
[20] Qi, G. Y.; Chen, G. R.; Du, S. Z.; Chen, Z. Q., Analysis of a new chaotic system, Physica A, 352, 295-308 (2005)
[21] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 3-22 (2002) · Zbl 1009.65049
[22] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer. Algorithms, 36, 31-52 (2004) · Zbl 1055.65098
[23] Li, C.; Peng, G., Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals, 22, 443-450 (2004) · Zbl 1060.37026
[24] Chen, J. H.; Chen, W. C., Chaotic dynamics of the fractionally damped van der Pol equation, Chaos Solitons Fractals, 35, 188-198 (2008)
[25] Sheu, L. J.; Chen, H. K.; Chen, J. H.; Tam, L. M., Chaos in a new system with fractional order, Chaos Solitons Fractals, 31, 1203-1212 (2007)
[26] Matignon, D., Stability results for fractional differential equations with applications to control processing, (Computational Engineering in Systems Applications. Computational Engineering in Systems Applications, IMACS, IEEE-SMC 2 (1996), Lille: Lille France), 963-968
[27] Anderson, B. D.O.; Bose, N. K.; Jury, E. I., A simple test for zeros of a complex polynomial in a sector, IEEE Trans. Automat. Control, 19, 437-438 (A1974)
[28] Davison, E. J.; Ramesh, N., A note on the eigenvalues of a real matrix, IEEE Trans. Automat. Control, 15, April, 252-253 (1970)
[29] Friedland, B., Advanced Control System Design (1995), Englewood Cliffs, Prentice-Hall: Englewood Cliffs, Prentice-Hall NJ
[30] Isidori, A., Nonlinear Control Systems (1995), Springer: Springer New York · Zbl 0569.93034
[31] Ahmed, E.; El-Sayed, A. M.A.; El-Saka, H. A.A., Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models, J. Math. Anal. Appl., 325, 1, 542-553 (2007) · Zbl 1105.65122
[32] Duan, Z. S.; Chen, G. R., Global robust stability and synchronization of networks with lorenz-type nodes, IEEE Trans. Circuits Syst. II, 56, 679-683 (2009)
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