×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (2000), 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 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 Yverdon · Zbl 0818.26003
[4] Pldlubny, I., Fractional differential equations, (1999), 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 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, (), 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 NJ
[30] Isidori, A., Nonlinear control systems, (1995), 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.