Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. (English) Zbl 1220.34060

Summary: In this Letter, a drive-response synchronization method with linear output error feedback is presented for “generalized projective synchronization” of a class of fractional-order chaotic systems via a scalar transmitted signal. This synchronization approach is theoretically and numerically studied. By using stability theory of linear fractional-order systems, the suitable conditions for achieving synchronization are given. Two examples are used to illustrate the effectiveness of the proposed synchronization method. Numerical simulations coincide with the theoretical analysis.


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34D06 Synchronization of solutions to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
93D15 Stabilization of systems by feedback
26A33 Fractional derivatives and integrals
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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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: Wiley-Interscience USA · 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] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[5] Hilfer, R., Applications of Fractional Calculus in Physics (2001), World Scientific: World Scientific New Jersey · Zbl 0998.26002
[6] Bagley, R. L.; Calico, R. A., J. Guid. Contr. Dyn., 14, 304 (1991)
[7] Sun, H. H.; Abdelwahad, A. A.; Onaral, B., IEEE Trans. Automat. Control, 29, 441 (1984) · Zbl 0532.93025
[8] Ichise, M.; Nagayanagi, Y.; Kojima, T., J. Electroanal. Chem., 33, 253 (1971)
[9] Heaviside, O., Electromagnetic Theory (1971), Chelsea: Chelsea New York · JFM 30.0801.03
[10] Laskin, N., Physica A, 287, 482 (2000)
[11] Kusnezov, D.; Bulgac, A.; Dang, G. D., Phys. Rev. Lett., 82, 1136 (1999)
[12] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., IEEE Trans. Circuits Syst. I, 42, 485 (1995)
[13] Grigorenko, I.; Grigorenko, E., Phys. Rev. Lett., 91, 034101 (2003)
[14] Peng, G. J., Phys. Lett. A, 363, 426 (2007) · Zbl 1197.37040
[15] Li, C.; Liao, X.; Yu, J., Phys. Rev. E, 68, 067203 (2003)
[16] Lu, J. G., Chaos Solitons Fractals, 27, 519 (2006) · Zbl 1086.94007
[17] Caputo, M., Geophys. J. R. Astron. Soc., 13, 529 (1967)
[18] Diethelm, K.; Freed, A. D.; Ford, N. J., Nonlinear Dynam., 29, 3 (2002) · Zbl 1009.65049
[19] Li, Z. G.; Xu, D. L., Phys. Lett. A, 282, 175 (2001) · Zbl 0983.37036
[20] Xu, D. L.; Li, Z. G., Chaos, 11, 439 (2001)
[21] Xu, D. L.; Chee, C. Y., Phys. Rev. E, 66, 046218 (2002)
[22] Rulkov, N. F.; Sushchik, M. M.; Tsimring, L. S.; Abarbanel, H. D.I., Phys. Rev. E, 51, 980 (1995)
[23] Kocarev, L.; Parlitz, U., Phys. Rev. Lett., 77, 2206 (1996)
[24] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S., Phys. Rep., 366, 1 (2002) · Zbl 0995.37022
[25] Yan, J. P.; Li, C. P., Chaos Solitons Fractals, 26, 1119 (2005) · Zbl 1073.65147
[26] Genesio, R.; Tesi, A., Automatica, 28, 531 (1992) · Zbl 0765.93030
[27] Arneodo, A.; Coullet, P.; Spiegel, E.; Tresser, C., Physica D, 14, 327 (1985) · Zbl 0595.58030
[28] Sprott, J. C.; Linz, S. J., Int. J. Chaos Theory Appl., 5, 3 (2000)
[29] Matignon, D., Stability Results of Fractional Differential Equations with Applications to Control Processing (1996), IMACS, IEEE-SMC: IMACS, IEEE-SMC Lille, France
[30] Chen, C. T., Linear System Theory and Design (1984), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York
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