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Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. (English) Zbl 1215.93114
Summary: Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.
MSC:
93C95Applications of control theory
37D45Strange attractors, chaotic dynamics
34H10Chaos control (ODE)
References:
[1]Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) · doi:10.1103/PhysRevLett.64.821
[2]Lakshmanan, M., Murali, K.: Chaos in Nonlinear Oscillators: Controlling and Synchronization. Singapore, World Scientific (1996)
[3]Han, S.K., Kerrer, C., Kuramoto, Y.: Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193 (1995) · doi:10.1103/PhysRevLett.75.3190
[4]Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological system. Nature 399, 354–359 (1999) · doi:10.1038/20676
[5]Yang, T., Chua, L.O.: Secure communication via chaotic parameter modulation. IEEE Trans. Circuits Syst. I 43, 817–819 (1996) · doi:10.1109/81.536758
[6]Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[7]Luo, A.C.J.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14, 1901–1951 (2009) · Zbl 1221.37218 · doi:10.1016/j.cnsns.2008.07.002
[8]Femat, R., Solis-Perales, G.: On the chaos synchronization phenomena. Phys. Lett. A 262, 50–60 (1997) · Zbl 0936.37010 · doi:10.1016/S0375-9601(99)00667-2
[9]Mossa Al-sawalha, M., Noorani, M.S.M.: Anti-synchronization of chaotic systems with uncertain parameters via adaptive control. Phys. Lett. A 373, 2852–2858 (2009) · Zbl 1233.93056 · doi:10.1016/j.physleta.2009.06.008
[10]Hua, W., Zheng-zhi, H., Wei, Z., Qi-yue, X.: Synchronization of unified chaotic systems with uncertain parameters based on the CLF. Nonlinear Anal. 10, 715–722 (2009) · Zbl 1167.37332 · doi:10.1016/j.nonrwa.2007.10.025
[11]Slotine, J.J.E., Lin, W.P.: Applied Nonlinear Control. Prentice-Hall, New York (1991)
[12]Chen, S., Lü, J.: Parameters identification and synchronization of chaotic systems based upon adaptive control. Phys. Lett. A 299, 353–358 (2002) · doi:10.1016/S0375-9601(02)00522-4
[13]Fotsin, H.B., Daafouz, J.: Adaptive synchronization of uncertain chaotic colpitts oscillators based on parameter identification. Phys. Lett. A 339, 304–315 (2005) · Zbl 1145.93313 · doi:10.1016/j.physleta.2005.03.049
[14]Huang, L., Wang, M., Feng, R.: Parameters identification and adaptive synchronization of chaotic systems with unknown parameters. Phys. Lett. A 342, 299–304 (2005) · Zbl 1222.93203 · doi:10.1016/j.physleta.2004.11.065
[15]Park, Ju H.: Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters. J. Comput. Appl. Math. 213, 288–293 (2008) · Zbl 1137.93035 · doi:10.1016/j.cam.2006.12.003
[16]Ge, Z.-M., Lee, J.-K.: Chaos synchronization and parameter identification for gyroscope system. Appl. Math. Comput. 163, 667–682 (2005) · Zbl 1116.70012 · doi:10.1016/j.amc.2004.04.008
[17]Rui-hong, L.: Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control. Commun. Nonlinear Sci. Numer. Simul. 14, 2757–2764 (2009) · Zbl 1221.93244 · doi:10.1016/j.cnsns.2008.10.006
[18]Manfeng, H., Zhenyuan, X.: Adaptive feedback controller for projective synchronization. Nonlinear Anal. 9, 1253–1260 (2008)
[19]Sudheer, K.S., Sabir, M.: Adaptive function projective synchronization of two-cell Quantum-CNN chaotic oscillators with uncertain parameters. Phys. Lett. A 373, 1847–1851 (2009) · Zbl 1229.92010 · doi:10.1016/j.physleta.2009.03.052
[20]Junwei, L., Xinyu, W., Yinhua, L.: How many parameters can be identified by adaptive synchronization in chaotic systems? Phys. Lett. A 373, 1249–1256 (2009) · Zbl 1228.34077 · doi:10.1016/j.physleta.2009.01.064
[21]Xiaoyun, C., Jianfeng, L.: Adaptive synchronization of different chaotic systems with fully unknown parameters. Phys. Lett. A 364, 123–128 (2007) · Zbl 1203.93161 · doi:10.1016/j.physleta.2006.11.092
[22]Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization of the complex Chen and Lü systems. Int. J. Bifurc. Chaos 17, 4295–4308 (2007) · Zbl 1146.93372 · doi:10.1142/S0218127407019962
[23]Mahmoud, G.M., Al-Kashif, M.A., Aly, S.A.: Basic properties and chaotic synchronization of complex Lorenz system. Int. J. Modern Phys. C 18, 253–265 (2007) · Zbl 1115.37035 · doi:10.1142/S0129183107010425
[24]Mahmoud, G.M., Aly, S.A., Al-Kashif, M.A.: Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. Nonlinear Dyn. 51, 171–181 (2008) · Zbl 1170.70365 · doi:10.1007/s11071-007-9200-y
[25]Mahmoud, G.M., Al-Kashif, M.A., Farghaly, A.A.: Chaotic and hyperchaotic attractors of a complex nonlinear system. J. Phys. A Math. Theor. 41(5), 055104 (2008). doi: 10.1088/1751-8113/41/5/055104
[26]Mahmoud, G.M., Abdusalam, H.A., Farghaly, A.A.: Chaotic behavior and chaos control for a class of complex partial differential equations. Int. J. Modern Phys. C 12(6), 889–899 (2001) · doi:10.1142/S0129183101002073
[27]Mahmoud, G.M., Bountis, T., AbdEl-Latif, G.M., Mahmoud, E.E.: Chaos synchronization of two different complex Chen and Lü systems. Nonlinear Dyn. 55, 43–53 (2009). doi: 10.1007/s11071-008-9343-5 · Zbl 1170.70011 · doi:10.1007/s11071-008-9343-5
[28]Mahmoud, G.M., Bountis, T., Al-Kashif, M.A., Aly, S.A.: Dynamical properties and synchronization of complex nonlinear equations for detuned lasers. Dyn. Syst. 24, 63–79 (2009). doi: 10.1080/14689360802438298 · Zbl 1172.34033 · doi:10.1080/14689360802438298
[29]Fowler, A.C., Gibbon, J.D., Mc Guinnes, M.T.: The real and complex Lorenz equations and their relevance to physical systems. Physica D 7, 126–134 (1983) · Zbl 1194.76087 · doi:10.1016/0167-2789(83)90123-9
[30]Gibbon, J.D., Mc Guinnes, M.J.: The real and complex Lorenz equations in rotating fluids and laser. Physica D 5, 108–121 (1982) · Zbl 1194.76280 · doi:10.1016/0167-2789(82)90053-7
[31]Ning, C.Z., Haken, H.: Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations. Phys. Rev. A 41, 3826–3837 (1990) · doi:10.1103/PhysRevA.41.3826