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Complex projective synchronization in coupled chaotic complex dynamical systems. (English) Zbl 1253.93060
Summary: In previous papers, the projective factors are always chosen as real numbers, real matrices, or even real-valued functions, which means that the coupled systems evolve in the same or inverse direction simultaneously. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, the projective synchronization with respect to a complex factor, called Complex Projective Synchronization (CPS), should be taken into consideration. In this paper, based on Lyapunov’s stability theory, three typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization. Further, an adaptive control method is adopted to design a universal controller for partially linear systems. Numerical examples are provided to show the effectiveness of the proposed method.

MSC:
93C15Control systems governed by ODE
93D05Lyapunov and other classical stabilities of control systems
93C40Adaptive control systems
34D06Synchronization
34H10Chaos control (ODE)
WorldCat.org
Full Text: DOI
References:
[1] Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821--824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[2] Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196--1199 (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[3] Pyragas, K.: Synchronization of coupled time-delay systems: analytical estimations. Phys. Rev. E 58, 3067--3071 (1998) · doi:10.1103/PhysRevE.58.3067
[4] 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
[5] Li, C., Liao, X., Wong, K.: Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication. Physica D 194, 187--202 (2004) · Zbl 1059.93118 · doi:10.1016/j.physd.2004.02.005
[6] Wu, X., Wang, H., Lu, H.: Hyperchaotic secure communication via generalized function projective synchronization. Nonlinear Anal., Real World Appl. 12, 1288--1299 (2011) · Zbl 1203.94128 · doi:10.1016/j.nonrwa.2010.09.026
[7] Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239--252 (2011) · Zbl 1215.93127 · doi:10.1007/s11071-010-9800-9
[8] Chen, T.P., Liu, X.W., Lu, W.L.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I, Regul. Pap. 54, 1317--1326 (2007) · doi:10.1109/TCSI.2007.895383
[9] Li, K.Z., Small, M., Fu, X.C.: Generation of clusters in complex dynamical networks via pinning control. J. Phys. A, Math. Theor. 41, 505101 (2008) · Zbl 1152.93002
[10] Zhang, H.F., Li, K.Z., Fu, X.C., Wang, B.H.: An efficient control strategy of epidemic spreading on scale-free networks. Chin. Phys. Lett. 26, 068901 (2009)
[11] Feng, J.W., Yam, P., Austin, F., Chen, X.: Synchronizing the noise-perturbed Rösler hyperchaotic system via sliding mode control. Z. Naturforsch. A 66, 6--12 (2011)
[12] Choi, Y.P., Haa, S.Y., Yunb, S.B.: Complete synchronization of Kuramoto oscillators with finite inertia. Physica D 240, 32--44 (2011) · Zbl 1217.34087 · doi:10.1016/j.physd.2010.08.004
[13] Li, X., Leung, A.C.S., Han, X., Liu, X., Chu, Y.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63, 263--275 (2011) · Zbl 1215.93074 · doi:10.1007/s11071-010-9802-7
[14] Wang, Z., Shi, X.: Anti-synchronization of Liu system and Lorenz system with known or unknown parameters. Nonlinear Dyn. 57, 425--430 (2009) · Zbl 1176.70037 · doi:10.1007/s11071-008-9452-1
[15] Suresh, R., Senthilkumar, D.V., Lakshmanan, M., Kurths, J.: Global phase synchronization in an array of time-delay systems. Phys. Rev. E 82, 016215 (2010) · Zbl 1270.34153
[16] Ghosha, D., Chowdhury, A.R.: Lag and anticipatory synchronization based parameter estimation scheme in modulated time-delayed systems. Nonlinear Anal., Real World Appl. 11, 3059--3065 (2010) · Zbl 1214.93109 · doi:10.1016/j.nonrwa.2009.10.025
[17] Wang, X., Wang, M.: Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems. Nonlinear Dyn. 62, 567--571 (2010) · doi:10.1007/s11071-010-9744-0
[18] Hu, M., Xu, Z.: Adaptive feedback controller for projective synchronization. Nonlinear Anal., Real World Appl. 9, 1253--1260 (2008) · Zbl 1144.93364 · doi:10.1016/j.nonrwa.2007.03.005
[19] Nie, H., Xie, L., Gao, J., Zhan, M.: Projective synchronization of two coupled excitable spiral waves. Chaos 21, 023107 (2011) · Zbl 1317.93140
[20] Yu, Y., Li, H.: Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design. Nonlinear Anal., Real World Appl. 12, 388--393 (2011) · Zbl 1214.34042 · doi:10.1016/j.nonrwa.2010.06.024
[21] Fowler, A.C., Gibbon, J.D., McGuinness, M.J.: The complex Lorenz equations. Physica D 4, 139--163 (1982) · Zbl 1194.37039 · doi:10.1016/0167-2789(82)90057-4
[22] Vladimirov, A.G., Toronov, V.Y., Derbov, V.L.: The complex Lorenz model: geometric structure, homoclinic bifurcation and one-dimensional map. Int. J. Bifurc. Chaos Appl. Sci. Eng. 8, 723--729 (1998) · Zbl 0938.34037 · doi:10.1142/S0218127498000516
[23] Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system. Chaos Solitons Fractals 33, 178--187 (2007) · Zbl 1152.37317 · doi:10.1016/j.chaos.2006.01.036
[24] Nian, F., Wang, X., Niu, Y., Lin, D.: Module-phase synchronization in complex dynamic system. Appl. Math. Comput. 217, 2481--2489 (2010) · Zbl 1207.93047 · doi:10.1016/j.amc.2010.07.059
[25] Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62, 875--882 (2010) · Zbl 1215.93114 · doi:10.1007/s11071-010-9770-y
[26] Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization for complex Chen and Lü systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 4295--4308 (2007) · Zbl 1146.93372 · doi:10.1142/S0218127407019962
[27] Hu, M., Yang, Y., Xu, Z., Guo, L.: Hybrid projective synchronization in a chaotic complex nonlinear system. Math. Comput. Simul. 79, 449--457 (2008) · Zbl 1151.93017 · doi:10.1016/j.matcom.2008.01.047
[28] Mahmoud, G.M., Bountis, T., AbdEl-Latif, G.M., Mahmoud, E.E.: Chaos synchronization of two different chaotic complex Chen and Lü systems. Nonlinear Dyn. 55, 43--53 (2009) · Zbl 1170.70011 · doi:10.1007/s11071-008-9343-5
[29] 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
[30] Liu, S., Liu, P.: Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters. Nonlinear Anal., Real World Appl. (2011). doi: 10.1016/j.nonrwa.2011.05.006 · Zbl 1231.37020
[31] Sun, Y.J.: Generalized projective synchronization for a class of chaotic systems with parameter mismatching, unknown external excitation, and uncertain input nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 16, 3863--3870 (2011) · Zbl 1223.65098 · doi:10.1016/j.cnsns.2011.01.025
[32] Ghosh, D.: Generalized projective synchronization in time-delayed systems: nonlinear observer approach. Chaos 19, 013102 (2009) · Zbl 1311.34111
[33] Cai, N., Jing, Y., Zhang, S.: Modified projective synchronization of chaotic systems with disturbances via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 15, 1613--1620 (2010) · Zbl 1221.37211 · doi:10.1016/j.cnsns.2009.06.012
[34] Wen, G.: Designing Hopf limit circle to dynamical systems via modified projective synchronization. Nonlinear Dyn. 63, 387--393 (2011) · doi:10.1007/s11071-010-9810-7
[35] Wu, Z., Fu, X.: Adaptive function projective synchronization of discrete chaotic systems with unknown parameters. Chin. Phys. Lett. 27, 050502 (2010)
[36] Du, H., Zeng, Q., Wang, C., Ling, M.: Function projective synchronization in coupled chaotic systems. Nonlinear Anal., Real World Appl. 11, 705--712 (2010) · Zbl 1181.37039 · doi:10.1016/j.nonrwa.2009.01.016