×

zbMATH — the first resource for mathematics

Lag synchronization of hyperchaotic complex nonlinear systems. (English) Zbl 1243.93044
Summary: In this paper, we study the Lag Synchronization (LS) of n-dimensional hyperchaotic complex nonlinear systems. The idea of the nonlinear control technique based on the complex Lyapunov function with lag in time is used to propose a scheme to investigate LS of hyperchaotic attractors of these systems. Both complex Lyapunov and control functions are introduced. For illustration, the scheme is applied to two hyperchaotic complex Lorenz systems. The real and complex control functions are derived analytically to achieve LS and to show that the complex error dynamical systems are globally stable. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve LS of two identical hyperchaotic attractors.

MSC:
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93D99 Stability of control systems
34D06 Synchronization of solutions to ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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
[2] 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
[3] Femat, R., Solis-Perales, G.: Synchronization of chaotic systems with different order. Phys. Rev. E 65, 036226 (2002) · Zbl 1245.93003
[4] 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 · Zbl 1131.37036
[5] Mahmoud, G.M., Ahmed, M.E., Mahmoud, E.E.: Analysis of hyperchaotic complex Lorenz systems. Int. J. Mod. Phys. C 19(10), 1477–1494 (2008) · Zbl 1170.37311 · doi:10.1142/S0129183108013151
[6] Mahmoud, G.M., Mahmoud, E.E., Ahmed, M.E.: On the hyperchaotic complex Lü system. Nonlinear Dyn. 58, 725–738 (2009). doi: 10.1007/s11071-009-9513-0 · Zbl 1183.70053 · doi:10.1007/s11071-009-9513-0
[7] Yan, Z., Yu, P.: Hyperchaos synchronization and control on a new hyperchaotic attractor. Chaos Solitons Fractals 35, 333–345 (2008) · Zbl 1132.93042 · doi:10.1016/j.chaos.2006.05.027
[8] Li, C., Liao, X., Wong, K.: Lag synchronization of hyperchaos with application to secure communications. Chaos Solitons Fractals 23, 183–193 (2005) · Zbl 1068.94004 · doi:10.1016/j.chaos.2004.04.025
[9] Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979) · Zbl 0996.37502 · doi:10.1016/0375-9601(79)90150-6
[10] Melchor-Aguilar, D., Niculescu, S.-I.: Estimates of the attraction region for a class of nonlinear time-delay systems. IMA J. Math. Control Inf. 24, 523–550 (2007) · Zbl 1133.93026 · doi:10.1093/imamci/dnm007
[11] Kharitonov, V.L., Melchor-Aguilar, D.: On delay-dependent stability conditions for time-varying systems. Syst. Control Lett. 46, 173–180 (2002) · Zbl 0994.93022 · doi:10.1016/S0167-6911(02)00124-X
[12] Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[13] Alvarez, G., Li, S., Montoya, F., Pastor, G., Romera, M.: Breaking projective chaos synchronization secure communication using filtering and generalized synchronization. Chaos Solitons Fractals 24, 775–783 (2005) · Zbl 1068.94002 · doi:10.1016/j.chaos.2004.09.038
[14] Juan, M., Xing-yuan, W.: Nonlinear observer based phase synchronization of chaotic systems. Phys. Lett. A 369, 294–298 (2007) · Zbl 1209.34076 · doi:10.1016/j.physleta.2007.04.102
[15] Zhan, M., Wei, G.W., Lai, C.H.: Transition from intermittency to periodicity in lag synchronization in coupled Rössler oscillators. Phys. Rev. E 65, 036202 (2002)
[16] Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag-synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78, 4193–4196 (1997) · Zbl 0896.60090 · doi:10.1103/PhysRevLett.78.4193
[17] Taherion, S., Lai, Y.C.: Observability of lag-synchronization in coupled chaotic oscillators. Phys. Rev. 59, 6247–6250 (1999)
[18] Shahverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Lag-synchronization in time-delayed systems. Phys. Lett. A 292, 320–324 (2002) · Zbl 0979.37022 · doi:10.1016/S0375-9601(01)00824-6
[19] 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
[20] Sun, Y., Cao, J.: Adaptive lag-synchronization of unknown chaotic delayed neural networks with noise perturbation. Phys. Lett. A 364, 277–285 (2007) · Zbl 1203.93110 · doi:10.1016/j.physleta.2006.12.019
[21] Miao, Q., Tang, Y., Lu, S., Fang, J.: Lag synchronizing of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57, 107–112 (2009). doi: 10.1007/s11071-008-9424-5 · Zbl 1176.34097 · doi:10.1007/s11071-008-9424-5
[22] Cheng, Z., Xin, Y., Li, X., Xing, J.: Synchronization and lag synchronization of chaotic network. In: Advances in Neural Networks (2009). ISNN 1197-1202
[23] Dehghani, M., Haeri, M.: Complete and lag synchronization of hyperchaos using nonlinear controller. In: SICE-ICASE International Joint Conference, Pusan, pp. 2252–2255 (2006). doi: 10.1109/SICE.2006.315812
[24] Chen, J., Liu, H., Lu, J., Zhang, Q.: Projective and lag synchronization of a novel hyperchaotic system via impulsive control. Commun. Nonlinear Sci. Numer. Simul. 16, 2033–2040 (2011) · Zbl 1221.93104 · doi:10.1016/j.cnsns.2010.07.027
[25] Wang, D., Zhong, Y., Chen, S.: Lag synchronizing chaotic system based on a single controller. Commun. Nonlinear Sci. Numer. Simul. 13, 637–644 (2008) · Zbl 1130.34322 · doi:10.1016/j.cnsns.2006.05.005
[26] 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
[27] 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(12), 4295–4308 (2007) · Zbl 1146.93372 · doi:10.1142/S0218127407019962
[28] Mahmoud, G.M., Al-Kashif, M.A., Aly, S.A.: Basic properties and chaotic synchronization of complex Lorenz system. Int. J. Mod. Phys. C 18, 253–265 (2007) · Zbl 1115.37035 · doi:10.1142/S0129183107010425
[29] 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
[30] 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
[31] Mahmoud, G.M., Bountis, T.: The dynamics of systems of complex nonlinear oscillators: a review. Int. J. Bifurc. Chaos 14(11), 3821–3846 (2004) · Zbl 1091.34524 · doi:10.1142/S0218127404011624
[32] Huang, L., Feng, R., Wang, M.: Synchronization of chaotic systems via nonlinear control. Phys. Lett. A 320, 271–275 (2004) · Zbl 1065.93028 · doi:10.1016/j.physleta.2003.11.027
[33] Park, J.H.: Chaos synchronization of a chaotic system via nonlinear control. Chaos Solitons Fractals 25(2), 699–704 (2005) · Zbl 1125.93469 · doi:10.1016/j.chaos.2004.11.031
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.