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Chaos synchronization of two different chaotic complex Chen and Lü systems. (English) Zbl 1170.70011
Summary: This paper investigates the chaos synchronization of two different chaotic complex systems of the Chen and Lü type [S. Chen and J. Lü, Chaos Solitons Fractals 14, No. 4, 643–647 (2002; Zbl 1005.93020)] via the methods of active control and global synchronization. In this regard, it generalizes earlier work on the synchronization of two identical oscillators in the cases where the drive and response systems are different, the parameter space is larger, and the dimensionality increases due to the complexification of dependent variables. The idea of chaos synchronization is to use the output of the drive system to control the response system, so that the output of the response system converges to the output of the drive system as time increases. Lyapunov functions are derived to prove that the differences in the dynamics of the two systems converge to zero exponentially fast, explicit expressions are given for the control functions, and numerical simulations are presented to illustrate our chaos synchronization techniques. We also point out that the global synchronization method is better suited for synchronizing identical chaotic oscillators, as it has serious limitations when applied to the case where the drive and response systems are different.
MSC:
70K55Transition to stochasticity (chaotic behavior)
70Q05Control of mechanical systems (general mechanics)
References:
[1]Agiza, H.N., Yassen, M.T.: Synchronization systems of Rössler and Chen chaotic dynamical systems using active control. Phys. Lett. A 278, 191–197 (2005) · Zbl 0972.37019 · doi:10.1016/S0375-9601(00)00777-5
[2]Chan, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9(7), 1465–1466 (1999) · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[3]Chen, H.K.: Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lü. Chaos Solitons Fractals 25(5), 1049–1056 (2005) · Zbl 1198.34069 · doi:10.1016/j.chaos.2004.11.032
[4]Chen, S., Lü, J.: Synchronization of an uncertain chaotic system via adaptive control. Chaos Solitons Fractals 14, 643–647 (2002) · Zbl 1005.93020 · doi:10.1016/S0960-0779(02)00006-1
[5]Dai, E.w., Lonngren, K.E., Sequencial synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 11, 1041–1044 (2000) · Zbl 0985.37106 · doi:10.1016/S0960-0779(98)00328-2
[6]Elabbasy, E.M., Agiza, H.N., El-Dessoky, M.: Synchronization of modified Chen system. Int. J. Bifurc. Chaos 14(11), 3969–3979 (2004) · Zbl 1090.37516 · doi:10.1142/S0218127404011740
[7]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
[8]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
[9]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
[10]Jiang, G.P., Tang, K.S., Chen, G.: A simple global synchronization criterion for coupled chaotic systems. Chaos Solitons Fractals 15, 925–935 (2003) · Zbl 1065.70015 · doi:10.1016/S0960-0779(02)00214-X
[11]Lei, Y., Xu, W., Shen, J., Fang, T.: Global synchronization of two parametrically excited systems using active control. Chaos Solitons Fractals 28, 428–436 (2006) · Zbl 1084.37029 · doi:10.1016/j.chaos.2005.05.043
[12]Li, D., Lü, J., Wu, X.: Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos Solitons Fractals 23, 79–85 (2005) · Zbl 1063.37030 · doi:10.1016/j.chaos.2004.03.027
[13]Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20(1), 130–141 (1963) · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[14]Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12(3), 659–661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[15]Lü, J., Chen, G., Zhang, S.: Dynamical analysis of a new chaotic attractor. Int. J. Bifurc. Chaos 12(5), 1001–1015 (2002) · Zbl 1044.37021 · doi:10.1142/S0218127402004851
[16]Lü, J., Chen, G., Zhang, S.: The compound structure of a new chaotic attractor. Chaos Solitons Fractals 14, 669–672 (2002) · Zbl 1067.37042 · doi:10.1016/S0960-0779(02)00007-3
[17]Lü, J., Zhou, T., Zhang, S.: Chaos synchronization between linearly coupled chaotic systems. Chaos Solitons Fractals 14, 524–541 (2002)
[18]Lü, J., Chen, G., Cheng, D.: A new chaotic system and beyond: the generalized Lorenz-like system. Int. J. Bifurc. and Chaos 14, 1507–1537 (2004) · Zbl 1129.37323 · doi:10.1142/S021812740401014X
[19]Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and global synchronization for complex Chen and Lü systems. Int. J. Bifurc. Chaos (2007, to appear)
[20]Mahmoud, G.M., Aly, S.A., Al-Kashif, M.A.: Dynamical properties of chaos synchronization of a new chaotic complex nonlinear system. J. Nonlinear Dyn. 51, 171–181 (2008) · Zbl 1170.70365 · doi:10.1007/s11071-007-9200-y
[21]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
[22]Park, J.H.: Chaos synchronization of a chaotic system via nonlinear control. Chaos Solitons Fractals 25(3), 579–584 (2005) · Zbl 1092.37514 · doi:10.1016/j.chaos.2004.11.038
[23]Park, J.H.: On synchronization of unified chaotic systems via nonlinear control. Chaos Solitons Fractals 25(2), 699–704 (2005) · Zbl 1125.93469 · doi:10.1016/j.chaos.2004.11.031
[24]Park, J.H.: Chaos synchronization between two different chaotic dynamical systems. Chaos Solitons Fractals 27, 549–554 (2006) · Zbl 1102.37304 · doi:10.1016/j.chaos.2005.03.049
[25]Rauh, A., Hannibal, L., Abraham, N.: Global stability properties of the complex Lorenz model. Physica D 99, 45–58 (1996) · Zbl 0887.34048 · doi:10.1016/S0167-2789(96)00129-7
[26]Tsonis, A.A.: Chaos from Theory to Applications. Plenum, New York (1992)
[27]Ucar, A., Lonngren, K.E., Bai, E.-W.: Synchronization of the unified chaotic systems via active control. Chaos Solitons Fractals 27(5), 1292–1297 (2005) · Zbl 1091.93030 · doi:10.1016/j.chaos.2005.04.104
[28]Ueta, T., Chen, G.: Bifurcation analysis of Chen’s attractor. Int. J. Bifurc. Chaos 10, 1917–1931 (2000)
[29]Verhulst, F.: Nonlinear Differential and Dynamical Systems. Springer, New York (1996)
[30]Vladimirov, A.G.: The complex Lorenz model: geometric structure, homoclinic bifurcation and one-dimensional map. Int. J. Bifurc. Chaos 8(4), 723–729 (1998) · Zbl 0938.34037 · doi:10.1142/S0218127498000516
[31]Wang, Y., Guan, Z.H., Wang, H.O.: Feedback and adaptive control for the synchronization of Chen system via a single variable. Phys. Lett. A 312, 34–40 (2003) · Zbl 1024.37053 · doi:10.1016/S0375-9601(03)00573-5
[32]Wu, X., Lü, J.: Parameter identification and backstepping control of uncertain Lü system. Chaos Solitons Fractals 18, 721–729 (2003) · Zbl 1068.93019 · doi:10.1016/S0960-0779(02)00659-8
[33]Yassen, M.T.: Feedback and adaptive synchronization of Lü system. Chaos Solitons Fractals 25(2), 379–386 (2005) · Zbl 1125.93473 · doi:10.1016/j.chaos.2004.11.042
[34]Yassen, M.T.: Chaos synchronization between two different chaotic systems using active control. Chaos Solitons Fractals 25, 131–140 (2005) · Zbl 1091.93520 · doi:10.1016/j.chaos.2004.03.038