Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system. (English) Zbl 1170.70365

Summary: In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.


70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
70K20 Stability for nonlinear problems in mechanics
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[1] Ning, C.Z., Haken, H.: Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations. Phys. Rev. A 41, 3826–3837, (1990)
[2] Rauh, A., Hannibal, L., Abraham, N.: Global stability properties of the complex Lorenz model. Physica D 99, 45–58 (1996) · Zbl 0887.34048
[3] Gibbon, J.D., McGuinnes, M.J.: The real and complex Lorenz equations in rotating fluids and laser. Physica D 5, 108–121 (1982) · Zbl 1194.76280
[4] Mahmoud, G.M., Bountis, T., Mahmoud, E.E.: Active control and globle synchronization for complex Chen and Lü systems. Int. J. Bif. chaos (in press) · Zbl 1146.93372
[5] Liu, W.B., Chen, G.R.: A new chaotic system and its generation. Int. J. Bif. Chaos 13(1), 261–267 (2003) · Zbl 1078.37504
[6] Sun, J.: Impulsive control of a new chaotic system. Math. Comput. Simul. 64, 669–677 (2004) · Zbl 1076.65119
[7] Lu, J., Chen, G., Cheng, D., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bif. Chaos 12(12), 2917–2926 (2002) · Zbl 1043.37026
[8] Chen, H.K., Lee, C.I.: Anti-control of chaos in rigid body motion. Chaos Solitons Fractals 21, 957–965 (2004) · Zbl 1046.70005
[9] Elabbasy, E.M., Agiza, H.N., El-Dessoky, M.: Global synchronization criterion and adaptive synchronization for new chaotic system. Chaos Solitons Fractals 23(10), 1299–1309 (2005) · Zbl 1086.37512
[10] Qi, G., Chen, G., Du, S., Chen, Z., Yuan, Z.: Analysis of a new chaotic system. Physica A 352, 295–308 (2005)
[11] Yassen, M.T.: Controlling chaos and ynchronization for new chaotic system using linaer feedback control. Chaos Solitons Fractals 26, 913–920 (2005) · Zbl 1093.93539
[12] Liu, W., Chen, G.: Can a three-dimensional smooth autonomous quadratic chaotic system generate a single four-scroll attractor? Int J. Bif. Chaos 14(4), 1395–1403 (2004) · Zbl 1086.37516
[13] Mahmoud, G.M., Aly, S.A., Farghaly, A.A.: On chaos synchronization of a complex two coupled dynamos system, Chaos Solitons and Fractals (in press) · Zbl 1152.37317
[14] Agiza, H.N., Yassen, M.T.: Synchronization systems of Rössler and Chen chaotic dynamical systems using active control. Phys. Lett. A 278, 191–197 (2001) · Zbl 0972.37019
[15] Ucar, A., Lonngren, K.E., Bai, E.-W.: Synchronization of the unified chaotic systems via active control. Chaos Solitons Fractals 27(5), 1292–1297 (2006) · Zbl 1091.93030
[16] 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
[17] Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985) · Zbl 0585.58037
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