Abstract
The adaptive synchronized problem of the four-dimensional (4D) Lü hyperchaotic system performed by Elabbasy et al. (Chaos Solitons Fractals 30:1133–1142, 2006) with uncertain parameters by applying the single control input is addressed in this article. Based on the Lyapunov theorem of stability, the single-input adaptive synchronization controllers associated with the adaptive update laws of system parameters are developed to make the states of two nearly identical 4D Lü hyperchaotic systems asymptotically synchronized. Numerical studies are presented to illustrate the effectiveness of the proposed chaotic synchronization schemes.
Similar content being viewed by others
References
Elabbasy, E.M., Agiza, H.N., El-Dessoky, M.M.: Adaptive synchronization of a hyperchaotic system with uncertain parameter. Chaos Solitons Fractals 30, 1133–1142 (2006)
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)
Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–11999 (1990)
Wang, Z.-L., Shi, X.-R.: Anti-synchronization of Liu system and Lorzen system with known or unknown parameters. Nonlinear Dyn. 57, 425–430 (2009)
Wang, Z.-L.: Projective synchronization of hyperchaotic Lü system and Liu system. Nonlinear Dyn. 59, 455–462 (2010)
Yan, Z.: Controlling hyperchaos in the new hyperchaotic Chen system. Appl. Math. Comput. 168, 1239–1250 (2005)
Yassen, M.T.: Synchronization hyperchaos of hyperchaotic systems. Chaos Solitons Fractals 37, 465–475 (2008)
Vincent, U.E.: Synchronization of identical and non-identical 4-D chaotic systems using active control. Chaos Solitons Fractals 37, 1065–1075 (2008)
Chen, C.H., Sheu, L.J., Chen, H.K., Chen, J.H., Wang, H.C., Chao, Y.C., Lin, Y.K.: A new hyper-chaotic system and its synchronization. Nonlinear Anal. RWA 10, 2088–2096 (2009)
Park, J.H.: Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos Solitons Fractals 26, 959–964 (2005)
Jia, Q.: Adaptive control and synchronization of a new hyperchaotic system with unknown parameters. Phys. Lett. A 362, 424–429 (2007)
Zhou, X., Wu, Y., Li, Y., Xue, H.: Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters. Appl. Math. Comput. 203, 80–85 (2008)
Zhou, X., Wu, Y.W., Li, Y.L., Xue, H.: Adaptive control and synchronization of a new modified hyperchaotic Lü system with uncertain parameters. Chaos Solitons Fractals 39, 2477–2483 (2009)
Li, X.F., Leung, A.C.S., Liu, X.J., Han, X.P., Chu, Y.D.: Adaptive synchronization of identical chaotic and hyper-chaotic systems with uncertain parameters. Nonlinear Anal. RWA 11, 2215–2223 (2010)
Zhang, H., Ma, X., Li, M., Zou, J.: Controlling and tracking hyperchaotic Rössler system via active backstepping design. Chaos Solitons Fractals 26, 353–361 (2005)
Laoye, J.A., Vincent, U.E., Kareem, S.O.: Chaos control of 4D chaotic system using recursive backstepping nonlinear controller. Chaos Solitons Fractals 39, 356–362 (2009)
Njah, A.N., Sunday, O.D.: Generalized on the chaos control of 4-D chaotic system using recursive backstepping nonlinear controller. Chaos Solitons Fractals 41, 2371–2376 (2009)
Peng, C.-C., Chen, C.-L.: Robust chaotic control of Lorenz system by backstepping design. Chaos Solitons Fractals 37, 598–608 (2008)
Jang, M.-J., Chen, C.-L., Chen, C.-K.: Sliding mode control of hyperchaos in Rössler systems. Chaos Solitons Fractals 14, 1465–1476 (2002)
Liu, L., Han, Z., Li, W.: Global sliding mode control and application in chaotic systems. Nonlinear Dyn. 56, 193–198 (2009)
Wang, H., Han, Z., Xie, Q., Zhang, W.: Finite-time chaos control of unified chaotic systems with uncertain parameters. Nonlinear Dyn. 55, 323–328 (2009)
Chen, C.-L., Yau, H.-T., Peng, C.-C.: Design of extended backstepping sliding mode controller for uncertain chaotic systems. Int. J. Nonlinear Sci. Numer. Simul. 8, 137–145 (2007)
Wang, F., Liu, C.: A new criterion for chaos and hyperchaos synchronization using linear feedback control. Phys. Lett. A 360, 274–278 (2006)
Rafikov, M., Balthazae, J.M.: On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun. Nonlinear Sci. Numer. Simul. 13, 1246–1255 (2008)
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)
Yassen, M.T.: Adaptive control and synchronization of a modified Chua’s circuit system. Appl. Math. Comput. 135, 113–128 (2003)
Agiza, H.N.: Chaos synchronization of Lü dynamical system. Nonlinear Anal. 58, 11–20 (2004)
Loría, A.: A linear time-varying controller for synchronization of Lü chaotic systems with one input. IEEE Trans. Circuit Syst. II, Express Briefs 56, 674–678 (2009)
Oancea, S., Grosu, F., Lazar, A., Grosu, I.: Master-slave synchronization of Lorenz systems using a single controller. Chaos Solitons Fractals 41, 2575–2580 (2009)
Shi, X.-R.: Bursting synchronization of Hind–Rose system based on a single controller. Nonlinear Dyn. 59, 95–99 (2010)
Yassen, M.T.: On hyperchaos synchronization of hyperchaotic Lü systems. Nonlinear Anal. 68, 3592–3600 (2008)
Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)
Khall, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002)
Slotine, J.E., Li, W.: Applied Nonlinear Control. Prentice Hall, New Jersey (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, CC. Adaptive synchronization of Lü hyperchaotic system with uncertain parameters based on single-input controller. Nonlinear Dyn 63, 447–454 (2011). https://doi.org/10.1007/s11071-010-9814-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9814-3