Chaos Synchronization of Uncertain Lorenz System via Single State Variable Feedback


This paper treats the problem of chaos synchronization for uncertain Lorenz system via single state variable information of the master system. By the Lyapunov stability theory and adaptive technique, the derived controller is featured as follows: 1) only single state variable information of the master system is needed; 2) chaos synchronization can also be achieved even if the perturbation is occurred in some parameters of the master chaotic system. Finally, the effectiveness of the proposed controllers is also illustrated by the simulations as well as rigorous mathematical proofs.

Share and Cite:

F. Chen and T. Zhang, "Chaos Synchronization of Uncertain Lorenz System via Single State Variable Feedback," Applied Mathematics, Vol. 4 No. 11B, 2013, pp. 7-12. doi: 10.4236/am.2013.411A2002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] W. Kinzel, A. Englert and I. Kanter, “On Chaos Synchronization and Secure Communication,” Philosophical Transactions of the Royal Society A, Vol. 368, No. 1911, 2010, pp. 379-389.
[2] S. Rasoulian and M. Shahrokhi, “Control of a Chemical Reactor with Chaotic Dynamics,” Iranian Journal of Chemistry & Chemical Engineering, Vol. 29, No. 4, 2010, pp. 149-159.
[3] X. Tan, J. Zhang and Y. Yang, “Synchronizing Chaotic Systems Using Backstepping Design,” Chaos, Solitons & Fractals, Vol. 16, No. 1, 2003, pp. 37-45.
[4] F. Sorrentino, G. Barlev, A. B. Cohen and E. Ott, “The Stability of Adaptive Synchronization of Chaotic Systems,” Chaos, Vol. 20, No. 1, 2010, Article ID: 013103.
[5] M. Haeri and M. Dehghani, “Impulsive Synchronization of Different Hyperchaotic (Chaotic) Systems,” Chaos, Solitons & Fractals, Vol. 38, No. 1, 2008, pp. 120-131.
[6] H.-T. Yau, “Design of Adaptive Sliding Mode Controller for Chaos Synchronization with Uncertainties,” Chaos, Solitons & Fractals, Vol. 22, No. 2, 2004, pp. 341-347.
[7] C.-C. Wang and J.-P. Su, “A New Adaptive Variable Structure Control for Chaotic Synchronization and Secure Communication,” Chaos, Solitons & Fractals, Vol. 20, No. 5, 2004, pp. 967-977.
[8] X. Yu and Y. Song, “Chaos Synchronization via Controlling Partial State of Chaotic Systems,” International Journal Bifurcation and Chaos, Vol. 11, No. 6, 2001, pp. 1737-1741.
[9] X.-F. Wang, Z.-Q. Wang and G.-R. Chen, “A New Criterion for Synchronization of Coupled Chaotic Oscillators with Application to Chua’s Circuits,” International Journal Bifurcation and Chaos, Vol. 9, No. 6, 1999, pp. 11691174.
[10] J. Cao, H. X. Li and D. W. C. Ho, “Synchronization Criteria of Lur’s Systems with Time-Delay Feedback Control,” Chaos, Solitons and Fractals, Vol. 23, No. 4, 2005, pp. 1285-1298.
[11] M. T. Yassen, “Feedback and Adaptive Synchronization of Chaotic Lu system,” Chaos Solitons and Fractals, Vol. 25, No. 2, 2005, pp. 379-386.
[12] J. Zhang, C. G. Li, H. B. Zhang and J. B. Yu, “Chaos Synchronization Using Single Variable Feedback Based on Backstepping Method,” Chaos Solitons and Fractals, Vol. 21, No. 5, 2004, pp. 1183-1193.
[13] J. A. Lu, X. Q. Wu, X. P. Han and J. H. Lü, “Adaptive Feedback Synchronization of a Unified Chaotic System,” Physics Letters A, Vol. 329, 2004, pp. 327-333.
[14] F. X. Chen, C. S. Zhang, G. J. Ji, S. Zhai and S. Zhou, “Chua System Chaos Synchronization Using Single Variable Feedback Based on Lasalle Invariance Principal,” Proceedings of the 2010 IEEE International Conference on Information and Automation, Harbin, 20-23 June 2010, pp. 301-304.
[15] P. Bergé, Y. Pomeau and C. Vidal, “Order within Chaos: Towards a Deterministic Approach to Turbulence,” John Wiley & Sons, New York, 1984.
[16] H. K. Khalil, “Nonlinear Systems,” Third Edition, Publish House of Electronics Industry, Beijing, 2007.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.