IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v39y2009i5p2491-2508.html
   My bibliography  Save this article

Chaos control and synchronization for a special generalized Lorenz canonical system – The SM system

Author

Listed:
  • Liao, Xiaoxin
  • Xu, F.
  • Wang, P.
  • Yu, Pei

Abstract

This paper presents some simple feedback control laws to study global stabilization and global synchronization for a special chaotic system described in the generalized Lorenz canonical form (GLCF) when τ=−1 (which, for convenience, we call Shimizu–Morioka system, or simply SM system). For an arbitrarily given equilibrium point, a simple feedback controller is designed to globally, exponentially stabilize the system, and reach globally exponent synchronization for two such systems. Based on the system’s coefficients and the structure of the system, simple feedback control laws and corresponding Lyapunov functions are constructed. Because all conditions are obtained explicitly in terms of algebraic expressions, they are easy to be implemented and applied to real problems. Numerical simulation results are presented to verify the theoretical predictions.

Suggested Citation

  • Liao, Xiaoxin & Xu, F. & Wang, P. & Yu, Pei, 2009. "Chaos control and synchronization for a special generalized Lorenz canonical system – The SM system," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2491-2508.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:5:p:2491-2508
    DOI: 10.1016/j.chaos.2007.07.029
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077907005462
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2007.07.029?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Liao, Xiaoxin & Yu, Pei, 2006. "Chaos control for the family of Rössler systems using feedback controllers," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 91-107.
    2. Čelikovský, Sergej & Chen, Guanrong, 2005. "On the generalized Lorenz canonical form," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1271-1276.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Gao, Tiegang & Chen, Zengqiang & Gu, Qiaolun & Yuan, Zhuzhi, 2008. "A new hyper-chaos generated from generalized Lorenz system via nonlinear feedback," Chaos, Solitons & Fractals, Elsevier, vol. 35(2), pages 390-397.
    2. Doungmo Goufo, Emile Franc, 2017. "Solvability of chaotic fractional systems with 3D four-scroll attractors," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 443-451.
    3. Falahpoor, M. & Ataei, M. & Kiyoumarsi, A., 2009. "A chattering-free sliding mode control design for uncertain chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1755-1765.
    4. Li, Damei & Wu, Xiaoqun & Lu, Jun-an, 2009. "Estimating the ultimate bound and positively invariant set for the hyperchaotic Lorenz–Haken system," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1290-1296.
    5. Mathale, D. & Doungmo Goufo, Emile F. & Khumalo, M., 2020. "Coexistence of multi-scroll chaotic attractors for fractional systems with exponential law and non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    6. Chen, Zengqiang & Yang, Yong & Yuan, Zhuzhi, 2008. "A single three-wing or four-wing chaotic attractor generated from a three-dimensional smooth quadratic autonomous system," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1187-1196.
    7. Huang, Kuifei & Yang, Qigui, 2009. "Stability and Hopf bifurcation analysis of a new system," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 567-578.
    8. Zhou, Xiaobing & Wu, Yue & Li, Yi & Wei, Zhengxi, 2008. "Hopf bifurcation analysis of the Liu system," Chaos, Solitons & Fractals, Elsevier, vol. 36(5), pages 1385-1391.
    9. Wang, Xia, 2009. "Si’lnikov chaos and Hopf bifurcation analysis of Rucklidge system," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2208-2217.
    10. El-Dessoky, M.M. & Agiza, H.N., 2009. "Global stabilization of some chaotic dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1584-1598.
    11. Zhiqin Qiao & Xianyi Li, 2014. "Dynamical analysis and numerical simulation of a new Lorenz-type chaotic system," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 20(3), pages 264-283, May.
    12. Qi, Guoyuan & Chen, Guanrong & Zhang, Yuhui, 2008. "On a new asymmetric chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 409-423.
    13. Qi, Guoyuan & van Wyk, Barend Jacobus & van Wyk, Michaël Antonie, 2009. "A four-wing attractor and its analysis," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 2016-2030.
    14. Sun, Fengyun & Zhao, Yi & Zhou, Tianshou, 2007. "Identify fully uncertain parameters and design controllers based on synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1677-1682.
    15. Xiong, Xiaohua & Wang, Junwei, 2009. "Conjugate Lorenz-type chaotic attractors," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 923-929.
    16. Yu, Simin & Tang, Wallace K.S., 2009. "Tetrapterous butterfly attractors in modified Lorenz systems," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1740-1749.
    17. Jiang, Yongxin & Sun, Jianhua, 2007. "Si’lnikov homoclinic orbits in a new chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 150-159.
    18. Zhou, Liangqiang & Chen, Fangqi, 2009. "Hopf bifurcation and Si’lnikov chaos of Genesio system," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1413-1422.
    19. Wang, Jiezhi & Chen, Zengqiang & Yuan, Zhuzhi, 2009. "Existence of a new three-dimensional chaotic attractor," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3053-3057.
    20. Dong, Chengwei & Liu, Huihui & Jie, Qi & Li, Hantao, 2022. "Topological classification of periodic orbits in the generalized Lorenz-type system with diverse symbolic dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:39:y:2009:i:5:p:2491-2508. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.