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Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization

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  • Shu, Yonglu
  • Xu, Hongxing
  • Zhao, Yunhong

Abstract

In this paper, we investigate the ultimate bound and positively invariant set for a new chaotic system via the generalized Lyapunov function theory. For this system, we derive a three-dimensional ellipsoidal ultimate bound and positively invariant set. In addition, the two-dimensional bound with respect to x-z and y-z are established. Finally, the result is applied to the study of completely chaos synchronization, an exact threshold is given with the system parameters. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.

Suggested Citation

  • Shu, Yonglu & Xu, Hongxing & Zhao, Yunhong, 2009. "Estimating the ultimate bound and positively invariant set for a new chaotic system and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2852-2857.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:5:p:2852-2857
    DOI: 10.1016/j.chaos.2009.04.003
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    References listed on IDEAS

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    1. Sun, Yeong-Jeu, 2009. "Solution bounds of generalized Lorenz chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 691-696.
    2. 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.
    3. Panchev, S. & Spassova, T. & Vitanov, N.K., 2007. "Analytical and numerical investigation of two families of Lorenz-like dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1658-1671.
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    Cited by:

    1. Huang, Jun & Han, Zhengzhi & Cai, Xiushan & Liu, Leipo, 2011. "Uniformly ultimately bounded tracking control of linear differential inclusions with stochastic disturbance," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(12), pages 2662-2672.
    2. Hou, Yi-You & Lin, Ming-Hung & Saberi-Nik, Hassan & Arya, Yogendra, 2024. "Boundary analysis and energy feedback control of fractional-order extended Malkus–Robbins dynamo system," Chaos, Solitons & Fractals, Elsevier, vol. 183(C).
    3. Zhang, Fuchen & Shu, Yonglu & Yang, Hongliang & Li, Xiaowu, 2011. "Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 137-144.
    4. Jian, Jigui & Wu, Kai & Wang, Baoxian, 2020. "Global Mittag-Leffler boundedness and synchronization for fractional-order chaotic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).

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