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Dynamics of the stochastic Lorenz-Haken system

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  • Li, Lijie
  • Feng, Yu
  • Liu, Yongjian

Abstract

In this paper, the dynamics of the stochastic Lorenz-Haken system are discussed, and some new results are presented. Firstly, the asymptotic behavior of the stochastic Lorenz-Haken system is analyzed. The interesting thing is that all of solutions of the system can tend to zero under some parameters conditions and never go through the hyper-plane x=0 as the large time. Secondly, the globally exponential attractive set and a four-dimensional ellipsoidal ultimate boundary are derived. The two-dimensional parabolic ultimate bound with respect to x−u is also established. The numerical results to estimate the ultimate boundary are also presented for verification. Finally, the random attractor set and the bifurcation phenomenon for the system are analyzed.

Suggested Citation

  • Li, Lijie & Feng, Yu & Liu, Yongjian, 2016. "Dynamics of the stochastic Lorenz-Haken system," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 670-678.
    • Handle: RePEc:eee:chsofr:v:91:y:2016:i:c:p:670-678
    DOI: 10.1016/j.chaos.2016.09.003
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    References listed on IDEAS

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    1. Liu, Meng & Wang, Ke, 2012. "Persistence and extinction of a single-species population system in a polluted environment with random perturbations and impulsive toxicant input," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1541-1550.
    2. Xu, Yong & Gu, Rencai & Zhang, Huiqing, 2011. "Effects of random noise in a dynamical model of love," Chaos, Solitons & Fractals, Elsevier, vol. 44(7), pages 490-497.
    3. Wang, Jingyue & Wang, Haotian & Guo, Lixin, 2014. "Analysis of effect of random perturbation on dynamic response of gear transmission system," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 78-88.
    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. Zhao, Yu & Yuan, Sanling, 2016. "Stability in distribution of a stochastic hybrid competitive Lotka–Volterra model with Lévy jumps," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 98-109.
    6. 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.
    7. Ye, Zhiyong & Ji, Huihui & Zhang, He, 2016. "Passivity analysis of Markovian switching complex dynamic networks with multiple time-varying delays and stochastic perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 147-157.
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