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Property of period-doubling bifurcation cascades of discrete dynamical systems

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  • Xu, Mingtian

Abstract

The period-doubling bifurcation process is one of the routes to chaos. During this process, when the system parameter reaches the bifurcation points, the periodic orbits lose their stability and double their periods. It is found that the periodic orbits born in this process exhibit an intimate relationship. On one hand, the orbits with relatively small periods can be approximately extracted from the orbit with a large period; on the other hand, the orbits with large periods can be approximately constructed by the orbit with relatively small period. Furthermore, our analytical results strongly suggest that the unstable periodic orbits originating from the period-doubling bifurcation process should play a big role in the ensuing chaos, at least at its early stage.

Suggested Citation

  • Xu, Mingtian, 2007. "Property of period-doubling bifurcation cascades of discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 455-462.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:2:p:455-462
    DOI: 10.1016/j.chaos.2006.01.022
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    References listed on IDEAS

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    1. Chen, Zhi-Min & Price, W.G., 2005. "Transition to chaos in a fluid motion system," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1195-1202.
    2. Wang, Liqiu & Xu, Mingtian, 2005. "Property of period-doubling bifurcations," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 527-532.
    3. Maize, S.M.A., 2006. "Self-pulsing and chaos of a two-photon optical bistable model in a ring cavity," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 590-600.
    4. Jing, Zhujun & Yang, Jianping & Feng, Wei, 2006. "Bifurcation and chaos in neural excitable system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 197-215.
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    Cited by:

    1. Vázquez-Medina, R. & Díaz-Méndez, A. & del Río-Correa, J.L. & López-Hernández, J., 2009. "Design of chaotic analog noise generators with logistic map and MOS QT circuits," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1779-1793.

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