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Multiple bifurcations and periodic “bubbling” in a delay population model

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  • Peng, Mingshu

Abstract

In this paper, the flip bifurcation and periodic doubling bifurcations of a discrete population model without delay influence is firstly studied and the phenomenon of Feigenbaum’s cascade of periodic doublings is also observed. Secondly, we explored the Neimark–Sacker bifurcation in the delay population model (two-dimension discrete dynamical systems) and the unique stable closed invariant curve which bifurcates from the nontrivial fixed point. Finally, a computer-assisted study for the delay population model is also delved into. Our computer simulation shows that the introduction of delay effect in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node→stable focus→an lower-dimensional closed invariant curve (quasi-periodic solution, limit cycle) or/and stable periodic solutions→chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling) and the sudden change between two different attractors, etc.

Suggested Citation

  • Peng, Mingshu, 2005. "Multiple bifurcations and periodic “bubbling” in a delay population model," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1123-1130.
  • Handle: RePEc:eee:chsofr:v:25:y:2005:i:5:p:1123-1130
    DOI: 10.1016/j.chaos.2004.11.087
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    Cited by:

    1. Peng, Mingshu & Yuan, Yuan, 2008. "Complex dynamics in discrete delayed models with D4 symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 393-408.
    2. Çelik, C. & Merdan, H. & Duman, O. & Akın, Ö., 2008. "Allee effects on population dynamics with delay," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 65-74.
    3. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Bifurcation control of the Hodgkin–Huxley equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 217-224.
    4. Merdan, H. & Duman, O. & Akın, Ö. & Çelik, C., 2009. "Allee effects on population dynamics in continuous (overlapping) case," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1994-2001.
    5. Gu, En-Guo & Hao, Yu-Dong, 2007. "On the global analysis of dynamics in a delayed regulation model with an external interference," Chaos, Solitons & Fractals, Elsevier, vol. 34(4), pages 1272-1284.
    6. Yang, Xiaozhong & Peng, Mingshu & Hu, Jiping & Jiang, Xiaoxia, 2009. "Bubbling phenomenon in a discrete economic model for the interaction of demand and supply," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1428-1438.
    7. Sen, Ayan & Mukherjee, Debasis, 2009. "Chaos in the delay logistic equation with discontinuous delays," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2126-2132.
    8. Avrutin, Viktor & Morcillo, Jose D. & Zhusubaliyev, Zhanybai T. & Angulo, Fabiola, 2017. "Bubbling in a power electronic inverter: Onset, development and detection," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 135-152.
    9. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Analysis and control of the bifurcation of Hodgkin–Huxley model," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 247-256.

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