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Zbl 1065.92035
Peng, Mingshu
Multiple bifurcations and periodic ``bubbling'' in a delay population model. (English)
[J] Chaos Solitons Fractals 25, No. 5, 1123-1130 (2005). ISSN 0960-0779

Summary: 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 explore the Neimark-Sacker bifurcation in the delay population model (two-dimensional 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 effects in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node $\to$ stable focus $\to$ lower-dimensional closed invariant curves (quasi-periodic solutions, limit cycles) or/and stable periodic solutions $\to$ chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling), the sudden change between two different attractors, etc.

MSC 2000:: *92D25 Population dynamics
39A10 Difference equations
37N25 Dynamical systems in biology

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