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On the periodic logistic equation. (English) Zbl 1109.90010
Summary: We show that the $$p$$-periodic logistic equation $$x_{n+1} = \mu _{n\,\text{mod\,} p}x_{n}(1 - x_{n})$$ has cycles (periodic solutions) of minimal periods $$1, p, 2p, 3p, \dots$$ Then we extend Singer’s theorem to periodic difference equations, and use it to show the $$p$$-periodic logistic equation has at most $$p$$ stable cycles. Also, we present computational methods investigating the stable cycles in case $$p = 2$$ and 3.

##### MSC:
 90B06 Transportation, logistics and supply chain management 39A11 Stability of difference equations (MSC2000) 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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