An extension of Sharkovsky's theorem to periodic difference equations (Q819695)
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scientific article; zbMATH DE number 5016171
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | An extension of Sharkovsky's theorem to periodic difference equations |
scientific article; zbMATH DE number 5016171 |
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An extension of Sharkovsky's theorem to periodic difference equations (English)
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29 March 2006
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The following system is studied: \[ x(n+1)=F(n,x(n))\tag{1} \] such that \(F(n+p,x)=F(n,x), p\) is a given positive integer. Geometric cycles of (1) are investigated. A simpler method for constructing \(p\)-periodic difference equations with \(r\)-periodic geometric cycles are given. A new ordering of positive integers (\(p\)-Sharkovsky ordering) is introduced. Both the Sharkovsky theorem and its converse are generalized to the system (1) [cf. \textit{A. N. Sharkovskij}, Ukr. Math. Zh. 16, 61--71 (1964; Zbl 0122.17504)].
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Sharkovsky theorem
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geometric cycles
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skew-product dynamical systems
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periodic orbits
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