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An extension of Sharkovsky’s theorem to periodic difference equations. (English) Zbl 1125.39001
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. A. N. Sharkovskij, Ukr. Math. Zh. 16, 61–71 (1964; Zbl 0122.17504)].

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
37C27 Periodic orbits of vector fields and flows
Full Text: DOI
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