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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)].

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