an:05016171
Zbl 1125.39001
Alsharawi, Ziyad; Angelos, James; Elaydi, Saber; Rakesh, Leela
An extension of Sharkovsky's theorem to periodic difference equations
EN
J. Math. Anal. Appl. 316, No. 1, 128-141 (2006).
00124052
2006
j
39A11 39A12 37C27
Sharkovsky theorem; geometric cycles; skew-product dynamical systems; periodic orbits
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)].
Ahmed Hegazi (Mansoura)
Zbl 0122.17504