Sharkovsky orderings of higher order difference equations
The authors invistigate the periodic nature of the \(k\)-th order nonlinear difference equation \(x_{n}=f(x_{n-k}).\) They introduce a new definition of the \(k\)-th Sharkovsky ordering, which involve a special case the Sharkovsky ordering with \(k=1\) and extending the Sharkovsky Ordering Theorem of the nonlinear difference equation \(x_{n}=f(x_{n-1}).\) They prove that the set of periods of a periodic solutions \(\{x_{n}\}\) of \(x_{n}=f(x_{n-k})\) is given by the \(k\)-th ordering set \(S_{k}(p),\) where \(p\) is a cycle of minimal period of \(x_{n}=f(x_{n-1})\). Some applications are given for the number of attracting cycles of \(x_{n}=f(x_{n-k})\).
scientific article; zbMATH DE number 2110776