Dynamics of \(x_{n+1}=\frac{x_{n-2k+1}}{x_{n-22k+1}+\alpha x_{n-2l}}\)
The paper deals with rational difference equation \[ x_{n+1} = \frac{x_{n-2k+1}}{x_{n-2k+1} + \alpha x_{n-2l}}, \;n= 0,1,\dots, \eqno(1) \] where \(k \in\{1,2,\dots\}, l \in\{0, 1,\dots\},\) and \(\alpha \neq 1.\) The authors prove that when \(\alpha < 1\) then every positive solution of (1) converges to the positive equilibrium \(\bar x = 1/(\alpha+1).\) When \(\alpha > 1\) eq. (1) possesses solutions that converges to the two-cycle \(\dots,0,1,0,1,\dots\). Some numerical examples are presented.
scientific article; zbMATH DE number 5132935
scientific article