Study of a cyclic system of difference equations with maximum - Anastasios Stoikidis; Garyfalos Papaschinopoulos (Electronic Journal of Qualitative Theory of Differential Equations, 2020)
In this paper we study the behaviour of the solutions of the following cyclic system of difference equations with maximum: \[ \begin{split} x_i(n+1)&=\max\left\{A_i,\displaystyle{\frac{x_i(n)}{x_{i+1}(n-1)}}\right\},\qquad i=1,2,\dots,k-1,\\ x_k(n+1)&=\max\left\{A_k,\displaystyle{\frac{x_k(n)}{x_1(n-1)}}\right\} \end{split} \] where $n=0,1,2,\dots$, $A_i$, $i=1,2,\dots,k,$ are positive constants, $x_i(-1), x_i(0)$, $i=1,2,\dots,k,$ are real positive numbers. Finally for $k=2$ under some conditions we find solutions which converge to periodic six solutions.