Global Dynamics and Bifurcation of a Higher Order Difference Equation

This study is devoted to dynamical analysis of following higher order difference equation \begin{eqnarray*}x_{n+1}=px_{n}+\frac{q}{rx_{n-k}^{2}},k\in \{1,2,...\}, \end{eqnarray*} where $p, q, r$ and the initial conditions are positive real numbers. In particular, we discuss the existence of periodic solutions of the difference equation. We also handle the boundedness, local and global stability of solutions of the difference equation. Moreover, we study the existence of Neimark-Sacker bifurcation of solutions of the difference equation for $k=1$ and also give an invariant curve of the difference equation. Finally, we provide some numerical examples to support our results and present some open problems for future works.