Boundedness of solutions and stability of certain second-order difference equation with quadratic term
We consider the second-order rational difference equation xn+1=γ+δxnxn−12,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$\end{document} where γ, δ are positive real numbers and the initial conditions x−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{-1}$\end{document} and x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{0}$\end{document} are positive real numbers. Boundedness along with global attractivity and Neimark–Sacker bifurcation results are established. Furthermore, we give an asymptotic approximation of the invariant curve near the equilibrium point.