Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms
We investigate global dynamics of the equation xn+1=xn-1axn2+exn-1+f,n=0,1,2,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x_{n+1}=\frac{x_{n-1}}{ax_{n}^{2}+ex_{n-1}+f},\quad n=0,1,2,\ldots , \end{aligned}$$\end{document}where the parameters a, e and f are nonnegative numbers with condition a+e+f>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a+e+f>0$$\end{document} and the initial conditions x-1,x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{-1},x_{0}$$\end{document} are arbitrary nonnegative numbers such that x-1+x0>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{-1}+x_{0}>0$$\end{document}. The global dynamics of this equation consists of three bifurcations, two exchange of stability bifurcations and one global period doubling bifurcation.