The stable equilibrium of a system of piecewise linear difference equations

In this article we consider the global behavior of the system of first order piecewise linear difference equations: xn+1=|xn|−yn+b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{n+1} = \vert x_{n}\vert - y _{n} +b$\end{document} and yn+1=xn−|yn|−d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$y_{n+1} = x_{n} - \vert y_{n}\vert - d$\end{document} where the parameters b and d are any positive real numbers. We show that for any initial condition in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R^{2}$\end{document} the solution to the system is eventually the equilibrium, (2b+d,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(2b + d, b)$\end{document}. Moreover, the solutions of the system will reach the equilibrium within six iterations.