By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , … , where the parameter β > 0 , and initial conditions x 0 and y 0 are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse’s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.

We investigate the global character of the difference equation of the form xn+1=f(xn,xn−1),n=0,1,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1} = f(x_{n}, x_{n-1}),\quad n=0,1, \ldots $$\end{document} with several period-two solutions, where f is increasing in all its variables. We show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium solutions or period-two solutions are in fact the global stable manifolds of neighboring saddle or non-hyperbolic equilibrium solutions or period-two solutions. As an application of our results we give the global dynamics of three feasible models in population dynamics which includes the nonlinearity of Beverton-Holt and sigmoid Beverton-Holt types.

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: , where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that

We present a complete local dynamics and investigate the global dynamics of the following second-order difference equation: , where the parameters , and are nonnegative numbers with condition , , and the initial conditions , are arbitrary nonnegative numbers such that

Consider the difference equation xn+1=α+∑i=0kaixn−i+∑i=0k∑j=ikaijxn−ixn−jβ+∑i=0kbixn−i+∑i=0k∑j=ikbijxn−ixn−j,n=0,1,…,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{n+1} = \frac{\alpha+ \sum_{i=0}^{k} a_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} a_{i j} x_{n-i} x_{n-j} }{\beta+ \sum_{i=0}^{k} b_{i} x_{n-i} + \sum_{i=0}^{k} \sum_{j=i}^{k} b_{ij} x_{n-i} x_{n-j}}, \quad n=0,1, \ldots, $$\end{document} where all parameters α, β, ai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}$\end{document}, bi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{i}$\end{document}, aij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{ij}$\end{document}, bij\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{ij}$\end{document}, i,j=0,1,…,k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i,j=0,1,\ldots, k$\end{document}, and the initial conditions xi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{i}$\end{document}, i∈{−k,…,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i \in\{-k, \ldots, 0 \}$\end{document}, are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

We investigate the global behavior of a cubic second order difference equation xn+1=Axn3+Bxn2xn−1+Cxnxn−12+Dxn−13+Exn2+Fxnxn−1+Gxn−12+Hxn+Ixn−1+J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{n+1}=Ax_{n}^{3}+ Bx_{n}^{2}x_{n-1}+Cx_{n}x_{n-1}^{2}+Dx_{n-1}^{3}+Ex_{n}^{2} +Fx_{n}x_{n-1}+Gx_{n-1}^{2}+Hx_{n}+Ix_{n-1}+J$\end{document}, n=0,1,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n=0,1,\ldots$\end{document} , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order difference equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order difference equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.

1Department of Health Informatics, College of Public Health and Health Informatics, King Saud Bin Abdulaziz University for Health Sciences, Riyadh 11481, Saudi Arabia 2Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al-Ain, UAE 3Department of Mathematics, National Tsing Hua University, Taiwan 4Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA

We consider the following system of difference equations: where , , , , are positive constants and are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at , which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at and thus describe the global dynamics of this system. Since the singular point at always possesses a basin of attraction this system exhibits Allee’s effect.