By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form xn+1=f(xn)xn−1,n=0,1,…$x_{n+1}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots $ , f:(0,+∞)→(0,+∞)$f:(0,+\infty )\to (0,+\infty )$, f is sufficiently smooth and the initial conditions are x−1,x0∈(0,+∞)$x_{-1}, x _{0}\in (0,+\infty )$. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α1$\alpha _{1}$. Then we apply the results to several difference equations.

By using KAM theory we investigate the stability of equilibrium points of the class of difference equations 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}=\frac{f(x _{n})}{x_{n-1}}, n=0,1,\ldots $\end{document} , f:(0,+∞)→(0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:(0,+\infty )\to (0,+\infty )$\end{document}, f is sufficiently smooth and the initial conditions are x−1,x0∈(0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x_{-1}, x _{0}\in (0,+\infty )$\end{document}. We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha _{1}$\end{document}. Then we apply the results to several difference equations.

In this paper, we consider the cooperative system [Formula: see text] where all parameters [Formula: see text] are positive numbers and the initial conditions [Formula: see text] are nonnegative numbers. We describe the global dynamics of this system in a number of cases. An interesting feature of this system is that it exhibits a coexistence of locally stable equilibrium and locally stable periodic solutions as well as the Allee effect.

We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

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 find an asymptotic approximations of the stable and unstable manifolds of the saddle equilibrium solution and the periodtwo solutions of the following difference equation xn+1 = p + xn−1/xn, where the parameter p is positive number and the initial conditions x −1 and x0 are positive numbers. These manifolds, which satisfy the standard functional equations of stable and unstable manifolds determine completely global dynamics 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.

We investigate global dynamics of the following systems of difference equations: {xn+1=b1xn2A1+yn2,yn+1=a2+c2yn2xn2,n=0,1,2,…, where the parameters b1, a2, A1, c2 are positive numbers and the initial condition y0 is an arbitrary nonnegative number and x0 is a positive number. We show that this system has rich dynamics which depends on the part of a parametric space. We find precisely the basins of attraction of all attractors including the points at ∞. MSC:39A10, 39A30, 37E99, 37D10.