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.

AbstractWe consider the second-order rational difference equation xn+1=γ+δxnxn−12,$$ {x_{n+1}=\gamma +\delta \frac{x_{n}}{x^{2}_{n-1}}}, $$ where γ, δ are positive real numbers and the initial conditions x−1$x_{-1}$ and x0$x_{0}$ 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.

Abstract A certain class of a host–parasitoid models, where some host are completely free from parasitism within a spatial refuge is studied. In this paper, we assume that a constant portion of host population may find a refuge and be safe from attack by parasitoids. We investigate the effect of the presence of refuge on the local stability and bifurcation of models. We give the reduction to the normal form and computation of the coefficients of the Neimark–Sacker bifurcation and the asymptotic approximation of the invariant curve. Then we apply theory to the three well-known host–parasitoid models, but now with refuge effect. In one of these models Chenciner bifurcation occurs. By using package Mathematica, we plot bifurcation diagrams, trajectories and the regions of stability and instability for each of these models.

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.