Extension, embedding and global stability in two dimensional monotone maps

Abstract

We consider the general second order difference equation \begin{document}$ x_{n+1} = F(x_n, x_{n-1}) $\end{document} in which \begin{document}$ F $\end{document} is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that \begin{document}$ F $\end{document} has a semi-convex compact invariant domain, then make an extension of \begin{document}$ F $\end{document} on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of \begin{document}$ F. $\end{document} Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.